Arithmetical Compactification of Mixed Shimura Varieties

Richard Pink

For Mohamed Elham Mohamed Sheiry,who was the first to teach me real mathematics.

1

Contents

0 Indroduction 3

0.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Equivariant families of mixed Hodge structures 17

2 Mixed Shimura data 29

3 Mixed Shimura varieties 41

4 Rational boundary components 54

5 Torus embeddings 72

6 Toroidal compactification 93

7 Stratification of toroidal compactification 117

8 Construction of ample line bundles 131

9 Algebraization of the toroidal compactification 143

10 Moduli schemes of abelian varieties 170

11 Canonical models 186

12 Canonical model of the compactification 196

Bibliography 213

2

Chapter 0

Indroduction

Let H ⊂ C denote the so-called “complex upper half plane”, i.e. the set of all z ∈ C with(strictly) positive imaginary part. The map

SL2(Z)×H → H, ((a bc d

), z) 7→ az+b

cz+d

defines a holomorphic left operation of SL2(Z) on H. For any subgroup of finite indexΓ ⊂ SL2(Z), the quotient Γ\H inherits the structure of a Riemann surface. This Riemannsurface is not compact, but by adjoining a finite number of additional points, the so-calledcusps, it can be embedded in a compact Riemann surface (see [Sh] ch.1). In particular,there is a smooth projective algebraic curve X over C, a non-empty Zariski-open subsetX ⊂ X, and an isomorphism X(C) ∼= Γ\H, all of this unique up to isomorphism.

Let us now assume that Γ is a congruence subgroup. This signifies that Γ contains thesubgroup

Γ(N) := (a bc d

)∈ SL2(Z) | a ≡ d ≡ 1, b ≡ c ≡ 0 mod N

for same integer N. Then Γ\H is a moduli space for elliptic curves with a certain levelstructure (which depends on Γ). For instance, if Γ = Γ(N), then Γ\H is in bijection with theset of all isomorphy classes of pairs (E, x, y), where E is an elliptic curve over C and x, yis a basis of the subgroup E[N ] of all N-torsion points of E, such that eN (x, y) = e2πi/N .Here eN denotes the canonical pairing E[N ]×E[N ]→ µN . The moduli problem associatedto Γ can be described in algebraic terms, and it can be shown in a purely algebraic waythat there exists a corresponding (coarse or fine) moduli scheme M over a certain explicitnumber field K. In the case Γ = Γ(N) we have K = Q(e2πi/N ) ⊂ C, while for arbitraryΓ ⊃ Γ(N) any subfield of Q(e2πi/N ) may occur.

This modular interpretation of Γ\H determines an isomorphism X ∼= M×K C. In otherwords, it defines a “model” of X over K. Let M be the (up to isomorphism) unique non-singular projective curve over K that contains M as an open dense subset. Then the aboveisomorphism extends to an isomorphism X ∼= M ×K C, that is, the model of X extendsto one of X. Thus the following problem arises naturally: Describe the structure of thismodel on, and near, the cusps X rX.

An answer could have the following form. Let D be the (unique) reduced closed sub-scheme of M with support in M rM . Via the isomorphism

D(Q) = D(C) ∼= X(C) rX(C) = X(Q) rX(Q)

3

induced from the above, the model defines an action of Gal (Q/K) on the cusps. Converselythis action determines the structure of D. More abstractly, it would be desirable to have anexplicit scheme D1 over K and an explicit isomorphism D ∼= D1 . This latter formulationgeneralizes to the infinitesimal neighborhoods: one would like to have an explicit formalscheme D1 over K, and an explicit isomorphism of D1 with the formal completion of Malong D.

In the case under consideration, such descriptions follow from a modular interpretationof X(C). This involves a degenerate version of elliptic curves with N -structure, the so-called generelized elliptic curves. For the formal completion, the Tate-curve can be used(see [DR], and 10.17–22 below).

In these notes we carry out the same program for arbitrary Shimura varieties. Toexplain the complications arising in the general case, fix a hermitian symmetric domain Hof non-compact type. Also fix a semisimple algebraic group G over Q, and an isomorphismfrom G(R)0 to the group of all holomorphic automorphisms of H. Let Γ be an arithmeticsubgroup of G(Q) ∩G(R)0, then Γ\H again carries a natural structure of normal complexspace. If Γ is sufficiently small, then Γ\H is already a complex manifold. For simplicity weassume this to be case.

In general, Γ\H is not compact. It possesses different possible compactifications asa normal complex space, with different advantages. Only in the case G ∼= PGL2,Q theyhappen to coincide. Logically and historically, the Baily-Borel compactification comes first.Its nice properties are that it depends on no extra data, and that it is uniquely characterizedas the minimal normal compactification. In any case, it can be defines intrinsically and isa projective variety. It possesses a natural stratification in terms of quotients Γ′\H′ forcertain other hermitian symmetric domains H′ (the so-called boundary components), andarithmetic subgroups Γ′ of Aut(H′) (see [BB], [AMRT], and 6.2–3 below).

Unfortunately, the Baily-Borel compactification may have serious singularities along theboundary. A nice resolution of these singularities is the toroidal compactification. It is de-fined locally as a torus embedding. The cone decompositions involved in this definition arenot canonical, so there are different toroidal compactifications associated to different sys-tems of cone decompositions. However, for suitable choices the resulting complex space isprojective and smooth, and the boundary is the union of smooth divisors with at most nor-mal crossings. The toroidal compactification dominates the Baily-Borel compactification,and possesses a natural stratification compatible with that of the Baily-Borel compactifi-cation. But the strata now have a more general form: they are torus-torsors over familiesof abelian varieties over the strata of the Baily-Borel compatification.

These compactifications carry a natural algebraic structure, and induce an algebraicstructure on Γ\H. The next step is to ask for a model of these algebraic varieties over anumber field. Here one gets stronger results if one considers not only Γ\H, but a certainfinite disjoint union of such quotients. For this, G is replaced by a central extension witha torus, H by a finite disjoint union of hermitian symmetric domains with a transitiveholomorphic action of G(R), and instead of Γ\H one considers a double quotient G(Q)\H×(G(Af )/Kf ). Here Af denotes the finite adeles of Q, Kf is an open compact subgroup ofG(Af ), and G(Q) acts on both factors. This double quotient is indeed of the desired form,and constitutes the set of complex points of a Shimura variety. Let us denote the associated

4

algebraic variety over C by MC, the Baily-Borel compactification by MC, and the toroidalcompactification associated to a system Σ of cone decompositions by MΣ,C.

The advantage of this adelic definition is that one gets a model over a number field thatis independent of Kf , called the reflex field. It is determined by G, H, and a little extradata concerning the center of G. Some Shimura varieties have a natural interpretation asmoduli spaces af abelian varieties with some extra structure. For instance, generalizing thespecial case considered above, the group of symplectic similitudes CSp2g,Q gives rise to theSiegel modular variety parametrizing principally polarized abelian varieties of demensiong with level structure. Such a modular interpretation induces a model M of MC over thereflex field. However, not every Shimura variety has such a purely algebraic modular inter-pretation. Nevertheless, following Shimura the models defined by modular interpretationcan be uniquely characterized in an intrinsic way (see [D1]). Taking this characterizationas a definition, one arrives at the so-called canonical model of an arbitrary Shimura variety.Its uniqueness is not too hard to prove ([D1]), but its existence in the most general casehas been established only recently (see [Mi1]).

The first of the questions we have to examine is: Under what condition does the canon-ical model extend to a model of MC or of MΣ,C? For MC it always extends uniquely toa model M , since the Baily-Borel compactification can be defined intrinsically. For thetoroidal compactification the answer depends on Σ, and is linked with the other questions.In any case, there is up to isomorphism at most one extension MΣ, so we do not get intotrouble if we directly turn to the next question: Which model is induced on the boundary?

In the case of the Baily-Borel compactification, the stratification of the boundary sug-gests a natural guess: the boundary of the extended canonical model M should possess astratification in terms of canonical models of other Shimura varieties. This indeed turnsout to the case (see our first main theorem 12.3). The complete answer includes an explicitdescription of the Shimura varieties that occur at the boundary (see 6.3).

For the toroidal compactification one has to work much harder to get the analogousresults. The chief reason for this is that, although the boundary strata are fibred over(other) Shimura varieties, they are more general objects. Thus, in order to be able todescribe the boundary within the same framework, one has to generalize the concept ofShimura variety.

A close look at the group theoretical data involved in the definition of the toroidalcompactification (see [AMRT] ch.III) suggests that the desired generalization involves anextension of a reductive group (associated to Shimura variety) by a unipotent group ofa certain type. Such groups arise naturally as normal subgroups of maximal parabolicsubgroups of our original group G. Along another line of thought, note that every Shimuravariety, even if it does not possess a (known) algebraic modular interpretation, at leastparametrizes, as a complex manifold, suitable variations of polarized pure Hodge structuresor combinations thereof. If the group is not reductive, these “combinations” can no longerbe direct sums, but should be mixed Hodge structures! These considerations naturallylead to a concept of mixed Shimura varieties that parametrize variations of mixed Hodgestructures, all of whose pure constituents are polarizable.

In the chapters 1–3 we develop a theory of mixed Shimura varieties. We consider mixedShimura varieties in their own right, not only insofar as they occur in the boundary of

5

other Shimura varieties. Nevertheless the desired properties (variation of mixed Hodgestructures, i.e. Griffiths’ transversality, and polarizability of all pure constituends) implythat they also are torus-torsors over families of abelian varieties over usual (henceforthcalled pure) Shimura varieties (see 3.12ff). They have a rich structure; in particular thegroup operations, torus actions, etc. on the fibres can all be described in terms of abstractlydefined morphisms of Shimura varieties. Moreover, for pure Shimura varieties that possessan interpretation as moduli schemes of polarized abelian varieties, the universal families(of abelian varieties, and of polarizing line bundles) themselves can be described as mixedShimura varieties (see 10.7). This phenomenon can be viewed as a modular interpretationfor these mixed Shimura varieties. A mixed Shimura variety is pure when the associatedalgebraic group is reductive. The treatment of chapter 1 follows [D2] § 1.

The next problem we have to deal with is how to make mixed Shimura varieties al-gebraic. Like for pure Shimura varieties, this problem can be solved by suitable projec-tive compactifications. Such compactifications are also interesting in connection with themodular interpretation: one can describe degenerations of abelian varieties in terms ofcompactifications of mixed Shimura varieties. As, for instance, [M4] suggests, these com-pactifications should be toroidal. In the special case of elliptic curves we indeed give suchdescription (see 10.17-22).

All in all, we see that toroidal compactifications should be constructed not only for pure,but for arbitrary mixed Shimura varieties. This is done in the chapter 4 and 6, which relyheavily on [AMRT]. Chapter 5 contains an assortment of results about torus embeddings,which will be needed throughout the remaining chapters. This construction is not morecomplicated than for pure Shimura varieties, on the contrary the general setting accountsfor a certain coherence. The essential new ingredient is an explicit way of associating certainmixed Shimura varieties to a given (possibly pure) one. These are closely related with whatin the usual theory is called rational boundary components, which is why we give themthe same name. The rational boundary components are in general “more mixed” than theoriginal mixed Shimura variety. The toroidal compactification is then constructed as in[AMRT]. First one considers for every rational boundary component a torus embedding,with respect to a certain cone decomposition and the torus that acts naturally on theunipotent fibre. If the cone decompositions have been chosen compatibly, these partialcompactifications can be glued together to form the desired toroidal compactification.

We carry out this construction with an eye toward the question for the model of theboundary. Since we eventually want to describe a stratification of the boundary in termsof other mixed Shimura varieties, in particular in adelic language, it is desirable that theconstruction already reflects this. Unfortunately this adds (mainly notational) complexityto the constructions of the chapters 6–9 and 12. The benefit is that we directly get a strat-ification of the toroidal compactification in terms of the mixed Shimura varieties occuringas rationl boundary components, and quotients thereof. This stratification is studied inchapter 7.

In order to algebraize mixed Shimura varieties and their compactifications, we have tofind compactifications that are projective. In other words we have to construct ample linebundles. While for pure Shimura varieties (in [AMRT] ch.IV §2) such ample line bundlescan be described in terms of “piecewise linear strictly convex rational functions” on the conedecompositions, the presence of abelian varieties in the fibres of mixed Shimura varieties

6

necessitates a more invariant construction (see 8.15, 9.39). As it turns out, one can use thetorus-torsor structure on mixed Shimura varieties to turn certain toroidal compactificationsof mixed Shimura varieties into line bundles. (In the case of a modular interpretation thishas been mentioned above.) In other words, certain morphisms M ′Σ′,C → MΣ,C of toroidalcompactifications of mixed Shimura varieties “are” line bundles in a canonical way (see8.6).

In chapter 8 we derive a criterion for such a line bundle to be ample (along the unipotentfibres). What remains to achieve the algebraization, is the construction of cone decompo-sitions satisfying this criterion. This is done in chapter 9. We prove that for every mixedShimura variety MC, there exists a mixed Shimura variety M ′C and cone decompositionΣ,Σ′, so that MΣ,C is compact, M ′Σ′,C →MΣ,C is a line bundle, and some combination withthe canonical sheaf is ample. Moreover, Σ can be chosen such that MΣ,C is smooth, thecomplement MΣ,C rMC is a union of smooth divisors with a most normal crossings, andsuch Σ can be made arbitrarily fine. In particular, both MC and such MΣ,C are algebraicvarieties. We also prove that “most” other toroidal compactifications MΣ,C are algebraicvarieties, and that all of them are algebraic spaces in the sense of [K].

Following the same program as above, we next have to define canonical models formixed Shimura varieties. This is done in the chapters 10–11. The definition and theproof of uniqueness are exactly as for pure Shimura varieties. The modular interpretationyields a canonical model for the mixed Shimura varieties associated to the moduli problemof polarized abelian varieties with level structure. Here, as in [D1], the main theoremof Shimura and Taniyama about abelian varieties with complex multiplication plays theessential role. Using this, along the lines of [D1], the existence of canonical models ingeneral is reduced to the case of pure Shimura varieties, where it is known by [Mi1].

Having developed the theory of mixed Shimura varieties thus far, we are now in theposition to formulate and to prove the main result for the toroidal compactification ofa mixed Shimura variety. Recall that chapter 7 gives a stratification in terms of othermixed Shimura varieties. By the results of chapter 9, this stratification is in fact algebraic.Just as for the Baily-Borel compactification, the main theorem (12.4) asserts that (if thecone decompositions satisfies a certain arithmeticity condition, see 6.4) the canonical modelextends to the toroidal compactification, and that this stratification descends to the reflexfield.

Since the neighborhoods of a stratum in the toroidal compactification are so nice, with-out much extra effort we get an even better assertion. By 7.17 some neighborhood of astratum is essentially isomorphic to a neighborhood of a stratum in a simple torus em-bedding of a rational boundary component. By 9.37, this description can be algebraized,yielding an isomorphism between the respective formal completions (over C). The last partof our main theorem (12.4 (c)) asserts that this isomorphism of formal schemes also de-scends to the reflex field. These results constitute a complete answer to the questions putforward above.

The main theorem is proved in chapter 12, in the following way. First, using the modulischeme of generalized elliptic curves (see 10.20), we prove it in the special case of ellipticmodular curves (associated to the group GL2,Q). (In particular, as mentioned above, theuniversal family of generalized elliptic curves “is” the canonical model of a certain toroidal

7

compactification of a mixed Shimura variety.) We then embed twists of such modular curvesinto arbitrary mixed Shimura variety (like Hirzebruch Zagier cycles in Hilbert modularsurfaces). Finally a density argument, much as in the intrinsic characterization of canonicalmodels, but this time for the formal neighborhood of a boundary component, extends theresult to the general case.

The result of these notes have several applications. Quite immediate are the conse-quences for q-expansions of (certain) automorphic forms. Consider, for a given mixedShimura variety M , a space of sections of a vector bundle F constructed out of sheaves ofdifferential forms, possibly with logarithmic poles at infinity, or with some other bound-ary condition. Our description of the formal neighborhood M of a boundary stratum M1

implies a similar description for the pullback F of F. Moreover, since M has an explicitdescription in terms of a torus embedding, there is a canonical decomposition of Γ(M,F)as a product of Γ(M1, Fν) for certain explicit sheaves Fν on M1. The map

Γ(MΣ, F )→ Γ(M, F )→∏ν

Γ(M1, Fν)

now associates to a section of F its “q-expansion coefficients.” These “coefficients” areanalog of Jacobi modular forms. Our results immediately imply a q-expansion principle forrationality of such automorphic forms over the reflex field: A section over C descends tothe reflex field if and only its q-expansion coefficients do so at a suitable set of boundarycomponents (see 12.18–20). In a similar form, for the Baily-Borel compactification, thisresult has been obtained by Harris [Ha1], [Ha2] for arbitrary vector-valued automorphicforms.

In another direction these results have consequences for the `-adic cohomology of (mixed)Shimura varieties. The explicit description of formal neighborhoods of the boundary shouldmake it possible to describe the pure constituents of, say H∗et(MQ,Q`) in terms of some purecohomology (e.g. intersection cohomology, cuspidal cohomology, or the like) of its boundarycomponents (including itself) with values in some other explicit `-adic sheaves. The authorplans to take this up in a sequel to these notes. This would of course be a prerequisite forstudying the mixed motives that occur in H∗et(MQ,Q`).

Two more technical remarks are in order. In the explanations above we have tacitlyused the following slight generalization of the definition of a pure Shimura variety. In theusual definition, H is a G(R)-conjugacy class of certain homomorphisms C× → G(R). Thegeneralization consists in allowing a finite G(R)-equivariant covering X → H, with G(R)acting transitively on X . This generalization makes a difference for the field of definition ofa connected component of a Shimura variety, and is necessary in order to stratify a normalcompactification in the same framework. From the point of view of global abelian classfield theory, it is in fact the more natural choice.

The other remark concerns questions of sign. In order to avoid a source of errors, wehave found it useful to work without any globally fixed choice of a square root of -1. In fact,complex analysis by no means depends on such a choice! We hope to have been successfulin this.

As explained above, our results rely on the modular interpretation of certain(mixed)Shimura varieties. This modular approach, of course, achieves much more: it can be carried

8

out not only over a number field, but over suitable rings of finite type over Z. This hasbeen done in several instances: Deligne and Rapoport [DR] for elliptic modular curves,Rapoport [Rap] for Hilbert modular varieties, and Chai [C] and Faltings [F2] for the Siegelmodular variety. A modular interpretation of certain mixed Shimura varieties (in termsof “1-motifs”) has been described in Brylinski’s thesis [Br]. In these cases, a part of theresults of this thesis have been known.

The toroidal compactification of pure Shimura varieties has been treated over C in[AMRT] and [N]. That is descends to Q has been proved in [F1], using a rigidity argument.After the present author had obtained his results, the following two manuscripts came tohis attention: (a) a survey article by Milne [Mi2] that independently suggests the conceptof mixed Shimura variety along a similar line of ideas; an (b) a manuscript by Harris [Ha3]containing similar results for pure shimura varieties.

Finally, the author has the pleasure to thank, most of all, Professor G. Harder forsuggesting the present research, for providing a creative atmosphere, and for his constantsupport. He is obliged to J.S. Milne for terminological and other suggestions, and to P.Deligne, Th. Hofer, N.M. Katz, and M. Rapoport for useful conversations. He is indebtedto the DAAD and the Universitat Bonn for financial support. Last, but not least, he isgrateful to S. Fette for friendly providing him with countless cups of tea.

9

“Just follow that line forever,” said the Mathemagician, “and when you reach the end,turn left. There you’ll find the land of Infinity, where the tallest, the shortest, the smallest,and the most and the least of everything are kept.” ... Up he went — very quickly at first— then more slowly — then in a little while even more slowly than that — and finally,after many minutes af climbing up the endless stairway, one weary foot was barely able tofollow the other. Milo suddenly realized that with all his efforts he was no closer to the topthan when he began, and not a great deal further from the bottom.

Norton Juster, The phantom tollbooth

L’arithmetiqueca fatigue,caenerve.

Eugene Ionesco, La lecon

10

0.1 Notations and conventions

General: The set of connected components of a topological space X is denoted by π0(X).For any two sets X, Y , the set-theoretic difference is denoted by X r Y . This is not to beconfused with elementwise difference or with left quotient: If (A,+) is an abelian group,a ∈ A and X, Y ⊂ A, then the elementwise sum or difference is denoted by X +Y , X −Y ,a−X, etc. Likewise for an arbitrary group (G, ·).

Fields and schemes: If T → S is a morphism of schemes, and X a scheme over S,then we write XT := X ×S T . If S = Spec(K) and T = Spec(L) for a field extension L/K,then we write this also in the form XL = X ×K L. If YL is a subscheme of XL which isderived from a subscheme of X by base extension, then by abuse of terminology we saythat it is defined over K. An “L-subobject” (e.g. “L-subgroup”, “L-parabolic subgroup”,“L-simple subgroup”) of an object X is a subscheme of XL that possesses the indicatedproperty.

Let K/L be a field extension and X a scheme over L. By a model of X over L we mean ascheme X0 over K together with an isomorphism X0,L

∼= X over L. The Weil restriction ofX with respect to L/K is denoted by RL/KX. If X is a group scheme (general, reductive,semisimple, or a torus), then so is RL/KX.

R is the field of real numbers, C a fixed algebraic closure of R. The symbol√−1 denotes

an arbitrary solution in C of the equation z2 + 1 = 0. We do not fix a particular choise of√−1. Wherever this symbol occurs, it will be clear that the whole assertion is independent

of such a choice. In the expression 2π√−1, the symbol π denotes the unique positive real

number for which 2π√−1 is a generator of kernel of the exponential function exp: C→ C×

(Clearly this does not depend on√−1).

If X is a variety over a field K that is contained in R, then we call X of compact typeif and only if X(R) is compact.

All number fields K are considered as subfields of C, in other words they are giventogether with a distinguished embedding into C. We denote by K the algebraic closure ofK in C, so that the absolute Galois group Gal(K/K) is a quotient of Aut(C). The ringof adeles of K is denoted by AK , the subring of finite adeles by AK,f , and we abbreviateA := AQ and Af := AQ,f .

Groups: Let G be a group and g, g′ ∈ G. The left action of G by conjugation on itselfis denoted by intG(g) : g′ 7→ g · g′ · g−1. The commutator is (g, g′) := g · g′ · g−1 · g′−1. Analmost direct (almost semidirect) product in a category of groups in any finite quotient ofthe usual direct (semidirect) product.

If A is a topological or algebraic group, then A0 is the connected component of theidentity of A. The center of A is denoted by Z(A), this is a closed, but not necessarilyconnected subgroup. If A is a topological group, then π0(A) is isomorphic to A/A0.

Let G be a linear algebraic group over a field K. By a representation of G we alwaysmean a finite dimensional representation over K. If L ⊂ Lie G is a K-sub Liealgebra, thenexp L denotes the unique connected K-subgroup of G with Lie algebra L.

11

By a reductive group over a field K we always mean a connected reductive linearalgebraic group over K. In particular, every semisimple group and every torus will beconnected. If G is a reductive group, then Gder = [G,G] and Gad = G/Z(G) denote itderived, resp. adjoint group. The adjoint representation of G on Lie G will be denoted byAdG, its derivative (the induced left action of Lie G on itself) by adG.

Let P be a connected linear algebraic group over K, and U its unipotent radical. ALevi-decomposition is a decomposition of P into a semidirect product P = U o G. Anytwo Levi-compositions are conjugate unter U(K). Also, let H be a reductive group overK, and ϕ1 , ϕ2 : H → P two homomorphisms so that the composites H → P ⇒ P/U areequal. Then ϕ1 and ϕ2 are conjugate under U(K).

Group actions: If a group G acts from the left hand side on a set X, we denote byG\X the set of all G-orbits in X. Likewise for X/H if a group H acts from the right handside, and consequently G\X/H if two such actions are given. Fortunately we shall not needactions from above or below. A left homogeneous space under G is a set X with a transitiveleft action of G. If G is a real Lie-group, then such X is a C∞-manifold in a canonical way.

Let Γ be a discrete group acting on a locally compact (hence Hausdorff) topologicalspace X. By definition Γ acts properly discontinuously on X if and only if for any twocompacts subsets K and K ′ of X the set γ ∈ Γ | γK ∩K ′ 6= ∅ is finite. An equivalentcondition is that any two points in X possess neighborhoods U and V such that the setγ ∈ Γ | γU∩V 6= ∅ is finite. If this condition holds, then Γ\X, endowed with the quotienttopology, is again Hausdorff (even locally compact) and locally isomorphic to a quotient ofX by a finite group. In particular the stabilizer of any point is finite.

Contrary to the usual terminology we also say that the group Γ acts properly discon-tinuously if the action factors through a quotient Γ which acts properly discontinuously inthe usual sense. The point is that the kernel of the action is allowed to be infinite.

Arithmetic subgroups: Let G be a linear algebraic group over Q. A subgroup ofG(Q) is called a congruence subgroup if and only if it is of the form G(Q) ∩Kf for someopen compact subgroup Kf ⊂ G(Af ). A subgroup of G(Q) is called an arithmetic subgroupif and only if it is commensurable with some (hence with every) congruence subgroup.

According to [B] 17.1 an element g ∈ GLn(Q) is called neat, if the subgroup of Q×thatis generated by the eigenvalues of g is torsion free. If G is a linear algebraic group, thenan element g ∈ G(Q) is called neat if its image in some faithful representation of G isneat. It is easy to check that this condition then holds for every, not necessarily faithfulrepresentation of G (see [B] 17.3). A subgroup of G(Q) is called neat if all its elements areneat. In particular such a group is torsion free. Every subgroup of a neat subgroup is neat,and the image of a neat subgroup under a homomorphism G → H is again neat. Finallyevery sufficiently small congruence subgroup of G(Q) is neat (by [B] 17.4, or 0.6 below).

Neatness and adelic groups: We want to extend the notion of neatness to subgroupsKf ⊂ G(Af ), with the same functorial properties, and so that G(Q)∩Kf is neat wheneverKf is neat. First consider an element gf = (gp)p ∈ GLn(Af ). For every p let Γp be the

subgroup of Q×p generated by all eigenvalues of gp. Let Q → Qp be some embedding,

12

and consider the torsion part (Q× ∩ Γp)tors. Since every subgroup of Q× consisting ofroots of unity is normalized by Gal(Q/Q), this group does not depend on the choice of theembedding Q → Qp. We now define gf to be neat if and only if⋂

p

(Q× ∩ Γp)tors = 1.

As before, if G is a linear algebraic group, we call an element gf ∈ G(Af ) neat if its imagein some faithful representation of G is neat. If ρ is a faithful representation of G, and ρ′

another representation of G, then the group Γp above for ρ is the same as that for ρ ⊕ ρ′,and contains that for ρ′. Thus if gf is neat, then its image in any representation of G isneat. A subgroup of G(Af ) is called neat if all its elements are neat. Clearly a subgroup ofa neat subgroup is again neat. If H is a linear algebraic subgroup of G, and Kf ⊂ G(Af )is neat, then H(Af ) ∩ Kf is neat. If ϕ : G → H is a homomorphism and Kf ⊂ G(Af )is neat, then ϕ(Kf ) is neat, since we can consider any faithful representation of H as arepresentation of G. The neatness property is invariant under all automorphisms of G, andall inner automorphisms of G(Af ). From the definition it is clear that if Kf ⊂ G(Af ) isneat, then G(Q) ∩Kf is neat in the usual sense.

Finally fix an integer d ≥ 3, and let Kf (d) ⊂ GLn(Z) be the subgroup of all elementsthat act trivially on Zn/d · Zn. We shall show that Kf (d) is neat. It follows that everysufficiently small open compact subgroup of a linear algebraic group is neat. The proof ofthe neatness of Kf (d) is standard. Note that d is divisible by some prime power pi ≥ 3.Consider an element gp ∈ GLn(Zp) that acts trivially on Znp/pi ·Znp , then it suffices to showthat the group Γp above does not contain a nontrivial root of unity. By assumption everyeigenvalue of gp in Zp (the algebraic closure) is congruent to 1 modulo pi · Zp. Thus thesame follows for every element of Γp, and we have to show that every root of unity ζ ∈ Zp,that is congruent to 1 modulo pi · Zp, is equal to 1. Since the residue class of ζ in F×p is 1,the order of ζ must be a power of p. Suppose this order is pj with j ≥ 1, then the followinginequalities for the p-adic valuation

vp(1− ζ) ≥ vp(pi) = i · vp(p) = i · pj−1 · (p− 1) · vp(1− ζ) > 0

imply i · (p− 1) ≤ i · pj−1 · (p− 1) ≤ 1, which contradicts the assumption that pi ≥ 3. Thusζ = 1, as desired.

13

List of frequently used symbols

A, Af , AK , AK,f adele ring 0.2AdG, adG adjoint operation 0.3A[d] group of d-division points 10.1B(X 0), B(X 0, P1) set of commutators 8.11β(P1,X1, P

′1,X ′1, pf ), β(P1,X1, P

′1,X ′1, pf )

certain maps in U , U 6.11, 6.15C set of complex numbers 0.2C(P,X ) conical complex 4.24C(X 0, P1) open cone associated to boundary component 4.15C∗(X 0, P1) union of all boundary components of C(X 0, P1) 4.22(CSP2g,H2g) Shimura data for the symplectic group 2.7∆1 normalizer of a boundary stratum in the Baily-

Borel compactification 6.3, 6.18, 7.3eσ splitting associated to a cone σ 5.11, 8.5expL subgroup associated to sub-Lie algebra 0.3E(P,X ) refelx field 11.1e(X) canonical pairing H(X)×H(X)→ Gm 10.2E(X ′0, P ′1) set of splittings 9.2FPMC Hodge filtration 1.1G semisimple part P/W 2.1Gad, Gder adjoint, derived group 0.3(Gm,Q,H0), (Gm,Q, h(H0))

standard Shimura data 2.8h homomorphism SC → PC 1.4h equivariant map X → Hom(SC, PC) 2.1H0, h0, h∞ reference group and homomorphisms 4.3H(X) group of translations normalizing X 10.1HW conjugacy class of h : SC → PC 1.6im imaginary part 4.14intG interior automorphism 0.3int(p) interior automorphism of (P,X ) 3.5Kf (d), KP

f (d), KUf (d), KW

f (d)special open compact subgroups 10.7, 10.15

LP canonical invertible sheaf on MKf (P,X )∗(C) 8.1Lχ invertible sheaf occuring in q-expansion 12.18λ isomorphism Z→ Z(1) 2.8, 3.16, 8.5Mp,q Hodge decomposition 1.1M(n) Tate twist 1.11

MKf (P,X )(C), MKfC (P,X ), MKf (P,X )

(mixed) Shimura variety 3.1, 9.25, 11.5

MKf (P,X )∗(C), MKfC (P,X )∗, MKf (P,X )∗

Baily-Borel compactification 6.2, 9.25, 12.3

MKf (P,X )(C), MKfC (P,X ), MKf (P,X )

14

toroidal compactification 6.24, 9.25, 9.34, 12.4–5MKf (P,X )+(C) union of MKf (P,X )(C) with all boundary strata of

codimension 1 8.2M(d), MV (d), MW (d), M0(d), M0

U (d)special mixed Shimura varieties 10.7, 10.15

Md moduli scheme of abelian varieties 10.6, 11.16M0

d, X0d “moduli scheme” of roots of unity 10.15

µ canonical cocharacter of S 1.3ordT homomorphism T (C)→ Y∗(T )R 5.8ωTX/S sheaf of invariant differentials 5.26

ω[dlog], ωXΣ[dlog], ωXΣ

/S[dlog]sheaf of differentials with at most logarithmicpoles along the boundary 5.26, 8.1

(P,X ), (P,X , h) (mixed) Shimura data 2.1(P,X )/P0 quotient mixed Shimura data 2.9(P1,X1)×(P,X ) (P2,X2)

fibre product 2.20(P0,X0) unipotent extension of (Gm,Q,H0) 2.24(P2g,X2g) unipotent extension of (CSP2g,H2g) 2.25(P[σ],X[σ]) mixed Shimura data associated to [σ] 7.1π the positive real number 0.2π, π′ projections P → G = P/W, P → P/U 2.1π0(X) set of connected components 0.1πσ canonical projection Tσ → Tσ 5.2π[σ] projection (P,X )→ (P[σ],X[σ]) 7.1[π]∗ projection to Baily-Borel compactification 6.24, 9.25, 12.4Q admissible Q-parabolic subgroup 4.5RL/KX Weil restriction 0.2R set of real numbers 0.2S, S1 Deligne torus 1.3Σ, Σ(X 0, P1, pf ) admissible (partial) cone decomposition 6.4Σ0 subset of all cones along the unipotent fibre 6.5Σ|(P1,X1) restriction to a boundary component 6.5Σ[σ] induced cone decomposition 7.7Σ(f) cone decomposition associated to a piecewise linear

convex rational function 5.20Σ+

Σ , Σ′Σ, ΣΣ cone decompositions associated to Σ 9.12|Σ| support of a cone decomposition 5.1σ0 interior of a cone 5.1σ0 standard cone 5.10, 8.5, 9.1σ dual cone 5.1[σ] double coset of cones 7.1Σ locally polyhedral subset 9.8TΣ torus embedding 5.3Tσ affine torus embedding 5.2

15

Tσ orbit in a torus embedding 5.2U weight -2 subgroup of P 2.1U , U(P1,X1, pf ) covering of MKf (P,X )(C) 6.10U , U(P1,X1, pf ) covering of MKf (P,X ,Σ)(C) 6.13V weight -1 subquotient of P 2.1V [d], U [d] standard Z/dZ-modules 10.3, 10.7w “weight” of S 1.3W unipotent radical of P 2.1WnM weight filtration 1.1Xd → Ad →Md universal family of abelian varieties 10.10, 11.16XΣ relative torus embedding 5.5X∗(T ) charcter group of a torus T 5.2X+ open subset of X that maps to X1 4.11X ∗ union of hermitain symmetric domain with all its

rational boundary components 6.2Y∗(T ) cocharacter group of torus T 5.2Z(n) Tate Hodge structure 1.11

∼ equivalence relation on U or U 6.10, 6.16√−1 the complex number 0.2

[ , ] commutator 0.3[ ·pf ], [ϕ] morphisms of mixed Shimura varieties

3.4, 6.2, 6.25, 9.25, 12.3, 12.4[ ·pf ]∗Σ, ϕ∗Σ pullback of admissible cone decomposition 6.5

16

Chapter 1

Equivariant families of mixedHodge structures

Usual Shimura varieties can be viewed analytically as moduli spaces for variations of certainpolarized pure Hodge structures, or combinations thereof. Thus, in analogy to [D2] §1.1, anatural starting point for us is to study how certain variations of mixed Hodge structurescan be expressed in group theoretic terms. For most of the chapter we consider the case ofrational weights, that is where the weight filtration is defined over Q. In 1.3–1.8 we describeequivariant families of rational mixed Hodge structures: these form naturally a complexmanifold (1.7). Then we analyze which of these satisfy the two requirements: (a) Griffiths’transversality (1.9–10), and (b) polarizability of all pure constituents (1.11–12). To avoidsome pathologies we replace the group in question by a smaller one (1.13–17). Thus weare lead naturally to the group theoretic data described in 1.18. Since the rationalityof the weight is a true restriction for usual Shimura varieties, it is desirable to drop thisrequirement. It turns out that to define mixed Shimura varieties it is enough to assume thatthe weight acts rationally on the “mixed” part of the data (see 3.3 (a)). The modification1.19 of the data of 1.18 achieves this.

1.1. Hodge structures: (compare [D3] §§2.1 and 2.3) Let M be a finite dimensionalQ-vector space. A pure Hodge structure of weight n ∈ Z on M is a decomposition MC =⊕

p+q=nMp,q into C-vector spaces, such that for all p, q ∈ Z with p + q = n one has

(Mp,q) = Mp,q. The associated (descending) Hodge filtration on MC is defined by F pMC :=⊕p′≥pM

p′,q. It determines the Hodge structure uniquely, because Mp,q = F pMC∩(F qMC).

A rational mixed Hodge structure on M consists of an ascending, exhausting, separatedfiltration WnMn∈Z of M by Q-vector spaces, called weight filtration, together with adescending, exhausting, separated filtration F pMCp∈Z of MC, called Hodge filtration,such that for all n ∈ Z the Hodge filtration induces a pure Hodge structure of weight n onGrWn M := WnM/Wn−1M . A pure Hodge structure of weight n is considered a special caseof a mixed Hodge structure by defining the weight filtration as Wn′M = M for n′ ≥ n, andWn′M = 0 for n′ < n.

The Hodge numbers are defined as hp,q := dimC(GrWp+qM)p,q. They satisfy hp,q = hq,p,almost all hp,q are zero, and their sum is equal to the dimension of M . If A ⊂ Z⊕ Z is an

17

arbitrary subset, then we call a Hodge structure of type A, if hp,q = 0 for all p, q 6∈ A. Theweights that occur in a Hodge structure are the numbers p+ q for all pairs (p, q), for whichhp,q = 0. The notions “of weight ≤ n” and “of weight ≥ n” are defined in the obvious way.

A morphism of rational mixed Hodge structures is a homomorphism f : M →M ′, suchthat f(WnM) ⊂ WnM

′ and f(F pMC) ⊂ F pM ′C for all n, p ∈ Z. By [D3] 2.3.5 (i) therational mixed Hodge structures from an abelian category with these morphisms. Givenrational mixed Hodge structures on M1 and M2, there are canonical rational mixed Hodgestructures on M1⊕M2, the dual (M1)∨, on Hom(M1,M2), and on M1⊗M2 (see [D3] 1.1.6,1.1.12 and 2.1.11).

A mixed Hodge structure on M splits over R, if there exists a decomposition MC =⊕p,qM

p,q, such that WnMC =⊕

p+q≤nMp,q, F pMC =

⊕p′≥pM

p′,q, and (Mp,q) = Mp,q.This decomposition is then uniquely determined by these properties. Every pure Hodgestructure splits over R, but not every mixed Hodge structure does. If one weakens therequirements, however, one can still associate to every mixed Hodge structure a canonicaldecomposition MC =

⊕p,qM

p,q, as in the following proposition.

1.2. Proposition: Fix a rational mixed Hodge structure on M .

(a) There exists a decomposition MC =⊕

p,qMp,q, such that WnMC =

⊕p+q≤nM

p,q and

F pMC =⊕

p′≥pMp′,q.

(b) The Hodge structure is uniquely determined by any such decomposition.

(c) There exists a unique decomposition as in (a), which also satisfies

(Mp,q) ≡Mp,qmod⊕p′<p,q′<q

Mp′,q′ .

Proof. (a) follows from (c), and (b) is obvious. For the existence in (c) see [D3] 1.2.11,for the uniqueness [CK] 2.2. q.e.d.

Remark. In general there exist different decompositions satisfying (a).

1.3. The Deligne-torus: (compare [D2] 1.1.1. Attention: Our conversation isopposite to that in [D1] 1.3 and [D3] 2.1.5.1.) Consider the torus S := RC/RGm,C. Over C itis canonically isomorphic to Gm,C×Gm,C, but the action of complex conjugation is twisted bythe automorphism c that interchanges the two factors. In particular S(R) = C× correspondsto the points of the form (z, z) with z ∈ C×. While the character group of Gm,C in Z in thestandard way, we identify the character group of S with Z⊕Z such that the character (p, q)maps z ∈ S(R) = C× to z−p · (z)−q ∈ C×. Under this identification the complex conjugationstill operates on Z ⊕ Z by interchanging the two factors. Consider the homomorphismsw : Gm,R → S, R× 3 t 7→ t ∈ C×; µ : Gm,C → SC, C× 3 z 7→ (z, 1) ∈ C× × C× = S(C); andthe norm N : S Gm,R, S(R) = C× 3 z 7→ z · z ∈ R×. The kernel S1 of N is anisotropicover R, and we have a short exact sequence 1→ S1 → S→ Gm,R → 1.

Let M be a finite dimensional Q-vector space. The choice of a representation k : SC →GL(MC) is equivalent to the choice of a decomposition MC =

⊕p,qM

p,q, where Mp,q isthe eigenspace in MC to the character (p, q). Like in 1.1 we call WnMC =

⊕p+q≤nMp,q

18

and F pMC =⊕

p′≥pMp′,q the associated weight filtration, respectly Hodge filtration, and

define the notions “of type A”, pure, etc. in the same way. These notions coincide withthose of 1.1, if the filtrations are those of a rational mixed Hodge structure on M . We nextstudy the question, under which condition on k this is the case.

1.4. Proposition: Let P be a connected linear algebraic group over Q. Let W bethe unipotent radical of P , let G := P/W and π : P → G the canonical projection. Leth : SC → PC be a homomorphism, such that the following conditions are satisfied:

(i) π h : SC → GC is already defined over R.

(ii) π h w : Gm,R → GR is a cocharacter of the center of P/W , that is already definedover Q.

(iii) Under the weight filtration on LieP defined by Adp h we have W−1(LieP ) = LieW .

Then:

(a) For every representation ρ : P → GL(M), defined over Q the homomorphism ρ h :SC → GL(MC) induces a rational mixed Hodge structure on M .

(b) The weight filtration on M is stable under P .

(c) For any p ∈ P (R) ·W (C) the assertions (a) and (b) also hold for int(p) h in place ofh. The weight filtration and the Hodge numbers in any representation are the same forint(p) h and for h.

Proof. Assertion (c) follows directly from (a) and (b), because the conditions (i) to(iii) are invariant under conjugation by an element of P (R) ·W (C). For (a) and (b) we firstuse induction to prove the statement: The weight filtration is defined over Q and stableunder P .

In the case dimM = 0 there is nothing to prove. Otherwise there exists a short exactsequence 0 → M ′ → M → M ′′ → 0 of representations of P , with M ′′ nontrivial andirreductible. Let MW be the space of W -coinvariants of M , then M ′′ is a quotient of MW .By definition MW is completely reductible under P , so there exists a P -invariant subspaceN ⊂ M with N + M ′′ = M and M ′′ ∼= im(N) ⊂ MW . If N is properly contained in M ,then the assertion follows from the formula WnMC = WnM

′C + WnNC for all n ∈ Z and

the inductive assumption. Otherwise we have M ′′ = MW . The differential of ρ inducesa P -equivariant linear map (LieW ) ⊗M → m, whose cokernel is precisely the space of(LieW )-coinvariants, so again equal to M ′′. Thus we obtain a surjective P -equivariant map(LieW ) ⊗M → M ′. Let n be the highest weight that occurs in M . Since by assumptionLieW is of weight ≤ −1, M ′ must be of weight ≤ n− 1. Thus the weight n does not occurin M ′, hence it must occur in M ′′. By condition (ii) every irreducible representation is pureof some weight, so M ′′ must be pure of weight n. For the weight filtration we thus get:WmMC = WmM

′C for m < n, and WmMC = MC for m ≥ n. Therefore the assertion follows

from the inductive assumption for M ′.

With this (b) is proved. For (a) it remains to show that for every n ∈ Z the Hodgefiltration induces a pure Hodge structure of weight n on GrWn M := WnM/Wn−1M . SinceGrWn M , as we have just proved, is again a representation of P over Q, it suffices to show:If M is already pure of weight n, then the decomposition MC =

⊕p+q=nM

p,q is a pureHodge structure of weight n. Let us again consider the map (LieW )⊗M → M of above.

19

The left hand side is mixed of weight ≤ n− 1, the right hand side pure of weight n. Thusthis map must be zero, which means that W operates trivially on M . By (i) the operationof SC on MC is therefore defined over R, so (Mp,q) = Mp,q for all p, q with p+ q = n. q.e.d.

1.5. Propositionen: Let M be a finite dimensional Q-vector space. A representationk : SC → GL(MC) defines a rational mixed Hodge structure on M , if and only if there existsa connected linear algebraic group P over Q, a representation ρ : P → GL(M) defined overQ, and a homomorphism h : SC → PC, such that k = ρ h and the conditions in 1.4 aresatisfied.

Proof. One implication is already contained in 1.4. So let us assume that k inducesa rational mixed Hodge structure on M . Let P ⊂ GL(M) be the stabilizer of the weightfiltration, this is a Q-parabolic, hence connected subgroup of GL(M). Let ρ : P → GL(M)be the inclusion, then k factors in exactly one way through a homomorphism h : SC → PC.By assumption GrWn M is a pure Hodge structure of weight n for every n ∈ Z. Thus forall p, q ∈ Z with p + q = n the space (GrWn M)p,q = (GrWn )p,q is the eigenspace in GrWn Massociated to the character c((p, q)) = (p, q), so the operation of S on GrWn MC is alreadydefined over R. Since furthermore Gm,R ⊂ S acts by scalars om GrWn MR, its operation isgiven by cocharacter Gm,Q → Z(GL(GrWn M)) that is defined over Q. The properties 1.4(i) and (ii) now follow from the isomorphism P/W ∼=

∏N GL(GrWN M .

Finally LieP is the successive extension of Hom(GrWn M,GrWmM) for all n ≥ m andLieW the successive extension of Hom(GrWn , Gr

WmM) for all n > m. In Hom(GrWn M,GrWmM)C

as representation of SC only characters (p, q) occur with p+ q = m− n, so LieW containsprecisely all eigenspaces in LieP associated to the characters (p, q) with p+ q ≤ −1. Thisimplies 1.4 (iii). q.e.d.

1.6. Equivariant families of Hodge structures: Generalizing [D2] 1.1.11 and1.1.12 we want to parametrize Hodge structures by a homogeneous space under a real Liegroup. Furthermore we want this group to come from a linear algebraic group over Q.Finally the homogeneous space should be described in terms of homomorphisms SC → PCand should carry a canonical complex structure.

More concretely let P be a connected linear algebraic group over Q. In a crucial case(see 4.7) P will be strongly related to a parabolic subgroup of another group, whence theunusual notation. Let W be the unipotent radical of P , and HW a P (R) ·W (C)-conjugacyclass in Hom(SC, PC). We assume that for one, hence by 1.4 (c) for all h ∈ HW the conditionsin 1.4 hold. We call a representation M of P pure of weight n, mixed of weight ≤ n, oftype A ⊂ Z ⊕ Z, etc. if and only if for some (⇔ by 1.4 (c) for all) h ∈ HW the inducedHodge structure on M has the respective property. Note that by 1.4 (ii) LieG = LieP/Wis pure of weight 0, so by 1.4 (iii) LieP is mixed of weight ≤ 0. The following propositions1.7, 1.10 and 1.12 are generalizations of [D2] 1.1.14.

1.7. Proposition: Let P and HW be as in 1.6, and let M be a faithful representationof P . Let φ be the obvious map

HW → rational mixed Hodge structures on M.

20

(a) There exists a unique structure on φ(HW ) as a complex manifold, such that the Hodgefiltration on MC depends analytically on φ(h) ∈ φ(HW ). This structure is P (R) ·W (C)-invariant, and W (C) acts analytically on φ(HW ).

(b) For any other representation M ′ of P the analogous map

φ′ : HW → rational mixed Hodge structures on M ′

factors through φ(HW ). The Hodge filtration on M ′ varies analytically with φ(h) ∈ φ(HW ).

(c) If in addition M ′ is faithful, then φ(HW ) and φ′(HW ) are canonically isomorphic andthe isomorphism is compatible with the complex structure.

Proof. (a) Since the weight filtration on M is constant, the Hodge filtration gives aninjective map φ(HW ) → Grass(M)(C) to a certain Grassmann variety. To prove the firstassertion we have show that its image is a locally closed analytic subvariety. Fix h ∈ HWand let C(h) be the centralizer of h in P (R) ·W (C), then φ factors through

HW ∼= P (R) ·W (C)/C(h)→ P (C)/expF 0(LieP )C → Grass(M)(C).

The map on the right hand side is a closed embedding of complex manifolds. The complexstructure in the middle is P (C)-invariant, and P (C) operates analytically. Thus it is enoughto show that the image ofHW is an open subset of P (C)/expF 0(LieP )C. Let L := (LieP )R+(LieW )C ⊂ (LieP )C, and let L0,0 ⊂ L be the eigenspace corresponding to the trivialoperation of AdP h. Then on the tangent space at h the map is given by

THW ,h∼= L/L0,0 → (LieP )C/F

0(LieP )C,

and we have to show that this map is surjective. This is equivalent to the equality (LieP )C =L + F 0(LieP )C. Since both sides of this contain (LieW )C, it suffices to show the equality(LieGC = (Lie)R + F 0(LieG)C. But LieG carries a pure Hodge structure of weight 0, sothis equality follows from (LieG)R ∩ F 0(LieG)C ⊂ (LieG)0,0 and dimension count.

(b) follows directly from (c), applied to M ⊕M ′ in the place of M ′. By part (b) of thefollowing lemma the fibres of φ do not depend on the choice of M , so the two maps φ andφ′ have identical fibres. This implies the first assertion of (c). Finally the proof shows thatthe complex structure on φ(HW ) can be defined independently on M . q.e.d.

1.8. Lemma: Let P , HW , M and φ be as in 1.7, and let h ∈ HW .

(a) The projection π : P → G induces an isomorphism

CentP (R)·W (C)(h)→ CentG(R)(π h).

(b)We haveStabP (R)·W (C)(φ(h) = exp(F 0(LieWC) o CentP (R)·W (C)(h).

Proof. (a) By 1.4 (ii) and (iii) the space of invariants in LieWC under (Ad |W )hw isthe zero subspace. Hence h defines a canonical Levi decomposition PC = WC o CentPC(h w). Since moreover π h is by 1.4 (ii) already defined over R, we have a correspondingdecomposition

P (R) ·W (C) = W (C) o CentP (R) ·W (C)(h w).

21

Thus CentP (R)·W (C)(h W ) ∼= G(R), so the centralizers of h in both these groups areisomorphic.

(b) First suppose that W = 1. Then there exists a unique decomposition M =⊕n∈ZMn into representions Mn of P = G that are pure of weight n. For all p, q we

therefore haveMp,q = F pMC ∩ F qmC ∩Wp+qMC,

so the decomposition into eigenspaces under ρ h is already uniquely determined by φ(h).Hence we have StabP (R)(φ(h)) = CentP (R)(h). In the general case this together with (a)implies

CentP (R)·W (R)(h) ⊂ StabP (R)·W (C)(φ(h) ⊂W (C) o CentP (R)·W (C)(h),

so it only remains to show that StabW (C)(φ(h)) = exp(F 0(LieW )C). This follows from

Stab(LieW )C(φ(h)) = (LieW )C ∩ F 0(End(M))C = F 0(LieW )C.

q.e.d.

1.9. Variation of Hodge structures: LetX be a complex manifold. A variationof rational mixed Hodge structures over X consists of a local system of finite dimensionalQ-vector spaces, together with a rational mixed Hodge structure on every fibre, such that:

(a) The weight filtration is locally constant and the Hodge filtration varies holomorphically.LetM be the locally free coherent OX -sheaf associated to the vector bundle, and let FPMbe the coherent subsheaves for all p ∈ Z that induce the Hodge filtration in each stalk.

(b) Transversality: For every p ∈ Z the canonical connection O : M→M⊗OX Ω1X maps

the subsheaf F pM to F p−1M⊗OX Ω1X .

1.10. Proposition: Let P , HW , M and φ be as in 1.7. Then we have a variationof rational mixed Hodge strutures on M over φ(HW ), if and only if for one (⇔ for all)h ∈ HW the Hodge structure on LieP is of type

(−1, 1), (0, 0), (1,−1) ∪ (−1, 0), (0,−1) ∪ (−1,−1).

Proof. With the notation of the proof of 1.7 the transversality condition in the pointφ(h) is equivalent to

im(dΦ) ⊂ F−1 End(M)C/F0 End(M)C.

Since this image equals (LieP )C/F0(LieP )C, this means F−1(LieP )C = (LieP )C. In other

words the Hodge numbers hp,q must vanish for all p, q ∈ Z with q < −1. If they vanish,then so do all hp,q with p < −1. Since LieP is mixed of weight ≤ 0, one easily finds thecondition equivalent to the stated type restriction. q.e.d.

1.11 Polarization of pure Hodge structures: Let√−1 be a primitive fourth root

of unity in C, then Z(1) := 2π√−1 ·Z is a submodule of C that is independant of the choice

of√−1. For all n ∈ Z and any Z-module M define M(n) := M ⊗ Z(1)⊗n. By the given

embedding Z(1) → C we get a canonical isomorphism M(n)C ∼= MC. The Tate Hodge

22

structure is the Q-vector space Q(1) with the pure Hodge structure of type (−1,−1). Forevery n ∈ Z we get a pure Hodge structure of type (−n,−n) on Q(n) = Q(1)⊗n, and againa canonical isomorphism Q(n)C ∼= C.

Let M be a Q-vector space and k : S→ GL(MR) a representation that induces a pureHodge structure of weight n on M . Let Ψ : M ⊗M → Q(−n) be a morphism of Hodgestructures. Choose a primitive fourth root of unity

√−1 ∈ C× = S(R), and consider the

bilinear form

MR ×MR → Q(−n)C = C, (m,m′) 7→ (2π√−1)n ·Ψ(m, k(

√−1)m′).

Claim: It does not depend on the choice of√−1, and takes values in R.

Proof. Since M is pure of weight n, we have k(−√−1) = k(−1) · k(

√−1) = (−1)n ·

k(√−1). Thus on replacing

√−1 by−

√−1 the bilinear form is multiplied by (−1)n·(−1)n =

1. So it does not depend on the choice of√−1. Also note that k(

√−1) lies in GL(M)(R), so

Ψ(m, k(√−1)m′) lies in Q(−n)R = (2π

√−1)−n · R ⊂ C. This implies the second assertion.

q.e.d.

Having shown this the following definition is meaningful. A polarizition of a rationalHodge structure pure of weight n is a morphism of Hodge structures Ψ : M ⊗M → Q(−n),such that the bilinear form

MR × MR → R, (m,m′) 7→ (2π√−1)n ·Ψ(m, k(

√−1)m′)

is symmetric and positive definite. Note that this pairing is symmetric if and only if n iseven and Ψ symmetric, or n odd and Ψ alternating.

1.12 Proposition: Let P and HW be as in 1.6. Let G1 be a normal subgroup ofG = P/W , defined over Q, which for one (⇔ for all) h ∈ HW contains the image of h(S1).The following assertions are equivalent:

(a) int(π(h√−1))) is a Cartan involution on G1 for (⇔ for all) h ∈ HW . In particular

the center of G1 is compact, hence splits over a CM -field.

(b) For every representation M of P which is pure of some weight n there exist

- a representation N of P , defined over Q, which factors through G/G1 and is pure oftype (n, n),

— a P -equivariant nondegenerate pairing Ψ : M ⊗M → N , and

— for every h ∈ HW a morphism of rational Hodge structures λh : N → Q(−n),

such that for every h ∈ HW the pairing λh Ψ : M ⊗M → Q(−n) is a polarization forthe Hodge structure on M defined by h.

Proof. Both conditions depend only on P/W , so we may assume that P = G. Fix apure representation ρ : G → GL(M). Fix h0 ∈ HW and a choice of

√−1, then we shall

show that condition (b) for the given representation M is equivalent to the following one:

(c) There exists a G1,R-invariant pairing Ψ′ : MR ×MR → R, defined over R, such that(m,m′) 7→ Ψ′(m, ρ h0(

√−1)m′) is symmetric and positive definite.

Clearly (b) implies (c) with Ψ′ : (2π√−1)n · λh0 Ψ. Conversely assume (c). Note first

that since (ρ h0(√−1))2 = ρ h0(−1) = (−1)n, the pairing Ψ′ is symmetric if n is even,

23

alternating if n is odd. So let N := (S2M)G1 if n is even, respectivley (∧2M)G1 if n is odd,

where (.)G1 denotes th G1-coinvariants. Let Ψ : M ⊗M → N be the canonical pairing,then by constructionΨ′ factors through N , hence Ψ′ = µ0 Ψ for some homomorphismµ0 : NR → R. Since G1 is a normal subgroup of G, N is a G-invariant quotient of M ⊗M ,hence necessarily pure of weight 2n. Since moreover G1 operates trivially on N , it is alreadypure of type (n, n). Since by assumption Ψ′ is nontrivial , so is Ψ.

By the construction of N for every µ ∈ Hom(NR,R) the pairing

MR ×MR → R, (m,m′) 7→ µ Ψ(m, ρ h0(√−1)m′)

is automatically symmetric. Thus the set of all µ, such that this pairing is symmetric andpositive definite, is open in the usual topology. Furthermore the existence of µ0 shows thatthis set is nonempty. Since the homomorphisms that are defined over Q are dense, we canreplace µ0 by a homomorphism N → Q that is defined over Q. Let λh0 := (2π

√−1)−1 ·

µ0 : N → Q(−n), this is automatically a morphism of rational Hodge structures, and byconstruction λh0 Ψ is a polarization for the Hodge structure on M defined by h0.

Now let g ∈ G(R) and h = int(g) h0. Denote by τG → GL(N) the representation ofG on N . Then

µ0 τ(g−1) Ψ(m, ρ h(√−1)m′) = µ0 Ψ(ρ(g−1)m, ρ h0(

√−1) · ρ(g−1)m′),

so after replacing µ0 by µ0 τ(g)−1, condition (c) holds for h in place of h0. Since the abovedefinitions of N and Ψ do not depend on h0, the desired λh exists for every h ∈ HW . Thus(c) implies (b).

By [D2] 1.1.15 condition (c) holds for all M if and only if int(h0(√−1)) is a Cartan

involution on G1,R. Clearly this second property is invariant under conjugation by G(R).It also implies that Z(G1)(R) is compact, so the torus Z(G1)0splits over a CM -field. Thus(c) is equivalent to (a). q.e.d.

1.13. Restriction of the group: In the situation of 1.12, we encounter the followingphenomenon. The quotient G/G1 is entirely arbitrary , but it contributes almost nothingto the family of Hodge structures. In particular the semisimple part of G/G1 is ballast. Onthe other hand, N is an arbitrary representation of G/G1, so the G-orbit generated by λhmay be disturbingly large. For our purposes we want to have that this orbit consists onlyof multiples of λh, at least for irreducible M . By 1.12, this clearly holds if G/G1 is a Q-splittorus. Thus, to solve both problems at the same time, we impose the following conditionon P . Let G2 be the smallest normal subgroup of G, defined over Q, which contains theimage π h(S) for one (⇔ for all) h ∈ HW . This group essentially determines the family ofHodge structures. We require:

(*) G/G2 is an almost direct product of a Q-split torus with a torus of compact type definedover Q.

Consider the situation of 1.12, with G1 the smallest normal Q-subgroup of G suchthat G/G1 is a Q-split torus. Clearly the condition (*) above together with 1.12 (a) isequivalent to: (a′) int(π(h(

√−1))) induces a Cartan involution on GadR , and Gad possesses

24

no nontrivial factors of compact type that are defined over Q. Furthermore the center of Gis an almost direct product of a Q-split torus with a torus of compact type defined over Q.

Assume that this condition holds. Then in 1.12 (b), N is a direct sum of onedimensionalrepresentations of G. If M is irreducible, then the space of G1-coinvariants of M ⊗ Mhas dimension ≤ 1, so we can choose N to have dimension 1. If M is arbitrary, butdim(G/G1) ≤ 1, then G operates by scalars on N through a character that depends onlyon n. ¿From the proof of 1.12 we see that again we can choose N to have dimension 1.Observe that if n 6= 0, the G-orbit generated by λh always has dimension ≥ 1. Moreover,as in the examples 2.7–8 below, the different λh need not have the same sign.

1.14. Proposition: Suppose that W = 1 and the conditions od 1.10 and 1.12 aresatisfied. Then every connected component of HW ∼= φ(HW ) is a hermitian symmetricdomain.

Proof. [D2] 1.1.14 (iii). q.e.d.

1.15 Replacing by smaller orbit: In general the map φ from proposition 1.7 is notinjective. We can, however, replace HW by an orbit under a subgroup of P (R) ·W (C), andstill get the same image under φ. Which subgroup we can choose is seen from 1.2 (c). Inparticular, if we want a subgroup of the form P (R) · U(C), where U ⊂ W is a subgroupdefined over Q, in general there is only the following possibility. Although in general theresulting map will still not be injective, it will be injective in the case that is of interest tous, namely when the condition of 1.10 is satisfied.

Explicitly let P and HW be as in 1.6. Let U ⊂ W be the unique connected subgroup,defined over Q, such that LieU = W−2(LieW ). By 1.4 (c) it does not depend on h ∈ HW .Let π′ be the canonical projection P → P/U . Let V := W/U , then LieV is pure of weight−1 and −2, V must be abelian. Consider instead of 1.4 (i) the following stronger conditionfor h ∈ Hom(SC, PC):

(i′)π′ h : SC → (P/U)C is already defined over R.

Finally letH := h ∈ HW |h satisfies (i′)

1.16 Proposition: With the notation on 1.15 we have: (a) H is a nonempty P (R) ·U(C)-orbit in Hom(SC, PC).

(b) For M und φ as in 1.7 we have φ(H) = φ(HW ).

(c) If foreover F 0(LieU)C = 0, then φ(H) ∼= H.

Proof. Since LieV is pure of weight −1, we have

F 0(LieV )C ∩ F 0(LieV )C = (LieV )R ∩ F 0(Lie v)C = 0,

and by dimension count we get (LieV )C = (LieV )R ⊗ F 0(Lie v)C = 0. This implies on theone hand (LieW )C = ((LieW )R + (LieU)C) + F 0(LieW )C, that is

W (C) = (W (R) · U(C)) · exp(F 0(LieW )C),

25

on the other hand

((LieW )R + (LieU)C) ∩ F 0(LieW )C = (LieU)C ∩ F 0(LieW )C = (LieU)C).

that is(W (R) · U(C)) ∩ exp(F 0(LieW )C) = exp(F 0(LieU)C).

As LieW is of weight ≤ −1, there exists no nontrivial group of W on which P operatestrivially. Thus the variety of all Levi decompositions of P is a principal homogeneous spaceunder W .

Let us first prove (b). We have to show that for every h ∈ HW there exists p ∈StabP (R)·W (C)(φ(h)), such that int(p) h ∈ H, that is π′ int(p) h : S→ (P/U)C is definedover R. As in the proof of 1.8 (a) consider the Levi decomposition of PC defined by h.Since there also exists a Levi decomposition that is defined over R, there exists w0 ∈W (C)such that int(w0) h w, hence also int(w0) h is defined over R. Write w0 = w′ · w′′ withw′ ∈W (R) ·U(C) and w′′ ∈ exp(F 0(LieW )C). Then π′ int(w′′)h = π′ int(w′)−1(w0)his defined over R, and by 1.8 (b) we have φ(int(w′′) h) = φ(h). Thus the assertion followswith p = w′′.

This also proves that H is nonempty. (a) now follows from the stronger statement:

(*) For any h ∈ H and p ∈ P (R) · W (R), we have int(p) h ∈ H if and only ifp ∈ P (R) · U(C).

To prove this we may assume without loss of generality that U is trivial. Writingp = p′ · w′ with p′ ∈ P (R) and w′′ ∈ W (C) we find that int(p) h is defined over R if andonly if int(w′)h is defined over R. That again is true if and only if the Levi decompositionof PC that is defined by int(w′) h w is defined over R. By the above remark the map

w′ 7→ (the Levi decomposition of PC defined by int(w′) h w)

is an algebraic isomorphism of W to the variety of Levi decompositions of P , and is definedover R since hw is defined over R. Thus the Levi decomposition associated to int(w′)hwis defined over R if and only if w′ ∈W (R), as desired.

For (c) consider an element h ∈ H. The fibre (φ|H)−1(φ(h)) is the StabP (R)·U(C)(φ(h))-orbit in H that is generated by h. If F 0(LieU)C = 0), then by part (b) of the followinglemma this fibre consists only of h. Thus φ|H is a bijection. q.e.d.

1.17. Lemma: Let P ,H, etc. be as in 1.15, and let h ∈ H. Then (a) CentP (R)·U(C)(h) =CentP (R)·W (C)(h) ∼= CentG(R)(π h), and

(b) StabP (R)·U(C)(φ(h)) = exp(F 0(LieU)C) o CentP (R)·U(C)(h).

Proof. (a) By 1.8 (a) it suffices to show that CentP (R)·W (C)(h) is contained in P (R) ·U(C). This follows from the assertion (*) in the proof of 1.16.

(b) The assertions (a) and 1.8 (b) imply

StabP (R)·U(C)(φ(h)) = (P (R) · U(C)) ∩ StabP (R)·W (C)(φ(h))

= (P (R) · U(C)) ∩ exp(F 0(LieW )C) o CentP (R)·W (C)(h))

= ((P (R) · U(C)) ∩ (exp(F 0(LieW )C)) o CentP (R)·W (C)(h)),

26

and in the proof of 1.16 we showed

(P (R) · U(C)) ∩ (exp(F 0(LieW )C) = (P (R) · U(C)) ∩ (exp(F 0(LieW )C)

= exp(F 0(LieU)C) o CentP (R)·U(C)(h)).

q.e.d.

1.18 Summary Putting together the results of 1.4, 1.7, 1.10, 1.12, 1.13 and 1.16we get the following, albeit somewhat complicated picture. Let P be a connected linearalgebraic group over Q. Let W be the unipoint radical of P , and U ⊂ W a subgroup thatis normal in P . Let V := W/U , G := P/W and π : P → G and π′ : P → P/U be thecanonical projections. Let H be a P (R) ·U(C)-orbit in Hom(SC, PC), such that for some (⇔for all) h ∈ H:

(i’) π′ h : SC → (P/U)C is already defined over R.

(ii) π h w : Gm,R → GR is a cocharacter of the center of G that is defined over Q.

(iii) Under the weight filtration on LieP that is defined by AdP h we have W−1(LieP ) =LieW and W−2(LieP ) = LieU .

(iv) The Hodge structure on Lie P is of type

(−1, 1), (0, 0), (1,−1) ∪ (−1, 0), (0,−1) ∪ (−1,−1).

(v) int(π(h(√−1))) induces a Cartan involution on GadR , and Gad possesses no nontrivial

factors of compact type that are defined over Q.

(vi) The center of G is an almost direct product of a Q-split torus with a torus of compacttype defined over Q.

Then

(a) H possesses a canonical P (R) · U(C)-invariant complex structure (1.7 and 1.16).

(b) For every represention M of P we have a variation of rational mixed Hodge structures(1.7 and 1.10).

(c) The operation of P (R) · U(C) on H can be extended canonically to an operation ofP (R) ·W (C). The complex structure is still invariant under this bigger group, and W (C)operates analytically on H (1.7 and 1.16).

(d) For every irreducible representation M of P that is pure of some weight n there existsa one dimensional representation of P on Q(−n), and a P -equivariant bilinear form Ψ :M × M → Q(−n), such that for all h ∈ HW either Ψ or −Ψ is a polarization of thecorresponding Hodge structure on M (1.12 and 1.13).

(e) If W = 1, then every connected componente of H is a hermitian symmetric domain(1.14).

Remark. We shall see later (2.19) that every connected component of H is (noncanon-ically) isomorphic to a holomorphic vector bundle on a hermitian symmetric domain.

1.19. Generalization to arbitrary weights: The cocharacter π h w above canbe considered as a weight, since it corresponds to the weight of the Hodge structure induced

27

on any representation of G. In this sense the above condition (ii) means that the weightis rational, i.e. defined over Q. Since both this requirement and condition (vi) above arenontrivial restrictions for Shimura varieties, we want to replace them by weaker conditions.For our purposes it will be enough to assume the same conditions for the image of P inAut(W ), or equivalently in Aut(U) × Aut(V ). Explicitly we replace (ii) and (vi) by theconditions:

parindent=0pt (ii’) π h w : Gm,R → GR is a cocharacter of the center of G.

(vi’) The center of G acts on U and on V through a torus that is an almost directproduct of a Q-split torus with a torus of compact type defined over Q.

Observe that these conditions imply that the weight acts on both U and V through arational scalar character.

It is clear that with these weakened conditions the conclusions of 1.18, except (d),still hold. It is important, however, that (d) does hold for M an arbitrary irreduciblesubquotient of V . Note also that the conclusion of 1.17 remain valid. Finally there is arelation with data as in 1.18. By 1.18 (iii) we have Z(P ) ∩W = 1, so Z(P ) is reductiveand isomorphic to the kernel of the action of Z(G) on LieU ⊗ LieV . Thus P1 := P/Z(P )together with the obvious choice for H1 satisfies the conditions of 1.18.

28

Chapter 2

Mixed Shimura data

In the first chapter we introduced certain P (R) ·U(C)-conjugacy classes of homomorphismsSC → PC as a natural generalization to nonreductive groups of the conjugacy classes ofhomomorphisms S → GR considered in [D2]. In this chapter we bring in yet anothergeneralization by considering finite coverings of such conjugacy classes. In various placeswe shall see the usefulness of this generalization (for instance in 3.9, 3.16–17), and indeedits necessity (see 4.11 (example 4.25–26), 7.1–2 or 8.5–6) for a unified formalism of thearithmetic structure of the boundary of Shimura varieties.

Definition 2.1 is a generalization of [D1] 1.5 resp. [D2] 2.1.1.1–3. In 2.3–10 we introducemore notions and some standard examples. In 2.11–12 we study the precise structure ofthe finite covering mentioned above. The largest part of this chapter, starting with 2.15,deals with properties and constructions related to the unipoint radical. In particular weprove a reduction lemma (2.26), by means of which any mixed Shimura data can (almost)be embedded into a direct product of usual pure ones (as in [D1] 1.5 or [D2] 2.1.1) togetherwith the standard examples described in this chapter.

2.1 Definiton: Let P be a connected linear algebraic group over Q. Let W be itsunipoint radical and U a subgroup of W that is normal in P . Let V := W/U , G :=P/W , and π : P → G and π′ : P → P/U be the canonical projections. Let X be a lefthomogeneous space under the subgroup P (R) ·U(C) ⊂ P (C). Let h : X → Hom(SC, PC) bea P (R) · U(C)-equivariant map, such that:

(i) Every fibre of h consists of at most finitely many points.

Furthermore for some (⇔ for all) x ∈ X we assume:

(ii) π′ hx : SC → (P/U)C is already defined over R.

(iii) π hx w : Gm,R → GR is a cocharacter of the center of G.

(iv) AdP hx induces on LieP a rational mixed Hodge structure of type

(−1, 1), (0, 0), (1,−1) ∪ (−1, 0), (0,−1) ∪ (−1,−1).

29

(v) The weight filtration on LieP is given by

Wn(LieP ) =

0 if n < −2,LieU if n = −2,LieV if n = −1,LieW if n ≥ 0.

(vi) int(π(hx(√−1))) induces a Cartan involution on GadR .

(vii) Gad possesses no nontrivial factors of compact type that are defined over Q.

(viii) The center of G acts on U and on V through a torus that is an almost direct productof a Q-split torus with a torus of compact type defined over Q.

Definition: We call such a triple (P,X , h) mixed Shimura data. If W = 1, then we alsocall it pure Shimura data, or just Shimura data.

Remarks: (a) The data (P,X , h) uniquely determines the rest, in particular U is char-acterized by (v). Thus the definition is meaningful. For abbreviation we shall mostly write(P,X ), since for every pair (P,X ), we shall always consider exactly one map h.

(b) It is easy to show that condition (iii) is a consequence of (iv).

(c) X consists of finitely many connected components. The map h : X → h(X ) is a localdiffeomorphism, so the complex structure that is given on h(X ) by 1.18 (a) induces acomplex structure on X .

(d) If W = 1 and X ∼= h(X ), then the definition is equivalent to [D1] 1.5 or to [D2] 2.1.1.1–3.

2.3. Definition: Let (P1,X1, h1) and (P2,X2, h2) be mixed Shimura data. Amorphism (P1,X1, h1) → (P2,X2, h2) consists of a homomorphism φ : P1 → P2 and aP1(R) · U1(C)-equivariant map Ψ : X1 → X2, such that following diagram commutes:

X1ψ−−→ X2

h1 ↓ ↓ h2

Hom(SC, P1,C)h7→φh−−−−−−→ Hom(SC, P2,C)

We call a morphism an embedding, and write (P1,X1, h1) → (P2,X2, h2), if both φ and Ψare injective.

2.4 Proposition: For any morphism (φ,Ψ) : (P1,X1)→ (P2,X2) the map Ψ : X1 →X2 is holomorphic. If (φ,Ψ) is an embedding, then Ψ is a closed embedding.

Proof. By 1.7 the associated map h1(X1)→ h2(X2) is holomorphic. From this the firstassertion follows, since Xi is locally isomorphic to hi(Xi) for i = 1, 2. The second assertionfollows from the fact, that P1(R) · U1(C) is a closed subgroup of P2(R) · U2(C). q.e.d.

2.5. Definition: For any two mixed Shimura data (P1,X1) and (P2,X2) we definetheir product as (P1,X1)× (P2,X2) := (P1×P2,X1×X2) with the obvious map X1×X2 →Hom(SC, (P1 × P2)C). Clearly this is again mixed Shimura data. In fact this is the directproduct in the category of mixed Shimura data, that is for every (P,X ) and any two

30

morphism (P,X ) → (Pi,Xi) there exists a unique morphism (P,X ) → (P1,X1) × (P2X2)such that the following diagram commutes:

(P1,X1) pr1

(P,X ) −→ (P1,X1)× (P2,X2) pr2

(P2,X2)

2.6. Example: Consider a torus T over Q, a homomorphism k : S → TR, and afinite set Y with a transitive action of π0(T (R)). With the constant map h : Y → k ⊂Hom(SC, TC) we then have mixed Shimura data (T,Y). In fact every mixed Shimura data(P,X ), where P is a torus or where X is a finite set, is of this form.

2.7. Example: the symplectic group: (Compare [D1] 1.6) Fix an integer g ≥ 1.Let V2g be a Q-vector space of dimension 2g and Ψ : V2g × V2g → Q a nondegener-ate alternating form. Let G2g := CSP2g,Q be the group of all f ∈ GL(V2g), such thatΨ(f(v), f(v′)) = λ ·Ψ(v, v′) for some scalar “multiplier” λ. This multiplier yields a surjec-tive homomorphism CSP2g,Q → Gm,Q, whose kernel is SP2g,Q = (CSP2g,Q)der. Let H2g bethe set of all homomorphisms k : S→ CSP2g,R that induces a pure Hodge structure of type(−1, 0), (0,−1) on V2g, and for which the pairing R2g×R2g → R, (v, v′) 7→ Ψ(v, k(

√−1)v′)

is symmetric and (positive or negative) definite. Note that this second condition does notdepend on the choice of

√−1. It is wellknown that CSP2g,Q is a reductive group and

operates transitively on H2g. Moreover H2g possesses exactly two connected components.The pair (CSP2g,Q,H2g) together with the inclusion h : H2g → Hom(SC, CSP2g,Q) is pureShimura data.

For g = 1 we have CSP2g,Q = GL2,Q and H2 can be identifies in the usual way with theunion of the complex upper and lower half planes. This, and more generally the unboundedrealization of H2g will be described in 4.25–26. Later in this chapter we shall give exampleswith nontrivial W , using some general constructions.

2.8. Example: For g = 0 we have the following analog of (CSP2g,Q,H2g). LetG0 := Gm,Q and k : S → G0,R, z 7→ z · z. Let H0 be the set of isomorphism Z → Z(1).Thus an element λ ∈ H0 corresponds to a unique choice of

√−1 by λ(1) = 2π

√−1. At

present we are only interested in the fact that H0 consists exactly two elements, but laterthe precise definition will become important. Consider the unique transitive operation ofπ0(G0(R)) = π0(R×) of H0, and let h : H0 → k ⊂ Hom(SC, G0,C) be the constant map.With these definitions (G0,H0) is pure Shimura data, as in 2.6.

For any g ≥ 1 consider the multiplier map φ : CSP2g,Q → Gm,Q. Then for everyx ∈ H2g we have φ hx = k. Define a map H2g → H0 ⊂ Hom(Q,Q(1)) by

x 7→ the unique λ ∈ H0 such that λ Ψ is a polarization of V2g.

Clearly this map is equivariant under CSP2g(R) → Gm(R), so we obtain a canonical mor-phism (CSP2g,Q,H2g)→ (Gm,Q,H0).

31

2.9. Proposition: Let (P,X ) be mixed Shimura data and P0 a normal subgroup of P .There exists a quotient mixed Shimura data (P,X )/P0 and a morphism (P,X )→ (P,X )/P0,unique up to isomorphism, such that every morphism (P,X )→ (P ′,X ′), where the homo-morphism P → P ′ factors through P/P0, factors in a unique way through (P,X )/P0:

P −→ P ′

↓ P/P0

=⇒(P,X ) → (P ′,X ′)↓

(P,X )/P0

Proof. The uniqueness follows from the universal property. First we construct (P,X )/P0.Let P1 := P/P0 and φ : P → P1 be the canonical projection. Then W1 = φ(W ) is theunipotent radical of P1, and U1 := φ(U) is normal in P1. Fixing x ∈ X we get a bijection

(P (R) · U(C)/ StabP (R)·U(C)(x) ∼−−→ X , p 7→ p · x.

LetX1 := (P1(R) · U1(C))/φ(StabP (R)·U(C)(x)), and

h1 : X1 → Hom(SC, P1,C), [p1] 7→ int(p1) φ hx.

By construction (P1,X1, h1) satisfies the conditions 2.1, so (P1,X1) is mixed Shimura data.Next define an equivariant map Ψ : X → X1 by p · x 7→ [φ(p)], then (φ,Ψ) : (P,X ) →(P1,X1) is a morphism of mixed Shimura data. It is now trivial to show that (P1,X1)possesses the desired universal property. q.e.d.

Remark. Let (P1,X1) := (P,X )/P0. By the proof above the map X → X1 is surjectiveif and only if the homomorphism P (R) · U(C) → P1(R) · U1(C) is surjective. While this isnot the case in general, it is so if P0 is unipotent.

2.10 Example: Let (P,X ) be mixed Shimura data.Since P acts nontrivially on everysubquotient of W , the commutator subgroup P der contains the unipotent radical of P . ThusP/P der is a torus. The property that X ∼= h(X ) is not invariant under forming quotients,as the following example shows. Let (CSP2g,Q,H2g)→ (Gm,Q,H0) be the morphism definedin 2.8. It is easy to verify that it induces an isomorphism

(CSP2g,Q,H2g)/SP2g,Q∼−−→ (Gm,QH0).

In this case H2g∼= h(H2g), but H0 6∼= h(H0). This shows one of the advantages of introduc-

ing finite coverings in 2.1.

Now let (P,X , h) be mixed Shimura data and ι : h(X ) → Hom(SC, PC) the inclu-sion. Then (P, h(X ), ι) is again mixed Shimura data, and there is a canonical morphism(P,X , h) → (P, h(X ), ι), or (P,X ) → (P, h(X )). We want to derive a relation between(P,X ), (P, h(X )) and (P,X )/P der. This relation will show that the generalization to finitecovering does not yield essentially new objects.

2.11. Proposition: For any mixed Shimura data (P,X ) the canonical morphism

(P,X ) −→ (P,X )/P der × (P, h(X ))

32

is an embedding.

Proof. (Compare [D1] prop. 2.6) Let T := P/P der, φ : P → T the canonical ho-momorphism, and (P,X )/P der = (T,Y). Fix x ∈ X , and define as abbreviation A :=StabP (R)·U(C)(x) and B := CentP (R)·U(C)(hx). The group A is a subgroup of finite index inB. As in the proof of 2.9 we identify X with (P (R)·U(C))/A. Since h(X ) ∼= (P (R)·U(C))/Bwe must prove that the map

(P (R) · U(C))/A→ T (R)/φ(A)× (P (R) · U(C))/B

is injective. This is equivalent to

A = φ−1(φ(A)) ∩B.

Now φ−1(φ(A))∩B = A ·P der(R) ·U(C)∩B = A · (P der(R) ·U(C)∩B), because A alreadylies in B. Since P der(R) · U(C) ∩B = CentP der(R)·U(C)(hx) the assertion is equivalent to

CentP der(R)·U(C)(hx) ⊂ A.

Now the connected component of identity of CentP der(R)·U(C)(hx) is contained in that of B,hence in A. Thus it is enough to show that CentP der(R)·U(C)(hx) is connected.

By 1.17 (a) we may now assume P = G. Being the centralizer of a torus, the algebraicsubgroup CentGderR

(hx) is connected. By 2.1 (vi) its Lie algebra is the subspace of invariants

in (LieG)R under AdG(h(√−1)), so CentGder(R)(hx) is compact. Since every algebraically

connected compact linear group is already topologically connected (see [BT] 14.3), we aredone. q.e.d.

2.12. Corollary: Let (P,X ) be mixed Shimura data. Every connected component ofX is mapped isomorphically to its image in Hom(SC, PC). If W = 1, then every connectedcomponent of X is a hermitian symmetric domain.

Proof. The first assertion follows directly from the injection of X → Y × h(X ) with afinite set Y. The second one follows from the first together with 1.14. q.e.d.

2.13. Covering by irreducible mixed Shimura data: Let (P,X ) be mixedShimura data. Let P1 be a normal subgroup of P , defined over Q, such that for some(⇔ for all) x ∈ X the homorphism hx factors through P1,C. Then W lies in P1, andP = P1 · π−1(Z(G)). Every P1(R) · U(C)-orbit X1 ⊂ X is the union of some connectedcomponents of X , since for any x ∈ X1 there is an isomorphism

X1∼= P1(R) · U(C)/ StabP1(R)·U(C)(x)

∼= (P1(R) · U(C) · StabP (R)·U(C)(x)/StabP (R)·U(C)(x),

and by 1.17 P1(R) ·U(C) ·StabP (R)·U(C)(x) is a subgroup of finite index in P (R) ·U(C). WithX1 3 x 7→ hx : SC → P1,C ⊂ PC we obtain mixed Shimura data (P1,X1) and a canonicalembedding (P1,X1) → (P,X ).

33

Writing X as a disjoint union of P1(R) · U(C)-orbits Xi we get a “covering” of (P,X )by these (P1,Xi). In this sence every mixed Shimura data is covered by others, for whichone can impose additional conditions.

In particular call mixed Shimura data (P,X ) irreducible if there does exist a propernormal subgroup of P , defined over Q, through which the homomorphism hx factors for any(⇔ for all ) x ∈ X . Any mixed Shimura data possesses a canonical covering by irreducibleones.

2.14. Proposition: Let (P,X ) be irreducible mixed Shimura data.

(a) P operates on LieU through a scalar character P → Gm,Q.

(b) There exists a character P → Gm,Q, a nondegenerate P -equivariant pairing Ψ : (LieV )⊗(LieV ) → Q, where P operates on Q through this character, and for every x ∈ X ahomomorphism λx ∈ Hom(Q,Q(1)), such that for every x ∈ X λx Ψ is a polarization ofthe Hodge structure on LieV defined by hx.

Proof. Without loss of generality we may replace (P,X ) by (P, h(X ))/Z(P ), then theconditons of 1.18 hold, in particular the weight is rational. Let G1 ⊂ G be the smallestnormal subgroup, defined over Q, such that for some (⇔ for all) x ∈ X the homomorphismh|S1 factors through G1. Then by the irreducibility of P (,X ) we have G = G1 · (hx w)(Gm,Q). The assertion (a) follows from the fact that G1 acts trivially on LieU , andthat the weight acts on LieU through a scalar character. For (b) let M := LieV and N ,Ψ : M ⊗M → N as in 1.12. By the remarks in 1.13 we may choose N of dimension 1, andthe rest follows. q.e.d.

2.15 Structure of the unipotent radical: Consider the short exact sequence1→ U →W → V → 1. Let [,] be the Lie bracket in LieP . Since LieW is mixed of weight≤ −1, the subalgebra [LieW,LieW ] ⊂ LieW is mixed of weight ≤ −2, hence contained inLieU . Furthermore [LieW,LieU ] is mixed of weight ≤ −3, hence 0, since in LieW only theweight −1 and −2 may occur. Thus W is a central extension of the two abelian unipotentgroups V and U . The commutator on W induces an alternating form Ψ : V × V → U ,which determines the extension up to isomorphism. In fact, given U , V and Ψ, one canreconstruct W for instance as follows. Let W := U × V as a variety over Q, and define(u, v) · (u′, v′) := (u+ u′ + 1

2 ·Ψ(v, v′), v + v′).

The operation of P by conjugation on U and on V factors through G, and the pairingin G-equivariant. Interpreting U and V as representations of G we have thus associated toP the data (G,V, U,Ψ). Up to isomorphism P is uniquely determined by this data. In factobserve that the extension 1→W → P → G→ 1 splits in any case, so only the operationof G on W is in question. If W is identified with U×V as above, then this operation may bedefined by g((u, v)) = (g(u), g(v)). Since G is reductive and no irreducible representationof G occurs in both U and V , this is the unique possible action.

Next we describe how to go back and forth between mixed Shimura data for some groupP and that for an extension of P by a unipotent group.

2.16. Partial inverse of 2.9: Consider mixed Shimura data (P,X ) and an extension1 → W0 → P1 → P → 1 of P by a unipotent group W0. We want to study all possible

34

mixed Shimura data (P1,X1) with an isomorphism (P1,X1)/W0∼= (P,X ) extending the

isomorphism P1/W0∼= P . The conditions 2.1 (iv) , (v) and (viii) give a necessary con-

dition for the existence of such (P1,X1). Observe that the adjoint action of P1 on everyabelian subquotient of W0 factors through P . Thus the necessary condition says that theLie Algebra of every irreducible subquotient of LieW0 must be type (−1, 0), (0,−1) or(−1,−1) as representation of G, and the center of G acts on it through a torus that isan almost direct product of a Q-split torus with a torus of compact type defined over Q.

2.17. Proposition: Let (P,X ), W0 and P1 be as in 2.16, and assume that LieW0

satisfies the condition given in 2.16. Then:

(a) There exists mixed Shimura data (P1,X1) and a morphism (P1,X1) → (P,X ) thatextends the given homomorphism P1 → P , such that (P1,X1)/W0

∼= (P,X ). They areuniquely determined up to isomorphism.

(b) For every morphism (P ′,X ′) → (P,X ) and every factorization P ′ → P1 → P thereexists exactly one extension (P ′,X ′)→ (P1,X1)→ (P,X ):

P ′ −→ P1

↓P

=⇒(P ′,X ′) → (P1,X1)

↓(P,X )

Definition: We call (P1,X1) a unipotent extension of (P,X ). Let U0 := W0 ∩ U1 andV0 := W0/U0, then we have a short exact sequence 1→ U0 →W0 → V0 → 1, which we callthe weight decomposition of W0.

Proof. Fix a Levi decomposition P1 = W1 o G, this induces a Levi decompositionP = W o G. Let x0 ∈ X such that hx0 factors through G ⊂ P . By assumption hx0

induces a rational Hodge structure of type (−1, 0), (0,−1), (−1,−1) on every irreducibleG-subspace of LieW1. Define U1 ⊂W1 by LieU1 = W−2(LieW1), this is a subspace definedover Q. As in 2.15 it follows that U1 is contained in the center of W1, so it is a normalsubgroup of P1. Now by definition the conditions 2.1 (iii)—(viii) hold for P1 and hx0 .Denote the given homomorphism P1 → P by φ. If desired (P1,X1) exists, then by 1.17for every x1 ∈ X1 that is mapped to x ∈ X we have StabP1(R)·U1(C)(x1) ∼= StabP (R)·U(C)(x).Thus X1 is isomorphic to

X1 := (x, k) ∈ X ×Hom(SC, P1,C such that hx = φ k, and

π′ k : SC → (P1/U1)C is already defined over R.

Taking this as definition of X1, together with the obvious map h : X1 → Hom(SC, P1,C,(x, k) 7→ k, all the conditions of 2.1 are satisfied, so we get mixed Shimura data (P1,X1) anda morphism (P1,X1)→ (P,X ), unique up to the isomorphism. Comparing this constructionwith that in 2.9 we find (P1,X1)/W0

∼= (P,X ). The universal property of (P1,X1) is obviousfrom its construction. q.e.d.

2.18. The structure of the fibres: Let (P1X1)→ (P,X ) be a unipotent extension.Then the map Ψ : X1 → X is a W0(R) · (W0 ∩ U1)(C)-torsor, in particular its fibres are

35

connected. Let x1 ∈ X1 and x := Ψ(x1), then x1 defines a Hodge structure on LieW0, andas in the proof of 1.16 it follows that the map

W0(R) · (W0 ∩ U1)(C) w0 7→w0·x1−−−−−−−−→ Ψ−1(x)o ↑ exp ↑

(LieW0)R + LieW0 ∩ U1)C ∼= (LieW0)C/F0(LieW0)C

is an holomorphic isomorphism.

Consider the constant vector bundle W0 on X1 with fibre (LieW0)C. The operation ofP1(R)·U1(C) on (LieW0)C makes it into an equivariant vector bundle. There exists a smoothsubbundle F0W0, whose fibre in x1 ∈ X1 equals F 0(LieW0)C for the Hodge structure definedby x1. This subbundle is invariant under the operation of W0(R) · (W0 ∩U1)(C) if and onlyif W0 is abelian. Then and only then F0W0 ⊂ W0 are pullbacks of vector bundless on X .

Suppose that the short exact sequence 1→W0 → P1 → P → 1 splits. By 2.17 (b) anysplitting yields a section ε : (P,X ) → (P1,X1). By pullback we obtain vector bundles onX , and the above isomorphism between a fibre Ψ−1(x) and LieW0)C/F

0(LieW0)C yieldsan identification of X1 → X with the holomorphic complex vector bundle ε∗(W0/F0W0).

Let us apply this to arbitrary mixed Shimura data (P1,X1) with W0 = W1. Then thesequence splits, and with 2.12 we have proved:

2.19. Proposition: Let (P,X ) be mixed Shimura data. Then every connectedcomponent of X is a holomorphic complex vector bundle on a hermitian symmetric domain.

2.20. Fibre product: Let (P1,X1)→ (P,X ) be a unipotent extension and (P2,X2)→(P,X ) an arbitrary morphism. Let P12 := P1 ×P P2 and X12 := X1 ×X X2 be the fibrepoducts. Since P1 → P is an extension by a unipotent group, P12 is a connected subgroupof P1 × P2 (this would not be true for an arbitrary morphism (P1,X1) → (P,X )). Defineh12 by the commutative diagram

X12h12−−−→ Hom(SC, P12,C)

∩↓

∩↓

X1 ×X2h1×h2−−−−−→ Hom(SC, P1,C × P2,C).

Since P12 → P2 is a unipotent extension, it follows at once (P12,X12) is mixed Shimuradata. We have a commutative diagram

(P12,X12) −→ (P1,X1)↓ ↓

(P2,X2) −→ (P,X )

Furthermore (P12,X12) possesses the universal property of the fibre product in the categoryof mixed Shimura data; this is also a trivial consequence of the construction. We write(P1,X1)×(P,X ) (P2,X2) for (P12,X12).

2.21. Operations on the fibre for split unipotent extensions: Consider mixedShimura data (P,X ) and a representation W0 of P that satisfies the condition in 2.16. Weconsider W0 as an (abelian) unipotent group over Q and define P1 := W0 o P with the

36

multiplication (w, p) · (w′, p′) = (w · p(w′), p · p′). By 2.17 we get a canonical morphism(P1,X1)→ (P,X ) together with a section (P,X )→ (P1,X1). For two such representationsW0 and W ′0 and a P -equivariant homomorphism f : W0 → W ′0 we define a homomorphism

P1 = W0oPf×id−−−−→W ′0oP = P ′1. By 2.17 we get a canonical morphism (P1,X1)→ (P ′1,X ′1)

of the corresponding mixed Shimura data. Thus for fixed (P,X ) we have a functor fromthe category of all dimensional representations of P over Q that satisfy 2.16 to the categoryof all mixed Shimura dater “over” (P,X ) together with a section. It is easily seen that thisfunctor maps direct sums to fibre products (2.20).

For instance the addition W0 ⊗ W0 → W0 yields a morphism m : (P1,X1) ×(P,X )

(P1,X1) → (P1,X1), which on the fibre of X1 ×X X1 → X corresponds to the additonon the corresponding vector bundle of 2.18. Thus (P1,X1) becomes a group object inthe category of all Shimura data over (P,X ). The endomorphism algeba EndP (W0) alsooperates an (P1,X1) over (P,X ). Both these operations are compatible with the “zero”section ε : (P,X ) → (P1,X1). There is also the following operation: For any w0 ∈ W0(Q)the conjugation by w0 induces an isomorphism

int(w0) : (P1,X1)(int(w0),(w0·))−−−−−−−−−−→ (P1,X1)

(P,X )

that transforms the zero section ε into int(w0) ε.

2.22. Operations on the fibre in an abelian unipotent extension: Let φ :(P1,X1)→ (P,X ) be a unipotent extention with W0 abelian. Here we do not assume thatthe extension splits. Still the operation of P1 on W0 factors through P , and the vectorbundles considered in 2.18 come from canonical vector bundles on X . So the operation ofW0(R) · (W0∩U1)(C) on X1 makes the bundle X1 → X into a holomorphic torsor under theadditive group of this vector bundle W0/F0W0 on X .

We can express this situation functorially as follows. For more generality let (P,X )→(P∗,X∗) be a morphism such that the operation of P on W0 factors through P∗. Let(P ′,X ′)→ (P∗,X∗) be the (splitting) unipotent extension with P ′ = W0 o P∗. Let P ′1,X ′1)be the fibre product (P ′,X ′)×(P∗,X∗) (P1,X1), then P ′1 = P ′ ×P∗ P1 = (W0 o P∗)×P∗ P1

∼=W0 o P1.

The mapP ′1∼= W0 o P1 → P1, (w, p1) 7→ w · p1

is a homomorphism, so by 2.17 (b) it extends canonically to a morphism

µ : (P ′,X ′)×(P∗,X∗) (P1,X1) = (P ′1,X ′1)→ (P1,X1).

2.23. Proposition: This morphism defines an operation of the group object (P ′,X ′)→(P∗,X∗) on (P1,X1) in the category of all mixed Shimura data over (P∗,X∗). Through this(P1,X1) becomes a (P ′,X ′)→ (P∗,X∗))-torsor over (P,X ).

Proof. The assertion is equivalent to the conjunction of the following two:

37

(a) The diagram

(P ′,X ′)×(P∗,X∗) (P ′,X ′)×(P∗,X∗) (P1,X1)id×µ−−−−→ (P ′,X ′)×(P∗,X∗) (P1,X1)

m×id ↓ ↓ µ(P ′,X ′)×(P∗,X∗) (P1,X1)

µ−−→ (P1,X1)

is commutative.

(b) The morphism

(P ′,X ′)×(P∗,X∗) (P1,X1)(pr2,µ)−−−−−−→ (P1,X1)×(P, CX) (P1,X1)

is well-defined and an isomorphism.

Since we are only dealing with unipotent extensions of (P,X ), it suffices to provethese assertions for the corresponding homomorphisms P ′ ×P∗ P1 → P1, etc. Consider thediagram in (a). Like for P ′ ×P∗ P1

∼= W0 o P1 there is a similar canonical isomorphismP ′ ×P∗ P ′ ×P∗ P1

∼= W0 o W0 o P1. One finds that an element (w,w′, p1) is mapped tow · w′ · p1 ∈ P1 through both corners of the diagram, whence (a). (b) is equivalent to theassertion that the diagram

(P ′1,X ′1)µ−−→ (P1,X1)

pr2 ↓ ↓ φ(P1,X1)

φ−−→ (P,X )

is cartesian. This is implied by the fact that the homomorphism W0 o P1 → P1 ×PP1, (w, p1) 7→ (w · p1, p1) is an isomorphism. q.e.d.

2.24. Example: Let (G0,H0) be as in 2.8. let U0 = Ga = Q with the standardoperation of G0 = Gm,Q. Clearly U0 is pure of type (−1,−1) as representation of G0, andG0 is itself Q-split , so the construction 2.17 yields a unipotent extension (P0,X0) of (G0,H0)with P0 = U0 oG0. We can also describe it as follows. Identify P0 with the subgroup of allmatices of the form

(∗ ∗0 1

)in GL2,Q, h(X0) with the set of all homomorphisms SC → P0,C of

the form S(C) ⊃ S(R) = C× 3 z 7→ ( zz0∗1), and X0 with h(X0)×H0.

Every connected component of X0 is isomorphic to C, and the group structure definedby 2.21 corresponds to that of the additive group C. As a variant we can also considerthe Shimura data P0, h(X0)), here h(X0) is connected, and (P0, h(X0)) is a group objectover (Gm,Q, k) = (G0, h(H0)) (see 2.8). Note that for every ((P0,X0) → (Gm,Q,H0))-torsor (P ′,X ′) → (P,X ) we have a canonical isomorphism between U0 and the kernel ofthe homomorphism π0 : P ′ → P .

2.25. Example: Let (CSP2g,Q,H2g) be the Shimura data defined in 2.7, and V2g

the standard representation of CSP2g,Q. The construction 2.17 yields mixed Shimura data(V2g o CSP2g,Q,H′2g). Let U2g := Ga = Q with the operation of CSP2g,Q by the multiplier.The given alternating form Ψ : V2g × V2g → Q = U2g is CSP2g,Q-equivariant. Let 1 →U2g → W2g → V2g → 1 be the central extension of unipotent groups defined by Ψ, and letP2g := W2g o CSP2g,Q. By 2.17 we obtain mixed Shimura data (P2g,X2g) and morphisms

(P2g,X2g)→ (V2g o CSP2g,Q,H′2g)→ (CSP2g,Q,H2g).

38

The unipotent extension (V2g oCSP2g,Q,H′2g)→ (CSP2g,Q,H2g) is an example for a groupobject. Let (CSP2g,Q,H2g)→ (Gm,Q,H0) be the morphism defined in 2.8. Since the opera-tion of CSP2g,Q on U2g factors through Gm,Q, every isomorphism U2g

∼= U0 gives a structureon (P2g,X2g) as a ((P0,X0)→ (Gm,Q,H0))-torsor over (V2g o CSP2g,Q,H′2g).

2.26. Reduction lemma: Let (P,X ) be irreducible mixed Shimura data. Then thereexists pure Shimura data (T,Y) and (G,H), where T is a torus and h : H → Hom(S, GC)injective, and a number n ≥ 0 such that

(a) If V = 1, then there exists an embedding

(P,X ) → (T,Y)× (G,H)×n∏i=1

(P0, h(X0)),

where (P0, h(X0)) is the mixed Shimura data defined in 2.24.

(b) If 2g = dim(V ) > 0, then there exists a morphism (P,X )→ (Gm,Q,H0), a ((P0,X0)→Gm,Q,H0))-torsor (P ′,X ′)→ (P,X ) and an embedding

(P ′,X ′) → (T,Y)× (G,H)×n∏i=1

(P2g,X2g),

where (Gm,Q,H0), (P0,X0), and (P2g,X2g) are mixed Shimura data defined in 2.8, 2.24 and2.25 respectively.

Proof. (a) By 2.14 (a) P operates through scalars on U . Let P1 ⊂ P be the kernel ofthis operation. Let λ1, . . . , λn be a basis for Hom(U,Q). For every i the quotient P/P1 ·Ker(λi) is isomorphic to the group P0 defined in 2.24. One sees easily that any suchisomorphism extends uniquely to a morphism (P,X ) → (P0, h(X0)). By construction thedirect product of these morphisms

(P,X )→n∏i=1

(P0, h(X0))

is injective on U . Thus the corresponding morphism

(P,X )→ (P,X )/U ×n∏i=1

(P0, h(X0))

is an embedding. Together with the embedding (P,X )/U → (T,Y) × (G,H) := (T,Y) ×(P, h(X ))/U from 2.11 we get the desired result.

(b) Let Ψ : V ×V → U be pairing induced from the commutator. Note that V is pureof weight −1, so according to 2.14 (b) we can fix a P -equivariant pairing Φ : V × V →U0 := Ga,Q = Q with a suitable operation of P on Q, such that for all x ∈ X there existsλ ∈ Hom(Q,Q(1)), such that λ Φ is a polarization of the Hodge structure on V defined byx. Define U ′ := U⊕Q and Ψ′ := (Ψ,Φ) : V ×V → U ′. By 2.15 the triple (U ′, V,Ψ′) definesa central extension 1 → U ′ → W ′ → V → 1, and W ′/U0 is isomorphic to W . Fixing suchan isomorphism we let P ′ := W ′ o G, this is then a unipotent extension of P by U0 = Q,

39

so we obtain a unipotent extension (P ′,X ′) → (P,X ) Let φ : P → Gm,Q be the characterthrough which P operates on U0. For every x ∈ X let λx ∈ H0 be the unique element, suchthat λx Φ is a polarization of the Hodge structure on V defined by x. Then (φ,Ψ) is amorphism (P,X ) → (Gm,Q,H0), and (P ′,X ′) → (P,X ) is a ((P0,X0) → (Gm,QH0))-torsorin a canonical way.

Since the irreducibility of (P,X ) implies that of (P ′,X ′), P ′ acts through scalars onU ′. Consider the set of all µ ∈ Hom(U ′,Q) such that for all x ∈ X ′ there exists λ ∈Hom(Q,Q(1)), so that the pairing λ µ Ψ′ is a polarization of V . This set is open in theusual topology, and nonempty since it contains pr2 : U ′ → U0 = Q. Therefore it containsa basis µ1 . . . µn of Hom(U ′,Q). Let 2g = dim(V ), and let (P2g,X2g), W2g, and V2g be asin 2.25. For every i = 1, . . . , n there exists an isomorphism V ∼= V2g such that the pairingµi Ψ′ corresponds to the pairing on V2g fixed in 2.7. Thus there is a homomorphismW ′ → W2g that induces an isomorphism V ∼= V2g and whose kernel on U ′ is equal to thekernel on µj . By the definition of (P2g,X2g) this gives a morphism (P ′,X ′) → (P2g,X2g).Taking together these n morphisms we obtain a morphism of Shimura data

(P ′,X ′)→n∏i=1

(P2g,X2g),

which by construction is injective on the unipotent radical W ′. Hence the correspondingmorphism

(P ′,X ′)→ (P ′,X ′)/W ′ ×n∏i=1

(P2g,X2g)

is an embedding. Together with the embedding (P ′,X ′)/W ′ → (T,Y)× (G,H) := (T,Y)×(P ′, h(X ′))/W ′ from 2.11 we obtain the desired result. q.e.d.

40

Chapter 3

Mixed Shimura varieties

In this chapter we begin with the definition of mixed Shimura varieties, and show (in 3.3)that these objects are nice complex spaces. Then we go on to study the morphisms andconstructions induced by those considered in chapter 2. Again most of the chapter (startingfrom 3.10) is devoted to the study of the unipotent fibres. It is remarkable that, under verymild conditions, the group objects and actions of 2.21—23 yield the same structures forthe associated mixed Shimura varieties (see 3.10—12). We prove that the unipotent fibreare torus-torsors over abelian varieties (3.14 and 3.22 (a)). In an important special casethis torus is canonically isomorphic to C× (3.16), in which case we are interested in theassociated line bundle (3.17). In 3.21 we give an ampleness criterion for this line bundle.

3.1. Definition: Let (P,X ) be mixed Shimura data, and Kf an open compactsubgroup of P (Af ). We define the corresponding mixed Shimura variety, as

MKf (P,X )(C) := P (Q)\X × (P (Af )/Kf ),

where P (Q) acts on both factors from the left hand side. If P is reductive, we also speakof pure Shimura variety, or just Shimura variety. The notation is chosen to the compatiblewith later chapters.

3.2. Elementary properties: Since Kf is open in P (Af ), the set P (Af )/Kf isa discrete topological space. We endow MKf (P,X )(C) with the quotient topology. Theset of connected components π0(MKf (P,X )(C)) is a quotient of π0(P (Q)\P (R) · U(C) ×P (Af )/Kf ) = π0(P (Q)\P (A)/Kf ), hence a finite discrete set (see [G] thm. 5). For everysystem of representatives (X 0, pf ), where X 0 is a connected component of X and pf ∈P (Af ), we have

MKf (P,X )(C) ∼= q Γ(pf )\X 0,

where Γ(pf ) = P (Q)∩ pf ·Kf · p−1f is a congruence subgroup of P (Q) depending on pf and

Kf . If P is a torus, then MKf (P,X )(C) is a finite set.

3.3. Proposition: (a) For any pair (X 0, pf ) the group Γ(pf ) operates properlydiscontinuously on X 0. There is a canonical structure of a normal complex space onMKf (P,X )(C), whose singularities are at most quotient singularities by finite groups.

41

(b) If Kf is neat, then MKf (P,X )(C) is a complex manifold.

Proof. (a) By 2.1 (viii) P/P der ·Z(P ) is an almost direct product of a Q-split torus witha torus of compact type defined over Q. Thus the image of Γ(pf ) in (P/P der · Z(P )(Q) isfinite, so it suffices to prove the assertion for Γ := Γ(pf )∩ (P der ·Z(P ))(Q). Now Z(P )(R)0

acts trivially on X 0, so in view of the convention 0.4 we may replace (P,X ) by (P,X )/Z(P ).Then Γ is an arithmetic subgroup of P der(Q). In particular it is a discrete subgroup ofP der(R), so it operates properly discontinuously on P∞ := P der(R)0 ·U(C). For any x ∈ X 0

we have a homomorphism X 0 ∼= P∞/K∞ with K∞ := StabP∞(x). But by 1.17 (a) and 2.1(vi) K∞ is compact, so Γ also operates properly discontinuously on X 0, as desired. Thecomplex structure on Γ(pf )\X 0 comes from the canonical P (R) · U(C)-invariant complexstructure on X .

(b) If Kf is neat, then so is Γ(pf ), hence it is already contained in (P der · Z(P ))(Q).Moreover its image in (P/Z(P ))(Q) is again neat, so in particular torsion free. By theargument in (a) the stabilizer of any point x ∈ X 0 in this image is finite, so it must betrivial. Thus Γ(pf )\X 0 is a complex manifold, as desired. q.e.d.

3.4. Morphisms on MKf (P,X )(C): (a) Let (P,X ) be mixed Shimura data pf ∈P (Af ), and Kf , K ′f ⊂ P (Af ) open compact subgroups, such that K ′f ⊂ pf ·Kf · p−1

f . Thenthe map

[ ·pf ] = [ ·pf ]K′f ,Kf : MKf (P,X )(C)→ (MKf (P,X )(C),

[(x, p′f )] 7→ [(x, p′f · pf )]

is well-defined. It is holomorphic, closed, finite, and surjective. For K ′f = pf ·Kf · p−1f it is

an isomorphism.

(b) Let φ : (P1,X1)→ (P2,X2) be a morphism of mixed Shimura data, andK1f ⊂ P1(Af )

and K2f ⊂ P2(Af ) open compact subgroups. If φ(K1

f ) ⊂ K2f , then the canonical map

[φ] = [φ]K1f ,K

2f

: MK1f (P1,X1)(C)→MK2

f (P2,X2)(C),

[(x1, p1f )] 7→ [(φ(x1), φ(p1

f ))]

is well-defined. By 2.4 and 3.3 it is holomorphic. If φ is an isomorphism and φ(K1f ) = K2

f ,then [φ] is an isomorphism.

3.5. Functoriality: The morphisms in 3.4 have the following functorial properties:

(a) Let (P,X ) be mixed Shimura data, pf , p′f ∈ P (Af ), and Kf , K ′f , K ′′f ⊂ P (Af ) open

compact subgroups, such that K ′f ⊂ pf · Kf · p−1f and K ′′f ⊂ p′f · K ′f · (p′f )−1. Then

K ′′f ⊂ (p′f · pf ) ·Kf · (p′f · pf )−1 and

[ ·(p′f · pf )]K′′f .Kf = [ ·pf ]K′f ,Kf [ ·p′f ]K′′f ,K′f.

(b) Let φ1 : (P1,X1) → (P2,X2) and φ2 : (P2,X2) → (P3,X3) be morphisms and K1f ⊂

P1(Af ), K2f ⊂ P2(Af ), and K3

f ⊂ P3(Af ), open compact subgroups, such that φ1(K1f ) ⊂ K2

f

and φ2(K2f ) ⊂ K3

f . Then (φ2 φ1)(K1f ) ⊂ K3

f and

[φ2 φ1]K1f ,K

3f

= [φ2]K2f ,K

3f [φ1]K1

f ,K2f.

42

(c) Let φ : (P1,X1) → (P2,X2) be a morphism, pf ∈ P1(Af ), and K1f , K1′

f ⊂ P1(Af )

and K2f , K2′

f ⊂ P2(Af ) open compact subgroups, such that φ(K1f ) ⊂ K2

f , φ(K1′f ) ⊂ K2′

f ,

K1,f ⊂ pf ·K

1f · p

−1f and K2′

f ⊂ φ(pf ) ·K2f · φ(pf )−1. Then

[φ]K1f ,K

2f [ ·pf ]K1′

f ,K1f

= [ ·φ(pf )]K2′f ,K

2f [φ]K1′

f ,K2′f.

(d) Let (P,X ) be mixed Shimura data, Kf an open compact subgroup of P (Af ), andkf ∈ Kf . Then [ ·kf ]Kf ,Kf is the identity on MKf (P,X )(C).

(e) Let (P,X ) be mixed Shimura data, p ∈ P (Q) and Kf , K ′f ⊂ P (Af ) open compact

subgroups, such that K ′f ⊂ p · Kf · p−1. Denote by int(p−1) : (P,X ) → (P,X ) the

morphism defined by P → P , p′ 7→ p−1 · p′ · p and X → X , x 7→ p−1 · x. Then the map[int(p−1)]K′f ,Kf is defined and equal to [ ·p]K′f ,Kf .

3.6. The projective system: For pf = 1 and K ′f ⊂ Kf the maps of 3.4 give thecanonical maps

[id] = [id]K′f ,Kf : MKf (P,X )(C)→MKf (P,X )(C),

[(x, p′f )] 7→ [(x, p′f )].

With these maps the MKf (P,X )(C) form a projective system of complex spaces, indexedby the system of all open compact subgroups of P (Af ). The morphisms 3.4 (a) definean operation of P (Af ) from the right hand side on this projective system. A morphismof mixed Shimura data induces an equivariant morphism of projective systems with thisright-P (Af )-operation.

3.7. Lemma: Let (P,X ) be mixed Shimura data, let ΓZ be an arithmetic subgroupof z ∈ Z(P )(Q) | z|X = id, and ΓZ be its closure in P (Af ).

(a) For any open compact subgroupKf ⊂ P (Af ) the projectionMKf (P,X )(C)→MΓZ ·Kf (P,X )(C)is an isomorphism.

(b) Let A be a compact subgroup of P (Af ). Then the obvious map

P (Q)\X × (P (Af )/ΓZ ·A) −→ lim←MKf (P,X )(C)

is bijective, where the limit is taken over all open compact subgroups Kf ⊂ P (Af ) thatcontain A.

Proof. (Compare [D1] 1.15.2)(a) follows from the isomorphisms

P (Q)\X × (P (Af )/Kf ) ∼= P (Q)\X × (ΓZ\P (Af )/Kf )

andΓZ\P (Af )/Kf

∼= P (Af )/ΓZ ·Kf∼= P (Af )/Γz ·Kf .

For (b) let Kαf be the system of all open subgroups that contain A. An element of the

inverse limit is a system of elements [(xα, pαf )] ∈MKα

f (P,X )(C), such that for all Kβf ⊂ K

αf

43

the image of [(xβ, pβf )] in MKα

f (P,X )(C) is equal to [(xα, pαf )]. Thus in particular the coset

P (Q) ·xα does not depend on α, so we may replace the representatives by others, for whichall xα are equal to a fixed x ∈ X . Having done this, let B := StabP (Q)(x), then we can stillreplace every pαf by an arbitrary element of the double coset B · pαf ·Kα

f . Since b · pαf ·Kαf is

open and compact for every b ∈ B, this double coset is closed. For all Kβf ⊂ Kα

f we have

B · pβf ·Kβf ⊂ B · pαf ·Kα

f , so (B · pβf ·Kβf ) ∩ pαf ·Kα

f , is nonempty and compact. Keepimgα fixed, these sets form a decreasing system of nonempty compact subsets of P (Af ). Thustheir intersection must be nonempty. This shows that there exists a common representativefor all [(xα, p

αf )], whence the desired surjective.

For the injectivity we may assume that A contains Γz. Let x and B be as before, andfix pf ∈ P (Af ), then we have to prove that

⋂αB · pf · Kα

f = B · pf · A. By the proof of

3.3 (a) ΓZ is of finite index in B ∩ pf · Kαf · p

−1f , so for all sufficently small Kα

f we have

B ∩ pf ·Kαf · p

−1f = B ∩ pf ·A · p−1

f . Fix such an α, then it follows:⋂β

B · pf ·Kβf =

⋃b∈B

⋃β

(B · pf ·Kβf ) ∩ (b · pf ·Kα

f )

=⋃b∈B

⋂β

b · (B ∩ pf ·Kαf · p−1

f ) · pf ·Kβf

= B ·⋂β

(B ∩ pf ·A · p−1f ) · pf ·Kβ

f

= B ·⋂β

(B · pf ·Kβf ) ∩ (pf ·Kβ

f )

= B ·⋂β

pf ·Kβf

= B · pf ·A,

as desired. q.e.d.

3.8. Proposition (compare [D1] 1.15): Let (P1,X1) → (P2,X2) be an embedding,and K1

f ⊂ P1(Af ), K2f ⊂ P2(Af ) open compact subgroups such that K1

f = φ−1(K2f ).

(a) The map [φ] : MK1f (P1,X1)(C)→MK2

f (P2,X2)(C) is finite and closed.

(b) For every K1f there exists K2

f such that this map is a closed embedding.

Proof. (a) Since Z(P1) is mapped to Z(P2), we may as in the proof of 3.3 (a) reduce

to the case where these subgroups are trivial. Since MK1f (P1,X1)(C) has only finitely many

connected components, it is enough to consider one such component. On it the map isgiven as Γ1\X 0

1 → Γ2\X 02 , where X 0

i are connected components of Xi and Γi arithmeticsubgroups of Pi(Q) with Γ1 = P1(Q) ∩ Γ2. Let x ∈ X 0

1 and write X 01∼= P i∞/K

i∞ as in the

proof of 3.3 (a), for i = 1, 2.

As in [Rag] 10.15 let ρ : P der2 → GLn,Q be a representation and v ∈ Qn, such that P der1

is the stabilizer of v in P der2 . Consider the continuous left-P 2∞-invariant map

Φ; P 2∞ → Cn, p 7→ ρ(p)v.

44

By construction Φ−1(m) = P 1∞. Since ρ(Γ2) stabilizes some Z-lattice in Qn, the set Φ(Γ2)

is a discrete subset of Cn. Like in [loc. cit.] it follows that Γ2 · P 1∞ is closed in P 2

∞and contains P 1

∞ as an open and closed subset. Consequently we have a closed embeddingΓ1/P

1∞ → Γ2\P 2

∞. SinceK2∞ is compact, the composition Γ1/P

1∞ → Γ2\P 2

∞ → Γ2\P 2∞/K

2∞

is also closed and thus also the map [φ].

Let p, p′ ∈ P 1∞, γ ∈ Γ2 and k ∈ K2

∞, such that p′ = γ · p · k. Then γ−1 · p′ = p · k,whence Φ(γ−1) = Φ(γ−1 · p′) ∈ Φ(p ·K1

∞) = ρ(p) · Φ(K1∞). The image of p′ in Γ1\P 1

∞/K1∞

depends only on Φ(γ−1) and Φ(k). But as Φ(K1∞) is compact, the set Φ(Γ2)∩ρ(p) ·Φ(K1

∞)is finite. Hence the fibre over [p] ∈ Γ2\P 2

∞/K2∞ is finite.

(b) (Compare [D1] 1.15) Without loss of generality we may assume K1f and K2

f neat.

By (a) it suffices to prove the injectivity. Fix K1f and let X(K2

f ) be the difference kernel ofthe two maps

[φ] pri : (MK1f (P1,X1)(C)2 →MK2

f (P2,X2)(C).

This is a closed analytic subvariety of MK1f (P1,X1)(C)×MK1

f (P1,X1)(C), finite and etaleover both factors, which decreases with K2

f . Thus X(K2f ) becomes constant for K2

f ⊃ K1f

sufficiently small. Hence it suffices to show the injectivity in the projective limit. By 3.7we have to show that the map

P1(Q)\X1 × (P1(Af )/K1f )→ P2(Q)\X2 × (P2(Af )/ΓZ2 ·K1

f )

is injective, where ΓZ2 is an arbitrary arithmetic subgroup of z2 ∈ Z(P2)(Q) | z2|X2= id.

Write Z(P2)0 is an almost direct product (Z(P2) ∩ P1)0 · T for some torus T , and chooseΓZ2 = ΓZ1 ·ΓT with ΓZ1 ⊂ P1(Q)∩K1

f and ΓT ⊂ T (Q)∩T (R)0 such that ΓT ∩P1(Q) = 1.Then we have ΓZ2 · P1(Af ) ∼= ΓT × P1(Af ). Using this relation it is straightforward toexplicitly verify the injectivity. q.e.d.

3.9. Proposition (compare [D1] 2.5 ): Let (P,X ) be mixed Shimura data withP der/W simply connected. Let (T,Y) = (P,X )/P der, and φ : P T the canonicalprojection. Then for every open compact subgroup Kf ⊂ P (Af ) the image φ(Kf ) ⊂ T (Af )is again open and compact, and the map [φ] : MKf (P,X )(C)→Mφ(Kf )(T,Y)(C) inducesa bijection of the connected components

π0(MKf (P,X )(C) ∼−−→Mφ(Kf )(T,Y)(C).

Proof. Since Kf is compact, its image φ(Kf ) is also compact. Its openness followsas in the proof of [D1] 2.4. Fix x ∈ X and let A be its stabilizer in P (R) · U(C), thenX ∼= P (R) · U(C)/A. We have

π0(MKf (P,X )(C) = π0(P (Q)\X × (P (Af )/Kf ))

= π0(P (Q)\(P (R) · U(C)/A)× (P (Af )/Kf ))

= π0(P (Q)\P (A)/(A×Kf ))

= π0(P (Q)\P (A)/Kf )/π0(A).

Since π0(P (Q)\P (A)/Kf ) = π0(P (Q)\P (A)/(P (R)0 × Kf )), and P (R)0 × Kf is open in

P (A), we have π0(P (Q)\P (A)/Kf ) = π0((P (Q) · P (R)0)\P (A)/Kf ). Since W (Q) ·W (R)

45

is dense in W (A), this set is equal to π0(G(Q)\G(A)/π(Kf )). Thus we may assume with-out loss of generality that P = G. In this case [D1] 2.5 asserts that the canonical mapπ0(G(Q)\G(A)/π(Kf ))→ π0(T (Q)\T (A/φ(Kf )) is bijective. Hence

π0(MKf (P,X )(C) ∼= π0(P (Q)\P (A)/Kf )/π0(A)∼−−→ π0(T (Q)\T (A)/φ(Kf ))/π0(A)

= π0(T (Q)\T (A)/φ(Kf ))/π0(φ(A)),

and the latter is equal to Kφ(Kf )(T,Y)(C), since by the construction of (P,X )/P der (see2.9) Y ∼= π0(T (R)/φ(A)). q.e.d.

3.10. Fibre product: Let (φ1,Ψ1) : (P1,X1) → (P,X ) be a unipotent extensionby W0, (φ2,Ψ2) : (P2,X2)→ (P,X ) an arbitrary morphism, and let (P12,X12) be the fibreproduct defined in 2.20. Let K1

f ⊂ P1(Af ) and K2f ⊂ P2(Af ) be open compact subgroups,

such that φ2(K2f ) ⊂ Kf := φ1(K1

f ) ⊂ P (Af ). Since P1 is isomorphic to W0 × P as an

algebraic variety, Kf is also open and compact. Let K12f := K1

f ×Kf K2f be the fibre

product, this is an open compact subgroup of P12(Af ). By 2.20 and 3.5 we now have acommutative diagram

MK12f (P12,X12)(C) −→ MK1

f (P1,X1)(C)↓ ↓

MK2f (P2,X2)(C) −→ MKf (P,X )(C)

This diagram induces a holomorphic map to the fibre product:

MK12f (P12,X12)(C)→MK1

f (P1,X1)(C)×MKf (P,X )(C)

MK2f (P2,X2)(C).

3.11. Lemma: In the situation of 3.10, asssume that Kf is neat, and that thehomomorphism

z2 ∈ Z(P2)(Q) | z2|X2= id ∩K2

f −→ z ∈ Z(P )(Q) | z|X = id ∩Kf

is surjective. Then the map in 3.10 is an isomorphism.

Remark. If Kf is neat, then the group z ∈ Z(P )(Q) | z|X = id ∩Kf is containedin T (Q), where T is the Zariski closure of any sufficiently small arithmetic subgroup ofZ(P )(Q). Thus the second condition follows from the first if Z(P ) is an almost directproduct of a Q-split torus with a torus of compact type defined over Q.

Proof. Since P1 → P is a unipotent extension, the homomorphisms P1(Af )→ P (Af ),P12(Af ) → P2(Af ), P1(Q) → P (Q), and P12(Q) → P2(Q) are surjective, and the homo-morphisms P12(Af ) → P1(Af ) ×P (Af ) P2(Af ) and P12(Q) → P1(Q) ×P (Q) P2(Q) are iso-morphisms. Furthermore the maps X1 → X and X12 → X2 are surjective, and by theconstruction 2.20 of P12,X12) we have X12

∼= X1 ×X X2. First we show the surjectivity.Consider points [(xi, p

if )] ∈ Pi(Q)\Xi × (Pi(Af )/Ki

f )) for i = 1, 2 which map to the samepoint in P (Q)\X × (P (Af )/Kf )). Then there exists p ∈ P (Q) and kf ∈ Kf such that

46

p · Ψ1(x1) = Ψ2(x2) and p · φ1(p1f ) · kf = φ2(p2

f ). Lift p and kf to p1 ∈ P1(Q), resp.

k1f ∈ K1

f , and replace (x1, p1f ) by p1 · (x1, p

1f · k1

f ). Then we have Ψ1(x1) = Ψ2(x2) and

φ1(p1f ) = φ2(p2

f ). Letting x12 := (x1, x2) and p12f := (p1

f , p2f ) we find that [(x12, p

12f )] maps

to ([(x1, p1f )], [(x2, p

2f )]) in MK12

f (P12,X12)(C). Thus the map is surjective.

For the injective consider x12 = (x1, x2), x′12 = (x′1, x′2) ∈ X12 and p12

f = (p1f , p

2f ),

p12′f = p1′

f , p2′f ) ∈ P12(Af ), such that [(x12, p

12f )] and x′12, p

12′f )] have the same image in

Pi(Q)\Xi × (Pi(Af )/Kif )) for both i = 1, 2. Then there exists pi ∈ Pi(Q) and kif ∈ Ki

f

such that pi · xi = x′i and pi · pif · kif = pi′f . Lift p2 and k2f to elements p12 ∈ P12(Q),

resp. k12f ∈ K12

f , and replace (x12, p12f ) by p12 · (x12, p

12f · k12

f ). Then we have x2 = x′2and p2

f = p2′f . This implies Ψ1(x1) = Ψ2(x2) = Ψ2(x′2) = Ψ1(x′1) and likewise φ1(p1

f ) =

φ1(p1′f ). Let x := Ψ1(x1) and pf := φ1(p1

f ), then these equalities show that φ1(p1) lies in

StabP (Q)(x) as well as in pf ·Kf ·p−1f . Since Kf is neat, the intersection of these two groups

is contained in Z(P )(Q). So this element is central and fixes one point of X , hence it actstrivially on the whole of X . Thus we get φ1(p1) ∈ z ∈ Z(P )(Q) | z|X = id ∩ Kf . Byassumption there exists z2 ∈ Z(P2)(Q) ∩ K2

f such that z2|X2= id and φ2(z2) = φ1(p1).

Define p′12 := (p1, z2) ∈ P12(Q) and k12′f := (k1

f , z−12 ), the latter is an element of K12

f since

φ1(k1f ) = φ1(p1

f )−1 · φ1(p1)−1 · φ1(p1′f ) = p−1

f · φ2(z2)−1 · pf = φ2(z−12 ). Then one easily

verifies that p′12 · (x12, p12f · k12′

f ) = (x′12, p12′f ), as desired. q.e.d.

3.12. Corollary: (a) (Compare 2.21) Let (P ′,X ′)→ (P,X ) be a unipotent extensionwith a fixed splitting P ′ ∼= W0 o P . Suppose that W0 is abelian , and let K ′f ⊂ P ′(Af ) be

an open compact subgroup of the form KWf oKf . If Kf is neat, then

MKf (P ′,X ′)(C) −→MKf (P,X )(C)

is a smooth holomorphic family of abelian complex Lie groups.

(b) (Compare 2.22) Let φ : (P1,X1)→ (P,X ) be a unipotent extension by an abeliangroup W0. Let φ∗ : (P,X ) → (P∗,X∗) be a morphism, such that the operation of Pon W0 factors through P∗, and such that Z(P∗) is an almost direct product of a Q-splittorus with a torus of compact type defined over Q (by 2.1. (viii) this is no big restriction).Let (P ′,X ′) → (P∗,X∗) be the splitting unipotent extension with P ′ = W0 o P∗. LetK1f ⊂ P1(Af ) be a neat open compact subgroup, Kf := φ(K1

f ), and KWf the image of

(z1 ∈ Z(P1)(Q) | z1|X1= id oW0(Af )) ∩ K1

f under the projection Z(P1) ×W0 → W0.

Furthermore let K∗f ⊂ P∗(Af ) be a neat open compact subgroup which normalizes KWf ,

such that φ∗(Kf ) ⊂ K∗f . Let K ′f := KWf oK∗f . Then

MK1f (P1,X1)(C) −→MKf (P,X )(C).

is in a canonical way a holomorphic torsor under the family of groups MKf (P ′,X ′)(C) −→MK∗f (P∗,X∗)(C).

Proof. (a) Consider the maps “zero section”, “inverse” and “group operation” on themixed Shimura varieties induced by the corresponding morphisms of 2.21. Of course the de-sired relations between these maps follow directly by functoriality. The only problem is that

47

the group operation induces only a map MKWf oKf (W0 o (P ′,X ′))(C) −→MK1

f (P1,X1)(C).Let T ⊂ Z(P ) be the Zariski closure on any sufficiently small arithmetic subgroup ofZ(P )(Q). Since (P ′,X ′) → (P,X ) is a unipotent extension and Kf is neat, it follows thatthe group z ∈ Z(P )(Q) | z|X = id∩Kf is contained in T (Q). By 2.1 (viii) T operates triv-ially on W0, so under the semidirect product decomposition z ∈ Z(P )(Q) | z|X = id∩Kf

lies in z′ ∈ Z(P ′)(Q) | z′|X ′ = id ∩ K ′f . Thus the conditions of 3.11 are satisfied, so

MKWf oKf (W0 o (P ′,X ′))(C) is isomorphic to the fibre product of MKf (P ′,X ′)(C) with

itself over MKf (P,X )(C). This yields the desired group structure. Since Kf is neat, the

map MK′f (P ′,X ′)(C) → MKf (P,X )(C) is locally isomorphic to the projection X ′ → X ,whence the smoothness.

(b) Let A := (z1 ∈ Z(P1)(Q) | z|X1= id×W0(Af ))∩K1

f , then pr1(A) is an arithmetic

subgroup of z1 ∈ Z(P1)(Q) | z1|X1= id. By 3.7 (a) we may therefore replace K1

f

by pr1(A) · K1f without changing the mixed Shimura variety MK1

f (P1,X1)(C). After this

modification A = (z1 ∈ Z(P1)(Q) | z1|X1= id × W0(Af ) ∩ K1

f ), so we have KWf =

W0(Af ) ∩ K1f . As in (a) the morphism µ : (P ′,X ′) ×(P∗,X∗) (P1,X1) → (P1,X1) of 2.22

induces the desired group action. Note that here the second condition of 3.11 trivially holds.It remains to prove that this defines a torsor. One easily verifies that the isomorphism(pr2, µ) : (P ′,X ′) ×(P∗,X∗) (P1,X1) → (P1,X1) ×(P,X ) (P1,X1) in the proof of 2.23 inducesan isomorphism of the associated mixed Shimura varieties. It thus remains to show thatthe mixed Shimura variety associated to (P1,X1)×(P,X ) (P1,X1) is isomorphic to the fibre

product of MK1f (P1,X1)(C) with itself over MKf (P,X )(C). This follows from 3.11 if we can

show that the homomorphism z1 ∈ Z(P1)(Q) | z1|X1= id ∩K1

f → z ∈ Z(P )(Q) | z|X =id ∩Kf is surjective. But with our hypothesis about A this is proved as in (a). q.e.d.

3.13. The fibre of a unipotent extension: Let (φ,Ψ) : (P1,X1) → (P,X ) be aunipotent extension. Let K1

f ⊂ P1(Af ) be an open compact subgroup and Kf := φ(K1f ) ⊂

P (Af ). Fix x1 ∈ X1, p1,f ∈ P1(Af ), x := Ψ(x1) and pf := φ(p1,f ). We want to describe the

fibre of the map [φ] : MK1f (P1,X1)(C) → MKf (P,X )(C) explicitly. Since W0(Q) is dense

in W0(Af ), the fibre over ([(x, pf )]) is equal to

[φ]−1([(x, pf )]) ∼= P1(Q)\P1(Q) · (Ψ−1(x)× φ−1(pf ) ·K1f/K

1f )

∼= StabP1(Q)(x)\(Ψ−1(x)× φ−1(pf ) ·K1f/K

1f )

∼= (StabP1(Q)(x) ∩ p1,f ·K1f · p−1

1,f )\Ψ−1(x)× p1,f ·K1f/K

1f

∼= (StabP1(Q)(x) ∩ p1,f ·K1f · p−1

1,f )\Ψ−1(x).

Since Ψ−1(x) is an orbit under W0(R) · (W0 ∩U1)(C) and this group is connected, all fibresof [φ] are connected. Now suppose that Kf is neat. Then it follows as in the proof of 3.11:

StabP1(Q)(x) ∩ p1,f ·K1f · p−1

1,f ⊂ z1 ∈ Z(P1)(Q) | z1|X1= id ×W0(Q).

Let Γ be the image of

(z1 ∈ Z(P1)(Q) | z1|X1= id ×W0(Q)) ∩ p1,f ·K1

f · p−11,f

48

under the projection Z(P1) × W0 → W0. For any splitting e : (P,X ) → (P1,X1), themap [w] 7→ [(w · e(x), pf )] is an isomorphism of the fibre over [(x, pf )] with Γ\W0(R) ·(W0 ∩ U1)(C). If there exists no splitting, then there are still such isomorphisms fibrewiseand noncanonically. If Z(P ) is an almost direct product of a Q-split torus with a torus ofcompact type defined over Q, then we already have

Γ ∼= W0(Q) ∩ p1,f ·K1f · p−1

1,f .

3.14. The group structure on the fibre: Consider the situation on 3.12 (a). The

fibre of MK′f (P ′,X ′)(C)→MKf (P,X )(C) is Γ\W0(R) ·W0 ∩ U1)(C), where Γ is the imageof (z′ ∈ Z(P ′)(Q) | z′|X ′ = id×W0(Q))∩pf (KW

f ) in W0(Q). This an arithmetic subgroup

of W0(Q), and W0(R) · (W0 ∩ U1)(C) is an abelian Lie group. The definition of the groupmultiplication 2.21 together with 3.12 (a) implies that the group structure on the fibre isthe natural one on Γ\W0(R) · (W0 ∩ U1)(C). The complex structure can be described bywriting this quotient in the form Γ\W0(C)/ exp(F 0(LieW0)C) following 2.18.

Let 1→ U0 →W0 → V0 → 1 be the weight decomposition of W0, and 1→ ΓU → Γ→ΓV → 1 the corresponding short exact sequence of arithmetic subgroups. Then we have ashort exact sequence

1→ ΓU\U0(C)→ Γ\W0(R) · U0(C)→ ΓV \V0(R)→ 1,

or, to indicate the complex structure:

1→ ΓU\U0(C)→ Γ\W0(C)/ exp(F 0(LieW0)C)→ ΓV \V0(C)/ exp(F 0(LieV0)C)→ 1.

Since ΓU is a lattice in U0(R), we have U0(C) ∼= ΓU ⊗ C, so ΓU\U0(C) ∼= ΓU ⊗ (Z\C) ∼=ΓU ⊗ C×. Thus this is an algebraic torus over C. Since ΓV is a lattice in V0(R), the groupΓV \V0(R) ∼= ΓV ⊗ (Z\R) is a compact complex torus. We shall see in 3.22 that it is anabelian variety.

3.15. Torsion points on the fibre: Consider a unipotent extension (φ,Ψ) :(P ′,X ′) → (P,X ) by W0, and neat open compact subgroups Kf = Ψ(K ′f ). Let KW

f :=

W0(Af ) ∩ K ′f and KW,+f be the normalizer of K ′f in W0(Af ). Since P ′ operates nontriv-

ially on every subquotient of W0, KW,+f is also an open compact subgroup of W0(Af ), so

KW,+f /KW

f is a finite group. For every wf ∈ KW,+f the map [ ·wf ]K′f ,K

′f

is well-defined,

and this defines an operation of KW,+f from the right hand side on MK′f (P ′,X ′)(C) over

MKf (P,X )(C). By 3.5 (d) this operation factors through KW,+f /KW

f . On a fibre Γ\Ψ−1(x)as in 3.13 this operation is given by

[ ·wf ]([x′]) = [w−1 · x′]

for any w ∈ W0(Q) such that p′−1f · w · p′f ≡ wf modKW

f . Clearly this is a free action if

Γ = W0(Q) ∩ p′f · K ′f · p′−1f , for instance if Z(P ) is an almost direct product of a Q-split

torus with a torus of compact type defined over Q.

49

Let us apply this in the situation of 3.12 (a). By the proof of 3.12 (a) the action is free.Using the given splitting e : (P,X ) → (P ′,X ′) we may associate to each [wf ] ∈ KW,+

f /KWf

the section [ ·wf ] [e] : MKf (P,X )(C)→MK′f (P ′,X ′)(C).

This definition induces a holomorphic monomorphism of groups over MKf (P,X )(C)

(KW,+f /KW

f )×MKf (P,X )(C) −→MK′f (P ′,X ′)(C).

On the fibre this monomorphism is given by

KW,+f /KW

f

pf ·∼−−−→ pf (KW,+

f )/pf (KWf )

−[id]∼←−−−−− (W0(Q) ∩ pf (KW,+

f ))/Γ∩↓

Γ\W0(R) · (W0 ∩ U ′)(C).

If for example KW,+f consists of all wf ∈ W0(Af ) such that d · wf ∈ KW

f , then this defines

a isomorphism of KW,+f /KW

f with the group of all d-division points on every fibre.

3.16. Example: Let (P0,X0) → (Gm,Q,H0) be the unipotent extension defined in2.24, and let Kf ⊂ Gm(Af ) and KU

f ⊂ U0(Af ) be open compact subgroups. Since H0

consists of two points, and Kf ⊂ Z×, for every y ∈ H0 the intersection StabQ×(y) ∩Kf istrivial. Thus in this case the group operation 3.12 (a) is defined without restriction on Kf .We want to define a canonical isomorphism

MKUf oKf (P0,X0)(C) −→ C× ×MKf (Gm,Q,H0)(C).

Since Kf ⊂ Z× and Gm(Af ) = Q× · Z×, the embedding H0 × Z× → H0 × Gm(Af ) inducesan isomorphism

±1\H0 × (Z×/Kf ) ∼−−→MKf (Gm,Q,H0)(C).

By definition U0 is Q interpreted as unipotent group. Thus U0(Q) = Q is dense in U0(Af ) =Af , and similary we have P0(Af ) = P0(Q) ·(KU

f oZ×). Let d be the unique positive rational

number such that U0(Q)∩KUf = d ·Z. The embedding X0× Z× → X0×P0(Af ) induces an

isomorphism

((d · Z) o ±1)\X0 × (Z×/Kf ) ∼−−→MKUf oKf (P0,X0)(C).

The projection P0 → Gm,Q and the given section ı : Gm,Q → P0 induce a canonicalisomorphism

X0 = h(X0)×H0∼= U0(C)×H0 = C×H0,

and the operation of P0(R) · U0(C) = C o R× 3 p = (u, x) on C is given by the affineautomorphism αp : z 7→ u + x · z. As ±1 operates nontrivially on C, it is possible todecompose ((d ·Z)o ±1)\C×H0× (Z×/Kf ) canonically into a product. But it turns outthat this is the same problem as the choice of a square root of −1, a choice that is necessaryto apply the exponential function in the situation.

50

Recall the definition H0 := Isom(Z,Z(1)) of 2.8. For every λ ∈ H0 and u ∈ C we get anelement λ(u) ∈ Q(1)⊗ C = C. Let exp : C→ C× be the exponential function, and considerthe map

C×H0 −→ C× ×H0, (z, y) 7→ (exp(1

d· λ(z)), y).

Since ((−1) · λ)((−1) · z) = λ(z), this map induces a holomorphic isomorphism

MKUf oKf (P0,X0)(C) ∼= ((d · Z) o ±1)\C×H0 × (Z×/Kf )

−→ C× × ±1\H0 × (Z×/Kf )

∼= C× ×MKf (Gm,Q,H0)(C).

It is easily verified that this isomorphism is compatible with the group structure.

Remark. We have seen that a canonical isomorphism can be obtained only becauseH0 consists of two points, and because they correspond to the two choises of

√−1. This

was the most important reason for our somewhat strange looking definition of H0 in 2.8.

3.17. C×-Torsors: Consider the situation of 3.12 (b) with (P ′,X ′) → (P∗C∗)taken to be the unipotent extension (P0,X0) → (Gm,Q,H0). Via the isomorphism 3.16

MK1f (P1,X1)(C) becomes a holomorphic C×-torsor over MKf (P,X )(C). Under the bijec-

tive correspondence between holomorphic C×-torsors and line bundles we therefore get acanonical line bundle on MKf (P,X )(C). (We shall discuss later (5.10 and 8.5–6) how thisline bundle can be constructed explictly as a torus embedding by adding the zero section)First we shall study the restriction of the C×-torsor to a fibre under a unipotent extension.We shall show particular that under certain conditions the associated line bundle defines apolarization on the fibre.

3.18. Restriction of a torsor to a fibre: Let P1 = W0 o P with a arbitraryW0 and let φ : (P1,X1) → (P,X ) be the associated unipotent extension. Let 1 → U0 →W0 → V0 → 1 be the weight decomposition of W0. Let K1

f ⊂ P1(Af ) be neat, and fix

[(x, pf )] ∈ Mφ(K1f )(P,X )(C). By 3.13 the fibre over [x, pf ] is canonically isomorphic to

ΓW \W0(R) · U0(C), where ΓW is a certain arithmetic subgroup of W0(Q). Let 1 → ΓU →ΓW → ΓV → 1 be the corresponding short exact sequence of arithmetic subgroups. By3.12 ΓW \W0(R) · U0(C) is a holomorphic ΓU\U0(C)-torsor over ΓV \V0(R).

Let U0(C) be the coherent sheaf on ΓV \V0(R) of all holomorphic maps to U0(C), and ΓUthe constant subsheaf with values in ΓU . Consider the short exact sequence on ΓV \V0(R):

1→ ΓU → U0(C)→ ΓU\U0(C)→ 1.

The isomorphism class of the torsor in question corresponds to a cohomology class inH1(ΓV \V0(R),ΓU\U0(C)). We want to determine its image in H2(ΓV \V0(R),ΓU ) under theboundary homomorphism.

Since ΓV \V0(R) is a K(ΓV , 1), there is a canonical isomorphism H2(ΓV \V0(R),ΓU ) ∼=H2(ΓV ,ΓU ). Let Alt2(ΓV ,ΓU ) be the group of all alternating forms ΓV × ΓV → ΓU .The cocycle condition implies directly that Alt2(ΓV ,ΓU ) is contained in the group of 2-cocycles Z2(ΓV ,ΓU ). By the lemma in [M1] §2 p.16 this inclusion induces an isomorphism

51

Alt2(ΓV ,ΓU ) → H2(ΓV ,ΓU ). The cohomology class thus corresponds canonically to analternating form. But we already have a canonical alternating form, namely that inducedby the commutator pairing Ψ : V0 × V0 → U0.

3.19. Lemma: The cohomology class of this torsor corresponds to the alternatingform −Ψ : ΓV × ΓV → ΓU .

Proof. We follow the proof in [M1] §2. Let φ : W0 → V0 = W0/U0 be the givenprojection, and choose a holomorphic section s : V0 → W0. The 1-cocycle that representsthe ΓU\U0(C)-bundle on ΓV \V0(R) comes from the map

ΓW × V0(R)→ U0(R), (ω, v) 7→ fω(v) := ω · s(v) · s(φ(ω) · v).

The corresponding 2-cocycle on ΓV with values in ΓU is induced from the map

ΓW × ΓW → ΓU ,

(ω1, ω2) 7→ F (ω1, ω2) := fω2(φ(ω1)) · fω1ω2(1)−1 · fω1(1).

An elementary calculation now shows

F (ω1, ω2) = [ω2, ω1] = −Ψ(φ(ω1), (φ(ω2)),

whence the assertion. q.e.d.

3.20. Construction of a polarization: Let (P2,X2) → (P,X ) be a unipotentextension by W0, and let 1 → U0 → W0 → V0 → 1 be the weight decomposition ofW0. Let (P1,X1) := (P2,X2)/U0, and suppose that (P2,X2) → (P1,X1) is given as a(P0,X0) → (Gm,Q,H0))-torsor with respect to a morphism (φ,Ψ) : (P,X ) → (Gm,Q,H0).In particular we are given an isomorphism U0

∼= Q, and for every x ∈ X we have an elementλΨ(x) ∈ Isom(Z,Z(1)). Let us assume that for every x ∈ X the pairing

λΨ(x) Ψ : V0 × V0 → U0∼= Q→ Q(1)

is a polarization of the Hodge structure on V0 defined by x. Then Ψ : V0 × V0 → U0 isnecessarily surjective. Let K2

f ⊂ P2(Af ) be a neat open compact subgroup, K1f its image

in P1(AF ), and Kf its image in P (Af ). Then by 3.17

MK2f (P2,X2)(C) −→MK1

f (P1,X1)(C)

is a C×-torsor. We are interested in its restriction to the fibres of

MK1f (P1,X1)(C) −→MKf (P,X )(C).

3.21. Proposition: In 3.20 the inverse of the associated line bundle is ample on

every fibre of MK1f (P1,X1)(C)→MKf (P,X )(C).

Proof. The complex structure on the fibre is given by 3.14. Let ρ : P → GL(V0)be the operation of P on V0. Since V0 is pure of weight −1, for every x ∈ X the complex

52

structure on V0(R) is given by C× V0(R)→ V0(R), (z, v) 7→ ρ hx(z)(v). By 3.16 and 3.19the Chern class of the line bundle is represented by the alternating form

E : ΓV × ΓV → 2π√−1 · Z, (v, v′) 7→ −1

d· λΨ(x) Ψ(v, v′).

The unique hermitian form V0(R) × V0(R) → Q(1) ⊗ C = C with respect to the complexstructure defined by x, whose imaginary part is equal to E, is given by

V0(R)× V0(R)→ C, (v, v′) 7→ E(v, v′) +√−1 · E(v, ρ hx(

√−1)v′).

Note that as in 1.11 this formula does not depend on the choice of√−1. The real part of

this hermitian form is

V0(R)× V0(R)→ R, (v, v′) 7→ −1

d·√−1 · λΨ(X) Ψ(v, ρ hx(

√−1)v′),

so the assumption about polarization implies that this hermitian form is negative definite.The theorem of Lefschetz (see [M1] §3 p. 29) now implies that the inverse of the line bundleis ample. q.e.d.

3.22. Further constructions (not needed in the sequel): (a) It is now easyto prove that the compact complex torus in 3.14 is an abelian variety. In fact, considera unipotent extension (P1,X1) → (P,X ) with P1 = V0 o P and V0 pure of weight −1.Without loss of generality one may assume that (P,X ) is irreducible. Then with 2.14 (b)one can, as in the proof of 2.26 (b), construct (P2,X2) such that the assumptions of 3.20are satisfied. Thus by 3.21 the fibre ΓV \V0(C)/ exp(F 0(LieV0)C) is an abelian variety. Thisargument will in fact be part of the proof of the algebraicity of an arbitrary mixed Shimuravariety. See chapter 10.

(b) As a variant of the construction in (a) it is a nice exercise to construct the dualabelian variety together with the C×-torsor associated to the Poincare-bundle. For this letU0 := Q, V ′0 := Hom(V0, U0), and let W0 be the central extension of V0⊕V ′0 by U0 with thecommutator pairing ((v, `), (v′`′)) 7→ `(v′) − `′(v). Assume that (P,X ) is irreducible, andlet P act on U0 through the character given by 2.14 (b). Then for suitable choices of opencompact subgroups the fibre of the mixed Shimura variety associated to V ′0 oP is the dualabelian variety, and W0 o P yields the Poincare-bundle.

(c) One can find a lot of other structures in the unipotent fibres. For instance fix (P,X )with Z(P ) an almost direct product of a Q-split torus with a torus of compact type definedover Q, and a neat open compact subgroup Kf ⊂ P (Af ). Then as in 2.21 we get for everysuitable representation W0 of P and open compact subgroup KW

f ⊂W0(Af ) a holomorphic

family of complex Lie groups over MKf (P,X )(C). Clearly this functor preserves exactsequences. Also the endomorphism algebra φ ∈ EndP (W0) | φ(KW

f ) = KWf acts on the

fibres. With 3.15 one can define level structures. It is nice that all these constructionscan be carried out within the framework of mixed Shimura varieties. We shall apply someof them in chapter 10 to describe a modular interpretation for the mixed Shimura varietyassociated to (P2g,X2g) as defined in 2.24–25.

53

Chapter 4

Rational boundary components

We now turn to the basic ingredient of our desired compactification, namely what we shallcall rational boundary components. This chapter deals only with group theoretic data, soit depends logically only on chapters 1 and 2. The central observation is that boundarycomponents encode a relation between a given equivariant family of mixed Hodge structuresas in 2.1, and another one, which is in some sense “more mixed” (see 4.12).

We begin with some facts for a certain class of parabolic subgroups of P , if (P,X ) ismixed Shimura data. This culminates in 4.6, which gives an equivalent characterization ofthis class through certain homomorphisms from a “reference group” H0 introduced in 4.3.In 4.7–11 this is then used to construct certain other mixed Shimura data associated to agiven (P,X ) and an admissible Q-parabolic subgroup Q. These we call rational boundarycomponents in (a somewhat loose) analogy to the usual terminology (see[AMRT] ch.III §3,and the remark 4.11 (i)). In 4.12–15 we generalize the realization of a hermitian symmetricdomain as a Siegel domain of the third kind, as carried out in [AMRT] ch.III §4. In 4.16–23we study various functional properties, in particular the hereditary behavior of rationalboundary components. At the end of the chapter we illustrate the general theory withsome examples.

4.1. Cocharacters and parabolic subgroups: LetG be a connected linear algebraicgroup over a field k. A subgroup P of G, defined over k is called a k-parabolic subgroup,if G/P is a projective variety (see [BR] 4.1).

First we suppose that G is reductive. Let P be a k-parabolic subgroup of G, B ⊂P a minimal k-parabolic subgroup, and T ⊂ B a maximal k-split torus. Let Φ be thecorresponding k-root system and ∆ ⊂ Φ the simple roots defined by B. Then LieP =LieT ⊕

⊕α∈Ψ gα for some subset Ψ ⊂ Φ. There exists a subset Σ ⊂ ∆ such that Ψ is the

set of all nonnegative linear combinations of elements of ∆ together with all nonpositivelinear combinations of elements of Σ (see [B–Lie] ch.VI 1.7 prop. 20). Let λα be thefundamental weight associated to α ∈ ∆, then

Ψ = β ∈ Φ|∀α ∈ ∆− Σ〈λα, β〉 ≥ 0.

Let λ :=∑

α∈∆−Σ λα, then this weight corresponds to a unique cocharacter Gm,k → T .The Lie algebra of a Levi component of P that contains T is generated by the centralizer

54

of T (this contains LieT ) and all gβ for which 〈λ, β〉 = 0. Thus all these β are orthogonalto λ, hence λ does not depend on the choice of B. Furthermore all maximal k-split tori ofP are conjugate under P , so λ : Gm,k → P is uniquely determined by P up to conjugationby P . Conversely P is uniquely determined by λ, because LieP is the direct sum of allnonnegative weight spaces in LieG under AdG λ. Note that λ factors through the subgroupP ∩Gder ⊂ G.

Let W be the unipotent radical of P and π : P → P/W the canonical projection. Thenthe composite homomorphism π λ : Gm,k → P/W is a cocharcter of the center of P/W ,and is uniquely determined by P . Every lifting of this cocharacter to P occurs as λ. Nowlet L/k be a field extension and λL : Gm,L → PL the cocharacter, defined over L, that isassociated to PL as an L-parabolic subgroup.

Claim. : π λL = π λ.

Indeed, choose a maximal L-split torus SL in PL that contains TL. Furthermore choosea minimal L-parabolic subgroup of GL that contains SL and is contained in BL. Withthese choices we have to prove that λL = λ. Let ΦL be the corresponding L-root system,∆L the set of L-simple roots, and ΣL ⊂ ∆L the subset associated to PL in the same way.Let ResT⊂S : ΦL → Φ ∪ 0 be the restriction map, then the image of ∆L lies in ∆ ∪ 0.By definition an λ-root α ∈ ∆L lies in ΣL if and only if ResT⊂S(α) ∈ Σ ∪ 0. Thus forall α ∈ ∆L we have 〈λ,ResT⊂S(α)〉 = 0 if α ∈ ΣL, and = 1 else. This implies λL = λ, asasserted.

This shows that the P -conjugacy class of λ does not depend on the field over which Pis considered. In particular it implies that the cocharacter π λ : Gm,k → P/W is definedover a subfield of k whenever P is defined over that field.

Now let P be a connected linear algebraic group, which we do not assume to be reduc-tive. Let W be its unipotent radical. Every parabolic subgroup Q of P contains W . Letλ : Gm,k → Q/W be the cocharacter of above for Q/W as parabolic subgroup of P/W .Suppose that we are given a cocharacter µ of the center of P/W , and let w be a lifting ofµ · λ to Q. The Q-conjugacy class of w is uniquely determined by Q and µ. If LieW hasonly nonnegative weights under Ad p w, then we again have that LieQ is the direct sumof all nonnegative weight spaces in LieP under AdG w.

4.2. Application: some special mixed Hodge structures: Let B be the Borelsubgroup of the form

(∗ ∗0 ∗)

of GL2,Q. Let λ : Gm,Q → B as in 4.1. Explicitly λ has the form

t 7→ ( t0∗t−1 ). Let µ : Gm,Q → GL(M), t 7→ t · id. For every representation ν : Gm → GL(M)

let W(ν)· be the filtration on M by negative weights under ν, that is W

(ν)n M is the direct

sum of the eigenspaces under ν associated to the weights −m for all m ≤ n. In particularconsider the standard representation M := Q2 of GL2,Q and its subspace M ′ :=

(∗0

). Then

we get explicitly

W (µ)n M =

0 for n ≤ −2,

M for n ≥ −1.

W (µ·λ)n M =

0 for n ≤ −2,M ′ for n = −1,−2,M for n ≥ −1.

Now consider a rational mixed Hodge structure on M , which possesses one of these filtra-tions as weight filtration, and for which the only nontrivial step in the Hodge filtration is

55

the term F 0MC. With the weight filtration W(µ)· this Hodge structure must be pure of

type (−1, 0), (0,−1), and F 0MC may be an arbitrary one dimensional subspace of MCthat is not defined over R. With the other weight filtration it must be mixed of type(0, 0), (−1,−1) and F 0MC may be an arbitrary one dimensional subspace of M ′C that isdifferent from M ′C. Thus we can make the following observation: For every pure Hodge

structure (W(µ)· , F ·) on M of type (−1, 0), (0,−1) the pair (W

(µ·λ)· , F ·) is a mixed Hodge

structure of type (0, 0), (−1,−1).The Hodge structure (W

(µ)· , F ·) corresponds to a unique homomorphism S → GL2,R.

For the Hodge structure (W(µ·λ)· , F ·) the decomposition MC = M ′C ⊕ F 0MC shows that

there exists only one decomposition as in 1.2 (a). Thus this Hodge structure correspondsto a unique homomorphism SC → BC. In this way we have associated to each homo-morphism S → GL2,R, which induces a pure Hodge structure of type (−1, 0), (0,−1) onM , a canonical homomorphism SC → BC that induces a mixed Hodge structure of type(0, 0), (−1,−1). The restriction of these two homomorphisms to Gm,R ⊂ S are preciselythe cocharacter µ, resp. µ · λ.

All pure Hodge structures on M of type (−1, 0), (0,−1) are conjugate under GL2(R).Note that the space of these Hodge structures possesses two connected components. TheIwasawa decomposition implies that they are already conjugate under B(R). Clearly the

map (W(µ)· , F ·) 7→ (W

(µ·λ)· , F ·) is equivariant under B(R). Let us also observe that both

Hodge structures induce the same Hodge structure on Λ2M , namely of type (−1,−1). Thiscorresponds to the fact that S operates on Λ2M in the same way through both homomor-phisms.

4.3. The reference group H0: Now we introduce our analog of the group “U1 ×SL(2,R)” considered in [AMRT] ch.III p.174ff. It embodies some of the features of Shimuradata, but is defined only over R. It will be used to study certain parabolic subgroups of Pfor mixed Shimurar data (P,X ). Consider the reductive group over R

H0 := (z, α) ∈ S×GL2,R|z · z = det(α).

Let B0 ⊂ H0 be the Borel subgroup of all elements of the form (∗, (x0∗∗)). Let M0 = R2

and N0 (= C as an R-vector space) be the representations of H0 over R that come from thetwo dimensional standard representation of GL2,R, respectively of S. Consider an arbitraryhomomorphism h0 : S → H0 that induces on both M0 and N0 a pure Hodge structure oftype (−1, 0), (0,−1). Clearly such a homomorphism exists. Let h∞ : SC → SC × BC bethe unique homomorphism that induces the same Hodge structure on N0 and the mixedHodge structure described in 4.2 on M0. Since S acts on Λ2M0 in the same way throughh0 and through h∞, the homomorphism h∞ factors through H0,C. Let U0 be the unipotentradical of B0, then modulo U0 h∞ is defined over R.

One easily sees that H0(R) is connected. Hence H0(R) does not operate transitively onthe set of all h0. But still B(R) operates transitively and the map h0 7→ h∞ is equivariantunder this group. From now on we fix one h0 and the corresponding h∞. We also defineλ1 : Gm,R → Hder

0 ∩B0 such that h∞ w = (h0 w) · λ1.

4.4. Lemma: Let (P,X ) be mixed Shimura data and ω : H0,C → PC a homomorphismsuch that πω is defined over R. Suppose that we are given an x ∈ X such that hx = ωh0.

56

Consider the operation of H0 an (LieU)R, (LieV )R and (LieG)R. We want to compare the“new” mixed Hodge structures on these spaces defined by Ad p ω h∞ with the “old”mixed Hodge structures defined by Ad p hx = Ad p ω h0.

(a) (LieU)R|H0is a direct sum of copies of Λ2M0. In the particular LieU remains pure of

type (−1,−1).

(b) (LieV )R|H0is a direct sum of copies of N0 and M0. In particular LieV becomes mixed

of type (0, 0), (−1, 0), (0,−1), (−1,−1).(c) LieG)R|H0

is a direct sum of irreducible components occuring in (N0)∨⊗N0, (N0)∨⊗M0

or in (M0)∨ ⊗M0. In the particular LieG becomes mixed of type

(1, 1), (1, 0), (0, 1), (1,−1), (0, 0), (−1, 1), (0,−1), (−1, 0), (−1,−1).

Proof. (Compare [AMRT] ch.III p.182 prop. 3) Every irreducible representation of H0

over C is isomorphic to SnM0,C⊗X , where n is a nonnegative integer and X a character of

H0,C. Since the composite homomorphism SCh0−−−→ H0,C (H0/H

der0 )C is a isomorphism,

the character group of H0,C is isomorphic to that of SC. Let (a, b) be the pair of integersrepresenting the character X h0, then SnM0,C ⊗ X is of type (a, b − n), (a − 1, b − n +1), . . . , (a− n, b) under X h0. By assumption (LieU)R is pure of type (−1,−1), (LieV )Rof type (−1, 0), (0,−1) and (LieG)R of type (−1, 1), (0, 0), (1,−1) under Ad p ω h0.This implies the assertion about the representation of H0. The type under Ad p ω h∞follows from this and form 4.2 and 4.3. q.e.d.

4.5. Definition: Let (P,X ) be mixed Shimura data. Let G := P/W , then everyQ-parabolic subgroup of P is the inverse image of a Q-parabolic subgroup of Gad. LetGad = G1 × . . . × Gr be the decomposition into Q-simple factors. Let Qi ⊂ Gi be a Q-parabolic subgroup for every i, and let Q ⊂ P be the inverse image of Q1× . . .×Qr. We callQ an admissible Q-parabolic subgroup of P , if every Qi is either equal to Gi or a maximalproper Q-parabolic subgroup of Gi.

4.6. Proposition: Let (P,X ) be mixed Shimura data and Q a Q-parabolic subgroupof P . As in 2.1 let π′ be the projection P → P/U . Then the following assertion areequivalent:

(a) Q is an admissible Q-parabolic subgroup of P .

(b) For every x ∈ X there exists a unique homomorphism ω = ωx : H0,C → PC such that:

(i) π′ ω : H0,C → (P/U)C is already defined over R.

(ii) hx = ω h0.

(iii) ω h∞ w : Gm,C → QC is of the form µ · λ as in 4.1 with µ = hx w, and (LieQ)Cis the direct sum of all nonnegative weight spaces in (LieP )C under Ad p ω h∞ w(w : Gm,R → S is defined in 1.3).

(c) There exists an x ∈ X and a homomorphism ω : H0,C → PC such that the threeconditions in (b) are satisfied.

Proof. (a) ⇒ (b): First assume W = 1. Let H1 be the subgroup S1 × SL2,R ⊂ H0

and X 0 the connected component of X that contains x. By [AMRT] ch.III p.220 no.2 the

57

parabolic subgroup Q corresponds to a unique rational boundry component of X 0, in theterminology of [loc.cit.]. Therefore by [AMRT] p.199 thm. 1 (v) and p.205 thm. 2 thereexists a unique homomorphism ω1 : H1 → GadR , such that hx|S1 = ω1 (h0|S1), and (LieQ)Ris the direct sum of all nonnegative weight spaces under Ad p ω1 λ1. Thus we have acommutative diagram

H1ω1−−−→ GadR

h0|S1 ∩↓ ↑

S1h0 H0 GR

∩↓ hx

S

and our aim is to find a homomorphism ω : H0 → GR which completes it to acommutaive diagram. Since H0 is generated by h0(S) and H1, such ω is uniquely determinedsuch if it exists. Moreover then the conditions (i) and (ii), and the second part of condition(iii) are automatically satisfied. The first part of (iii) is equivalent to the assertion thatω1 λ1 is a λ as in 4.1. This follows from the explicit description of the R-root system of(P/W )ad in [AMRT] ch.III and the construction of ω1 in the proof of [loc. cit.] p.199 thm. 1(v). It thus remains only to construct some ω, with which the diagram stays commutative.

First we shall lift ω1 to a homomorphism ω1 : H1 → G. Let us write h0|S1 = (e1, e2),where e1 : S1 → S1 is the identity and e2 is a homomorphism S1 → SL2,R. Since SL2,Ris simply connected, we can lift ω1 uniquely on the second factor to a homomorphismω1 : SL2,R → GderR . On the first factor we then define the lifting by z 7→ h0(z) · ω1(e2(z))−1.With this definition we obtain another commutative diagram

H1h0|S1 ∩

↓ ω1

S1h0 H0 GR

∩↓ hx

S

Consider the homomorphism

Gm,R ×H1 = Gm,R × S1 × SL2,R → H0, (t, z, g) 7→ (t · z, t · g).

This is an isogeny of degree 2, whose kernel is ±(1, 1, id). Since by 2.1 (iii) the image ofhx|Gm,R lies in the center of GR, we can define a homomorphism

ω : Gm,R × S1 × SL2,R → GR, (t, z, g) 7→ hx(t) · ω1((z, g)).

Now the desired homomorphism ω exists if and only if ω factors through the above isogeny.But this follows from the calculation

ω((−1,−1,−id)) = hx(−1) · ω((−1,−id))

= hx(−1) · ω1(h0(−1))

= hx(−1) · hx(−1)

= 1.

Thus the assertion is proved in the case W = 1.

58

Now let P be arbitrary. Then we already have a unique homomorphism H0 → GR. Notethat hx w determines a unique Levi decomposition of PC, which is defined over R moduloUC. The desired homomorphism ω : H0,C → PC must map into this Levi component, sincethe image of h0 w lies in the center of H0. Thus it is uniquely determined and we candefine it directly using this Levi decompositon. Condition (i) then follows from the sameproperty of the decomposition. Condition (ii) holds by construction. Likewise (iii) followsafter we have shown that LieW is of weight ≥ 0 under AdP ω h∞ w. But this followsfrom 4.4 (a) and (b). This finishes the proof of the implication (a) ⇒ (b).

The implification (b) ⇒ (c) is trivial. For (c) ⇒ (a) we may assume P is to besemisimple after having devided by the radical of P as in 2.9. Then the assertion followsfrom [AMRT] ch.III, in particular p.174 cor., p.180 prop. 2 and the construction of thestandard parabolic subgroups of P in [loc. cit.] ch.III §3. q.e.d.

4.7. A canonical subgroup of an admissible Q-parabolic subgroup: We nowwant to use the data given by 4.6 to construct certain other mixed Shimura data associatedto a given (P,X ) and an admissible Q-parabolic subgroup Q. The first step is to definethe group involved. Let (P,X ), Q, x and ωx be as in 4.6. We define P1 to be the smallestnormal subgroup of Q, defined over Q, such that ωx h∞ : S→ QC factors through P1,C.

Claim. (a) ωx h∞ : S→ QC does not depend on the choice of h0.

(b) P1 depends only on (P,X ) and on Q.

(c) The map X → Hom(SC, P1,C), x 7→ ωx h∞ is Q(R) · U(C)-equivariant.

Proof. Fix x ∈ X . By 4.2 every other h′0 is of the form int(b) h0 for some b ∈ B(R),and then the corresponding h′∞ equals int(b) h∞. The homomorphism ω′x := ωx int(b−1)satisfies the conditions of 4.6 with h0 replaced by h′0. The assertion (a) now follows fromthe equation ω′x h′∞ = ωx int(b−1) int(b) h∞ = ωx h∞. In particular P1 does notdepend on the choice of h0. By the Iwasawa decompositon for G the group Q(R) · U(C)already operates transitively on X . Let q ∈ Q(R) ·U(C), then int(q) ωx satisfies the sameconditions of 4.6 as ωq·x. This implies (b) and (c), since q normalizes P1,C. q.e.d.

4.8. Lemma: Consider the situation of 4.7. Let W1 be the unipotent radical of P1.Then there exists a subgroup U1 ⊂ W1, defined over Q, normal in Q, and depending onlyon (P,X ) and Q, such that (P1,X1), W1, U1 together with the homomorphism ωx h∞ :SC → P1,C satisfy the conditions 2.1 (ii)–(v).

Proof. Conditions 2.1 (iii) and (iv) do not depend on the choice of U1 and are easilyproved. In fact, (iii) is a direct consequence of 4.6 (iii) and the same assertion for (P,X ).For (iv) observe that LieP1 is contained in LieQ, and the latter is of weight ≤ 0 by 4.6(iii), so the condition follows from 4.4.

Next we have to construct U1. Before we can do this, however, we have to provethat AdQ ωx h∞ induces a rational mixed Hodge structure on LieQ, and that its weightfiltration is stable under Q. Then it is also independant of x. To prove this we may replace(P,X ) by (P,X )/Z(P ), then the homomorphism π hx w : Gm,R → GR of 2.1 (iii) isalready defined over Q. It suffices to show the conditions 1.4 (i)–(iii) for the group P1 andthe homomorphism ωx h∞. Denote the unipotent radical of Q by WQ.

59

In the case W = 1 we show the conditions a fortiori for Q instead of P1. By 4.6 (b) (i)ωx is already defined over R. Since LieU0 is of weight −2 under AdB0 h∞, it is mapped toWQ under ωx. Now h∞ is defined over R modulo U0, hence modulo WQ the homomorphismωx h∞ is defined over R. This is condition 1.4 (i). For 1.4 (ii) we have to show thatmodulo WQ the homomorphism ωx h∞ w is central and defined over Q. By 4.6 (b) (iii)this follows from the corresponding assertion for hx w (see above and 2.1 (iii) for (P,X ))and for λ (see 4.1). For 1.4 (iii) we have to prove that LieWQ is of weight ≤ −1 underAdQ ωx h∞. But this follows again from 4.6 (b) (iii), since the centralizer of wx h∞ wis a Levi component of Q.

Let us now consider the general case. The conditions 1.4 (i) and (ii) carry over directly,so it remains to study LieWQ. Condition 1.4 (iii) follows from P1 if we can prove the equalityLieW1 = W−1(LieWQ). We may fix a splitting P = W o G and assume without loss ofgenerality that hx, hence also ωx, factors through GR. Since we have already proved theconditions of 1.4 for Q/W ∼= G∩Q, we have a rational mixed Hodge structure on LieQ thatis stable under G∩Q. By 4.6 (b) (iii) LieWQ is of weight ≤ 0. Thus [LieWQ,W−1(LieWQ)]is contained in W−1(LieWQ), so W−1(LieWQ) is an ideal of LieWQ. Since it is alreadystable under G ∩Q, it is already an ideal of LieQ. Thus W 1

Q := exp(W−1(LieW0)) ⊂ WQ

is a normal subgroup of Q. By the preceding paragraph we have W ·W 1Q = WQ, hence

WQ/W1Q is isomorphic to W/(W ∩ W 1

Q). By 4.4 (a) U must be contained in W 1Q, so

WQ/W1Q is isomorphic to a quotient of V . Thus it is abelian, so the adjoint operation of

Q on Lie(WQ/W1Q) factors through Q/WQ. Moreover 4.4 (b) implies that Lie(WQ/W

1Q) is

pure of type (0, 0). This means that the image of ωx h∞ lies in the kernel of the adjointrepresentation of Q on Lie(WQ/W

1Q). By definition P1 must also lie in this kernel, so it

operates trivially on WQ/W1Q. We have now proved that the Levi component of the group

(P1 ·WQ)/W 1Q operates trivially on the unipotent radical. But such a group possesses only

one Levi decomposition, which is a direct product. Again from the definition of P1 it followsthat (P1 ·W 1

Q)/W 1Q is the unique Levi component of (P1 ·WQ)/W 1

Q, whence W1 ⊂W 1Q, as

desired. The reverse inclusion follows from the fact that P1 operates nontrivially on everysimple subquotient of W 1

Q.

If we now define U1 := exp(W−2(LieW1)), then by the above this subgroup is normalin Q and independant of x. Condition 2.1 (v) holds by construction and by the result inthe preceding paragraph. Finally 2.1 (ii) follows with the same argument as above: Sinceby 4.4 (a) U ⊂ U1, we may without loss of generality assume that U = 1. Then by 4.6 (b)(i) ωx is already defined over R. Since LieU0 is of weight −2, it is mapped to U1 under ωx.Because h∞ is defined over R modulo U0, it follows that modulo U1 the homomorphismωx h∞ is also defined over R. q.e.d.

4.9. Lemma: Consider the situation of 4.8. Let P11 ⊂ P1 ∩ P der be the smallestnormal subgroup such that (P1 ∩ P der)/P11 is a Q-split torus.

Let G1,C be the centralizer of ωx h∞ w in P1,C, and let G11,C := G1,C ∩ P11,C. TheseLevi components of P1,C resp. of P11,C. Then

(a) G11,C commutes with ωx(Hder0,C ) in PC. Thus G11,C · ωx(Hder

0,C ) is a reductive subgroup ofPC.

60

(b) The subset G1,C · ωx(Hder0,C ) is a reductive subgroup of PC. It contains G11,C · ωx(Hder

0,C )as a normal subgroup, and it contains ωx(H0,C) as a not necessarily normal subgroup. Itssemisimple part is the same as that of G11,C · ωx(Hder

0,C ).

(c) If ωx is defined over R, then each of the groups G11,C, G11,C ·ωx(Hder0,C ) and G1,C ·ωx(Hder

0,C )is defined over R.

Proof. Since all the groups mentioned lie inside the centralizer of hx w in PC, which isa Levi component of PC, both (a) and (b) follow directly from the corresponding assertionfor (P,X )/W instead of (P,X ). The same holds for (c), since this Levi component is definedover R if ωx is defined over R. Thus we may assume that W = 1, so ωx is defined over R.

(a) Since U0 is of weight −2 under Ad h∞, it is mapped to U1,R under ωx. Since(LieU1)R|AdP ωxh∞ is pure of type (−1,−1), S operates through scalars on (LieU1)R.Hence by definition P1 also operates through scalars on LieU1. Thus P11 operates triviallyon U1, and in particular G11,C commutes with ωx(U0,C). By construction G11,C also com-mutes with the image of ωx h∞ w. Now the equation h∞ w = (h0 w) · λ1 of 4.3,and the fact that the image of ωx h0 w = hx w lies in the center of P , imply thatG11,C commutes with the image of ωx λ1. Hence it commutes with the Borel subgroupωx(Hder

0 ∩ B0)C of ωx(Hder0,C ). This implies, for instance by the lemma 1 of [B-Lie] ch.VIII

§11.1, that G11,C commutes with the whole of ωx(Hder0,C ), as desired.

(b) It is easy to see that the assertions are invariant under replacing (P,X ) by (P,X )/Z(P ),so we may assume that P is semisimple. Then by the definition of P1 we have P1,R = P11,R ·ωxh∞w(Gm,R), whence G1,C = G11,C ·ωxh∞w(Gm,R). Since the center of P = G is triv-ial, we have wxh0w = hxw = 1, hence by 4.3 ωxh∞w = (ωxh0w)·(ωxλ1) = ωxλ1.But λ1 factors through Hder

0,C , so we even get G1,C · ωx(Hder0,C ) = G11,C · ωx(Hder

0,C ). Thus the

desired assertions are abvious, except that it contains ωx(H0,C). But by definition G1,Ccontains the image of ωx h∞, so this follows from the fact that H0 is generated by Hder

0

together with the image of h∞.

(c) Let b ∈ (Hder0 ∩B0)(C) such that int(b)h∞ : SC → H0,C is defined over R. Then by

definition both int(ωx(b))(G1,C) and int(ωx(b))(G11,C) are defined over R. Since ωx(b) liesin ωx(Hder

0 )(C), the assertion follows for the groups G1,C · ωx(Hder0,C ) and G11,C · ωx(Hder

0,C ).

For G11,C it follows since by (a) this subgroup commutes with ωx(b). q.e.d.

4.10. Corollary: In 4.8 the conditions 2.1 (vi)–(viii) are also satisfied.

Proof. For 2.1 (vi) and (vii) we may assume that P is semisimple. Consider thegroup G11,R of 4.9, which by 4.9 (c) is defined over R. By 4.9 (b) it is a normal subgroup ofG1,C ·ωx(H0,C), so in particular it is normalized by the image of ωx h0 = hx. The condition2.1 (vi) for (P,X ) thus implies that int(hx(

√−1)) induces a Cartan involution on G11,R.

By definition h0(√−1) ≡ h∞(

√−1) modulo Hder

0 , so int(ωx h∞(√−1)) also induces a

Cartan involution on G11,R. This implies 2.1 (vi) for P1. By the definition of P1 the imageof ωx h∞(SC) in any Q-direct factor of (P1/W1)ad is nontrivial, so it also implies 2.1 (vii).

To prove 2.1 (viii) we may replace (P,X ) by (P,X )/Z(P ). Then the center of G isan almost direct product of a Q-split torus with a torus of a compact type defined over Q.It suffices to prove the same for the center of P1/W1, and to do this we may assume thatW = 1. Then we may again devide by Z(P ), after which we find ourselves in the situation

61

above. Since P1/P11 is a Q-split torus, it suffices to study the center of P11/W1. By theabove int(ωx h∞(

√−1)) induces a Cartan involution (P11/W1)R, but since the image of

ωx h∞(√−1) in (P1/W1)(R) commutes with the center of (P11/W1)R, the latter is already

of compact type. q.e.d.

4.11. Rational boundary components: Consider the situation of 4.8. Let q ∈Q(R)0 ·U(C), then by 4.7 (c) we have ωq·xh∞ = int(q)ωxh∞. Since Q(R)0 ·U(C) inducesonly inner automorphisms on P1(R)/W1(R), there exists p1 ∈ P1(R) such that ωq·x h∞ ≡int(p1) ωx h∞ modulo W1(C). But the Levi decomposition of P1,C defined by ωq·x h∞and by ωx h∞ are modulo U1 both defined over R, and any two such Levi decompositionsare conjugate under W1(R) · U1(C). Hence we may even find p1 ∈ P1(R) · U1(C) such thatωq·x h∞ = int(p1) ωx h∞. Since Q(R)0 · U(C) · x is a connected component of X , thisshows that the map X → Hom(SC, P1,C), x 7→ ωx h∞ maps every connected component ofX to a P1(R) · U1(C)-orbit.

Through π0(P1(R) · U1(C)) = π0(P1(R))[inclusion]−−−−−−−−→ π0(P (R)) the group P1(R) · U1(C)

operates on π0(X ) in a canonical way. Consider the Q(R) · U(C)-equivariant map

X → π0(X )×Hom(SC, P1,C), x 7→ ([x], ωx h∞),

where we denote by [x] the connected component of X that contains x. Let X1 be a P1(R) ·U1(C)-orbit in π0(X ) × Hom(SC, P1,C) that contains an element of the form ([x], ωx h∞).The image of the above map is contained in finitely many such X1. Let X+ ⊂ X be theinverse image of X1. From 4.8, 4.10 and the definition of P1 we now get:

(P1,X1) is irreducible mixed Shimura data. It depends only on (P,X ), Q and the choiceof a P1(R)-orbit in π0(X ). There is a canonical P1(R) · U(C)-equivariant map X+ → X1,which induces a bijection on the connected components π0(X+) ∼−−→ π0(X1). We call(P1,X1) a rational boundary component of (P,X ). We say that (P1,X1) is a proper rationalboundary component if Q is a proper parabolic subgroup of P . Otherwise we call (P1,X1)an improper boundary component. Note that (P,X ) is a rational boundary component ofitself if and only if it is irreducible.

Remarks: (i) This name is somewhat misleading, but it was chosen for the followingreason. Let W = 1, then the pure Shimura data (P1,X1)/W1 corresponds to some boundarycomponents in the Baily-Borel compactification of MKf (P,X )(C) (see 6.2–3), while (P1,X1)really corresponds to a formal neighborhood of this. For arbitrary P we have the followingsituation. The boundary strata in a toroidal compactification of MKf (P,X )(C) correspondto a mixed Shimura varieties associated to certain quotients (P1,X1)/U2 for U2 ⊂ U1, andagain (P1,X1) itself corresponds to a formal neighborhood of this (see 7.3 and 7.17).

(ii) With our definition a rational boundary component is always irreducible. This isnot strictly necessary, since other choices for P1, which differ from our definition only inthe center of P1/W1, also lead to mixed Shimura data. For our purposes, however, theirreducibility requirement seems the most practical.

4.12. Proposition Consider mixed Shimura data (P,X ) and a rational boundarycomponent (P1,X1) of (P,X ). Let M be a representation of P . Let x ∈ X and x1 its image

62

in X1. By 1.4 (a) both x and x1 define a rational mixed Hodge structure on M . The Hodgefiltration of these two Hodge structures coincide.

Remarks: (i) By 4.6 (iii) the weight filtration of the Hodge structure defined by x1

depends only on (P,X ) and Q. The proposition thus shows how the “new” Hodge structurecan be obtained from the “old” one. By 1.16 this Hodge structure is described by a uniquehomomorphism SC → P1,C. Thus with this result one could define (P1,X1) and the mapX+ → X1 directly.

(ii) In particular the proposition implies that the “old” Hodge filtration together withthe weight filtration defined by Q define a mixed Hodge structure. According to M.Rapoport this fact has already been known for reductive P . I could, however, not finda reference.

Proof. Let ωx be as in 4.6, then the two Hodge filtrations are defined by ωx h0 and byωx h∞. Thus it suffices to show the analogous statement for every representation of H0,C.As in the proof of 4.4 every irreducible representation of H0,C is of the form SnM0,C ⊗ χfor a nonnegative integer n and a linear character χ of H0,C. In particular every irreduciblerepresentation occurs in some M0,C⊗n ⊗χ. Since h∞ and h0 do not differ modulo Hder

0 , thedesired assertion holds for every one dimensional representation. By the definition of h∞in 4.3 it also holds for M0,C. Since the Hodge filtration on a tensor product is given by theformula F p(M1⊗M2) =

∑p1+p2=p F

p1M1⊗F p2M2, in which the weight filtration does notoccur, the assertion also follows for M0,C⊗n ⊗ χ. q.e.d.

4.13. Corollary: The map X+ → X1 is injective and holomorphic.

Proof. Let M be a faithful representation of P . By 4.12 the Hodge structure on Mdefined by x ∈ X+ is already uniquely determined by the image of x in X1. By 1.16 and2.12 the injectivity follows on each connected component of X+. By constructuion the mapis bijective on the set of connected components, hence it itself is injective. The holomorphyfollows from 1.7, 1.16 and 4.12. q.e.d.

4.14. Imaginary part: Let (P,X ) be mixed Shimura data. As in 1.11 let U(R)(−1) =(2π√−1)−1 · U(R) inside the C-vector space U(C). This notation for the imaginary part

may seem somewhat arbitrary, but we shall see in 6.6 why the Tate twist by −1 is thenatural choice. Now every element of P (R) ·U(C) can be written uniquely in the form u · pwith u ∈ U(R)(−1) and p ∈ P (R). Since P (R)U(C) operates transitively on X , and forsome x0 ∈ X the homomorphism hx0 is defined over R, for every x ∈ X there exists anelement ux ∈ U(R)(−1) such that int(u−1

x ) hx is defined over R. By 1.17 (a) this elementis uniquely determined. Thus we have a canonical map

im : X → U(R)(−1), x 7→ im(x) := ux,

which we call the projection to the imaginary part. Let P (R) · U(C) act on U(R)(−1)through the isomorphism U(R)(−1) → P (R) · U(C)/P (R), u 7→ [u]. In other words we letP (R) act by conjugation, and U(R)(−1) by translation on itself. Clearly im is equivariantand surjective. Moreover this construction is functorial. In particular, if P is a normalsubgroup of a connected group Q, then Q(R)0 ·U(C) operates again equivariantly on X and

63

on U(R)(−1). Also, for a morphism (P,X ) → (P ′,X ′) we have a P (R) · U(C)-equivariantcommutaive diagram:

X −→ X ′↓ ↓

U(R)(−1) −→ U ′(R)(−1).

4.15. Proposition: Consider mixed Shimura data (P,X ) and a rational boundarycomponent (P1,X1) of (P,X ). Let X+ ⊂ X be as in 4.11. Let X 0 be a connected componentof X+, and X 0

1 the corresponding connected component of X1.

(a) The map X+ → X1 is an open embedding.

(b) The image of X 0 in X 01 is the inverse image of an open convex cone C : +C(X 0, P1) ⊂

U1(R)(−1) under the map im|X 01

: X 01 → U1(R)(−1) defined in 4.14.

(c) This cone is an orbit in U1(R)(−1) under translation by U(R)(−1) and conjugation byQ(R)0. It is also invariant under translation by (U1 ∩W )(R)(−1).

(d) Modulo (U1 ∩W )(R)(−1) the cone C is a nondegenerate homogeneous selfadjoint cone(in the sense of [AMRT] ch.II p.57 §1.1).

Remark. In general C(X 0, P1) does depend on the connected component X 0. See forinstance 4.26.

Proof. In the case W = 1 these are the assertions of [AMRT] ch.III p.227 thm. 1and the lemma p.236. In the general case let (P ′,X ′) := (P,X )/W and (P ′1,X ′1) :=(P1,X1)/(W ∩W1), then in the obvious way (P ′1,X ′1) is a rational boundary componentof (P ′,X ′). The maps of 4.11 and 4.14 yield a commutative equivariant diagram

X 0 −→ X 01

im−−−→ U1(R)(−1)↓ ↓ ↓X ′0 −→ X ′01

im−−−→ U ′1(R)(−1)

Let us first prove that the image of X 0 in X 01 is stable under W (R) · (W ∩U1)(C). For this

consider a faithful representation M on P . By 1.17 (b) the set of mixed Hodge structureson M defined by X 0 is invariant under W (C), hence also under W (R) · (W ∩ U1)(C). By4.12 the map X 0 → X 0

1 is equivariant under this group, which proves the desired assertion.

By definition we have X ′01∼= X 0

1 /(W ∩W1)(R) · (W ∩ U1)(C) and we know that theimage of X 0 is invariant under (W ∩W1)(R) ·(W ∩U1)(C). This shows that left hand squarein the above diagram is cartesian. Let C(X 0, P1) be the inverse image of C(X ′0, P ′1) underthe projection U1(R)(−1)→ (U1/(W ∩ U1))(R)(−1) = U ′1(R)(−1). Then all the assertions,except the first statement of (c), follow directly from the same assertions for (P ′,X ′).Finally we already know that (Q/W )(R)0 acts transitively on X ′0, and W (R) · U(C) actstransitively on the fibres of the projection X 0 → X ′0. Thus Q(R)0 · U(C) = U(R)(−1) oQ(R)0 acts transitively on X 0, whence also on C(X 0, P1). As explained in 4.14, Q(R)0 actsby conjugation, and U(R)(−1) by translation on U1(R)(−1). This finishes the proof of (c).q.e.d.

4.16. Functoriality: Let (φ, ψ) : (P,X )→ (P ′,X ′) be a morphism of mixed Shimuradata and Q an admissible Q-parabolic subgroup of P . Let x ∈ X and ωx : H0,C → PC as in

64

4.6. Let Q′C be the unique connected subgroup of P ′C whose Lie algebra is the direct sum ofall nonnegative weight spaces in (LieP ′)C under AdP ′ φ ωx h∞ w. It follows from 4.4(a) and (b) that Q′C contains the unipotent radical of P ′C, so this is a parabolic subgroup.By construction it contains φ(QC). The cocharacter φωx h∞ w is modulo the unipotentradical of Q independant of x and h0 and defined over Q. Thus Q′C depends only on Q andis defined over Q. By 4.6 Q′ is an admissible Q-parabolic subgroup of P ′.

Let P1 ⊂ Q and P ′1 ⊂ Q′ be defined as in 4.7. By definition φ induces a homomorphismφ1 := φ|P1

: P1 → P ′1. Let X1 and X+ be as in 4.11. Let (X ′)+ ⊂ X ′ be the P ′1(R) ·Q′(R)0 ·U ′(C)-orbit in X ′ that is generated by φ(X+), and X ′1 the corresponding P ′1(R) · U ′1(C)-orbit in π0(X ′) × Hom(SC, P ′1,C). Thus we have associated to (P1,X1) a canonical rationalboundary component (P ′1,X ′1) of (P ′,X ′). Furthermore φ1 extends uniquely to a morphismof mixed Shimura data (φ1, ψ1) : (P1,X1) → (P ′1,X ′1), such that the following diagram iscommutative and P1(R) · U(C)-equivariant:

X+ ψ−−→ (X ′)+

∩↓

∩↓

X1ψ1−−−→ X ′1

Consider connected components X 0 of X+ and (X ′)0 of X ′ such that X 0 is mappedto (X ′)0. Let C(X 0, P1) ⊂ U1(R)(−1) and C((X ′)0, P ′1) ⊂ U ′1(R)(−1) be the correspondingcones according to 4.15. Then C(X 0, P1) is mapped to C((X ′)0, P ′1) under φ.

Let (φ, ψ) be an automorphism of (P,X ), and (P1,X1) a rational boundary componentof (P,X ). Then denote by (φ(P1), ψ(X1)) the unique rational boundary component (P ′1,X ′1)of above. This applies especially to interior automorphisms (int(p), p· ) for p ∈ P (Q).

4.17. Hereditary properties: Let (P,X ) be mixed Shimura data, (P1,X1) a rationalboundary component of (P,X ), and (P2,X2) a rational boundary component of (P1,X1).We want to show (P2,X2) can be identified canonically with a rational boundary componentof (P,X ).

Let Q1 ⊂ P be the parabolic subgroup belonging to (P1,X1), and Q12 ⊂ P1 thatbelonging to (P2,X2). Let X ⊃ X+ → X1 and X1 ⊃ X+

1 → X2 be as in 4.11, and letX ⊃ X++ be the inverse image of X+

1 in X+. Let x ∈ X++, x1 its image in X+1 , and x2 its

image in X2. Let hx : SC → PC, hx1 : SC → P1,C and hx2 : SC → P2,C be the correspondinghomomorphisms. Let ωx : H0,C → PC and ωx1 : H0,C → P1,C be as in 4.6. Then we havehx = ωx h0, hx1 = ωx h∞ = ωx1 h0 and hx2 = ωx1 h∞.

Consider the following reductive group over R

H00 := (z, α, β) ∈ S× (GL2,R)2 | z · z = det(α) = det(β),

and the homomorphisms φ1, φ2, φ3 : H0,C → H00,C that are defined by φ1(z, α) :=(z, pr2 h0(z), α), φ2(z, α) := (z, α, pr2 h∞(z)) and φ3(z, α) := (z, α, α). Observe therelations φ1 h∞ = φ2 h0 and φ2 h∞ = φ3 h∞. The homomorphisms φ1, φ2|Hder

0and

φ3 are defined over R.

4.18. Lemma: In the situation of 4.17 there exists a unique homomorphism Ωx :H00,C → PC such that:

65

(i) π′ Ωx : H00,C → (P/U)C is defined over R.

(ii) Ωx φ1 = ωx.

(iii) Ωx φ2 = ωx1 .

Proof. We shall construct Ωx explicitly. The map

H0 ×Hder0 → H00, ((z, α), (1, β)) 7→ φ1(z, α) · φ2(1, β)

= (z, pr2 h0(z) · β, α)

is an isomorphism of algebraic varieties. This and the conditions (ii) and (iii) impliy thatΩx must be given as follows:

H00,C∼←−− (H0 ×Hder

0 )C −→ PC,

(z, α, β) 7→ ((z, β), (1, pr2 h0(z−1) · α)) 7→ ωx(z, β) · ωx1(1, pr2 h0(z−1) · α).

With this definition (ii) is automatic. For (iii) we have

Ωx φ2(z, α) = ωz(z, pr2 h∞(z)) · ωx1(1, pr2 h0(z−1) · α)

= ωx h∞(z) · ωx1 h0(z−1) · ωx1(z, α)

= hx1(z) · hx1(z−1) · ωx1(z, α)

= ωx1(z, α),

as desired. For (i) note that by assumption π′ ωx is defined over R. Thus it suffices toshow that ωx1|Hder

0is defined ovr R. Now there exists an element u1 ∈ U1(C) such that

int(u−11 )ωx1 is defined over R. By 4.4 (a) ωx1(Hder

0 ) commutes with U1. Therefore ωx1|Hder0

itself is defined over R. Thus we have shown that the map Ωx satisfies the conditons (i) to(iii), but we stille have to prove that it is a homomorphism.

To do this first assume that W = 1. For any t ∈ Gm,R ⊂ S the element Ωx(t, t, t) =Ωx φ1 h0 w(t) = hx w(t) lies in the center of P . Thus every such element commuteswith Ωx(S1 × (SL2,R)2), so that it suffices to prove that Ωx|S1×(SL2,R)2 is a homomorphism.Since this group is connected, it is enough to prove this for the composite of this map witheach of the projections P → P/P der and P → P ad. The first case follows from condition(ii). The second follows from [AMRT] ch.III p.201 prop. 1 and p.207 cor. Now let W bearbitrary. Then hxw defines a Levi decomposition od PC. The homomorphism ωx respectsthis decomposition, hence so does hx1 = ωx h∞. This in turn implies that ωx1 respectsthis decompostion as well, hence so does Ωx. Thus the assertion follows from that in thecase W = 1. q.e.d.

4.19. Lemma: Consider the situation of 4.17.

(a) Let ω′x := Ωx φ3 : H0,C → PC. There exists a unique admissible Q-parabolic subgroupQ2 ⊂ P such that the conditions 4.6 (b) hold for Q2 and ω′x. It depends only on (P1,X1)and (P2,X2).

(b) Q12 = P1 ∩Q2, and Q1 = (Q1 ∩Q2) · P1.

(c)Let (P3,X3) be the unique rational boundary component of (P,X ) that belongs to Q2

and for which x is mapped to X3. Then P2 = P3 as subgroups of P , and there exists a unique

66

isomorphism X2 → X3, which together with the identity on P2 yields an isomorphism ofmixed Shimura data (P2,X2) ∼−−→ (P3,X3) such that the following diagram is commutive:

X++ → X+1 → X2

‖ ↓ oX++ −→ X3.

Proof. (a) Define Q2,C ⊂ PC by the properties 4.6 (b). By 4.4 (a) and (b) this is aparabolic subgroup. We want to show that it is defined over Q. By assumption ωx(U0,C) iscontained in U1,C and ωx1(U0,C) in U2,C. Since by 4.4 (a) U1,C is contained in U2,C, we get

ω′x(U0,C) ⊂ Ωx(1 ×(

1 ∗0 1

)2)ωx(U0,C) · ωx1(U0,C) ⊂ U2,C.

Furthermore we have

ω′x h∞ = Ωx φ3 h∞ = Ωx φ2 h∞ = ωx1 h∞ = hx2 .

Thus U2,C lies in the unipotent radical of Q2,C and ω′x h∞ is defined over Q modulo U2.Hence Q2,C is already defined over Q. Now 4.6 implies that Q2 is an admissible Q-parabolicsubgroup of P that does not depend on x. This proves (a).

(b) By 4.6 (b) (LieQ2)C is the direct sum of all nonnegative weight spaces in (LieP )Cunder ω′x h∞ w. Likewise (LieQ12)C is the direct sum of all nonnegative weight spaces in(LieP1)C under ωx1 h∞ w. As we have just seen, these cocharcters are equal, whence thefirst equality. For the second it suffices to prove the inclusion Q1 ⊂ Q2 · P1. Observe that,by the definition 4.7 of P1, the adjoint action of this group on (LieQ1)/(LieP1) is trivial.In particular, this quotient is purely of type (0, 0) under ωx1 h∞. By the definition of Q2,this implies LieQ1 ⊂ LieQ2 + LieP1, whence Q1 ⊂ Q2 · P1.

(c) The second assertion follows from the first and the relation ω′x h∞ = ωx1 h∞. Forthe equation P2 = P3 we can replace P by (P/W )ad without loss of generality, so we mayassume P to be semisimple and adjoint. In this case the assertion follows from [AMRT]ch.III thm. 3 p.240. q.e.d.

4.20. Corollary: Every rational boundary component of a rational boundary com-ponent of (P,X ) is itself in a canonical way a rational boundary component of (P,X ).

Proof. 4.18 and 4.19. q.e.d.

Remark. (i) This corollary implies that the natural order relation between rationalboundary components is indeed a partial order.

(ii) There exists only finitely many P (Q)-conjugacy classes of rational boundary com-ponents. This follows at once from the corresponding fact for admissible Q-parabolic sub-groups and the construction in 4.11. Observe also that since Q1(R) acts transitively onπ0(X ), the conjugacy class of (P1,X1) is already determined by P1 or Q1 alone. If (P/W )ad

is Q-simple, then by the explicit description of admissible Q-parabolic subgroups in [AMRT]ch.III or [BB] the conjugacy classes are in total order. In general there still exists uniqueminimal, resp. maiximal conjugacy classes. The maximal conjugacy class consists of courseof all irreducible components of (P,X ).

67

(iii) Fix a rational boundary component (P1,X1) of (P,X ), then two rational boundarycomponents of (P1,X1) are conjugate under P (Q) if and only if they are conjugate underP1(Q). In fact, it suffices to check this assertion in the case where (P/W )ad is Q-simple.Then by the explicit description of the Q-root system of (P1/W1)ad in [AMRT] ch.III §3.5p.220 this is also Q-simple. But the total order relation is of course preserved.

4.21. Proposition: Consider the situation of 4.17. Let X 0 ⊂ X++, X 01 ⊂ X

+1

and X 02 ⊂ X2 be connected components that are mapped to each other by the canonical

inclusion X++ → X+1 → X2. Recall that U1 ⊂ U2 (by 4.4 (a)). Then:

(a) C(X 01 , P2) = C(X 0, P2) + U1(R)(−1) = C(X 0, P2) + (W1 ∩ U2)(R)(−1).

(b) C(X 0, P1) is contained in the closure of C(X 0, P2).

(c) Let (P3,X3) be another rational boundary component of (P,X ) that lies “between”(P2,X2) and (P,X ). Then C(X 0, P1) and C(X 0, P3) have a point in common if and only if(P3,X3) = (P1,X1).

Proof. By 4.15 (b) we have X 01 = U1(R)(−1) · image(X 0), which implies the equation

C(X 01 , P2) = C(X 0, P2) + U1(R)(−1). The second equation of (a) follows from 4.15 (c).

This proves (a).

If W = 1, then the assertion (b) and (c) follow from [AMRT] ch.III p.240 thm. 3. In general,the rational boundary component of (P,X )/W corresponding to (P1,X1) is canonicallyisomorphic to (P1,X1)/(W1 ∩ W ). Moreover, C(X 0/W,P1/(W1 ∩ W )) is just the imageof C(X 0, P1), and by 4.15 (c), the latter is the full inverse image of the former under theprojection U1(R)(−1) → (U1/U1 ∩W )(R)(−1). The same holds for (P2,X2), and (b) and(c) follow in the general case. q.e.d.

4.22. Definition-Proposition: Let (P1,X1) be a rational boundary component of (P,X )and X 0 a connected component of X+. We define C∗(X 0, P1) ⊂ U1(R)(−1) to be the unionof the cones C(X 0, P2) for all rational boundary components (P2,X2) between (P,X ) and(P1,X1), including (P,X ) and (P1,X1).

(a) C∗(X 0, P1) is a convex cone.

Fix a rational boundary component (P2,X2) between (P1,X1) and (P,X ). Then

(b) C∗(X 02 , P1) = C∗(X 0, P1) + U2(R)(−1), and

(c) C∗(X 0, P2) = C∗(X 0, P1) ∩ U2(R)(−1) = C∗(X 0, P1) ∩ (U1 ∩W2)(R)(−1).

Proof. (a) Since C∗(X 0, P1) is a union of cones, it remains to prove the convexity. Thisfollows from the stronger Claim: Let u2 ∈ C(X 0, P2) ⊂ C∗(X 0, P1) and u′2 ∈ C(X 0, P ′2) ⊂C∗(X 0, P1), then the open line segment λ · u2 + (1 − λ) · u′2 | 0 < λ < 1 is contained inC(X 0, P3) for a unique rational boundary component of both (P2,X2) and (P ′2,X ′2) suchthat (P1,X1) is a rational boundary component of (P3,X3). If W = 1, then this followsfrom [AMRT] ch.II together with ch.III p.240 thm. 3. The general case follows from this,together with the second assertion of 4.15 (c).

(b) The inclusion “⊂” follows from 4.21 (a). For the other inclusion let u1 ∈ C∗(X 0, P1)and u2 ∈ C(X 0, P2). By the claim above, u1 + u2 ∈ C(X 0, P3) for a rational boundarycomponent (P3,X3) between (P1,X1) and (P2,X2). Thus u1 ∈ C(X 0, P3) + U2(R)(−1) =C(X 0

2 , P3) ⊂ C∗(X 02 , P1) by 4.21 (a), as desired.

68

(c) The inclusion “⊂” are obvious. Let u1 ∈ C(X 0, P3) ∩ (U1 ∩W2)(R)(−1) for some(P3,X3) between (P1,X1) and (P,X ). Let u2 ∈ C(X 0, P2), then by the claim above, u1 +u2 ∈ C(X 0, P4) for some (P4,X4) between (P1,X1) and (P2,X2), and also between (P1,X1)and (P3,X3). In particular, we have C(X 0, P4) ∩ (U4 ∩W2)(R)(−1) 6= ∅. By 4.21 (a), thisimplies that C(X 0

2 , P4) = C(X 0, P4) + (U4 ∩W2)(R)(−1) contains the origin, so by 4.15we must have (P4,X4) = (P2,X2). Hence (P3,X3) is between (P2,X2) and (P,X ), andC(X 0, P3) ⊂ C∗(X 0, P2), as desired. q.e.d.

4.23. Proposition: Let (P1,X1) be a rational boundary component of (P,X )and (Pi,Xi) for i = 2, 3 rational boundary component between (P1,X1) and (P,X ). Let(P,X ) → (P ′,X ′) be an embedding, and (P ′i ,X ′i ) be the corresponding rational boundarycomponents of (P ′,X ′). Then (P2,X2) is a rational boundary component of (P3,X3) if andonly if (P ′2,X ′2) is a rational boundary component of (P ′3,X ′3).

Proof. Fix ui ∈ C(X0, Pi), then by the proof of 4.22, u2 + u3 ∈ C(X 0, P4) for some(P4,X4) between (P1,X1) on one hand, and both (P2,X2), (P3,X3) on the other hand.Thus it suffices to show that (P4,X4) = (P3,X3) if and only if (P ′4,X ′4) = P ′3,X ′3). Now, bydefinition, (P4,X4) = (P3,X3) if and only if the corresponding homomorphism ω : H0,C →P3,C of 4.6, resp. 4.11, is trivial on Hder

0,C . By the injectivity of P3 → P ′3, the equivalencefollows. q.e.d.

4.24. The conical complex: The convex cone C∗(X 0, P1) considered in the previouslemma is in general neither open nor closed in U1(R)(−1). By 4.21 it is stratified by theboundary components lying in between. With respect to the Q-structure U1(R)(−1) =U1(Q)(−1)⊗ R each such stratum is an open convex cone in a subspace of U1(R)(−1) thatis defined over Q. Define

C(P,X ) :=∐

(X 0,P1)

C∗(X 0, P1)/ ∼,

whence∼ is the equivalence relation generated by the graph of all embeddings C∗(X 0, P1) →C∗(X 0, P2), where (P2,X2) is a rational boundary component of (P1,X1) and (P1,X1) oneof (P,X ). As a set C(P,X ) is the disjoint union of all C(X 0, P1), but endowed with thequotient topology the C(X 0, P1) form a locally closed stratification of C(P,X ). The closureof any C(X 0, P1) is isomorphic to C∗(X 0, P1). This is the conical complex associated to(P,X ).

By 4.16 the conical complex is functorial in (P,X ). In particular the automorphismsof (P,X ) act from the left hand side on C(P,X ). There is also the following relation withthe conical complex for a rational boundary component (P1,X1) of (P,X ). Consider thesets C∗(X 0, P2) for all rational boundary components (P2,X2) of (P1,X1). We can considertheir union C(P,X ), but by 4.22 (b) we can also consider their union in C(P1,X1). Thesetwo are isomorphic. In C(P,X ) they form a closed, in C(P1,X1) and open subset.

In the rest of this chapter we shall describe some examples.

4.25. Example: Let (P,X ) be the pure Shimura data (CSp2g,Q,H2g) defined in 2.7.Let V ′ ⊂ V := V2g be a totally isotropic subspace with respect to the alternating form Ψ,

69

and let Q := StabP (V ′). For V ′ 6= 0 this is a maximal proper Q-parabolic subgroup, andevery such is obtained in this way. Thus V ′ 7→ Q defines a bijection between the totallyΨ-isotropic subspaces of V and the admissible Q-parabolic subgroups of CSp2g,Q. We shallnow keep V ′ fixed.

Let (P1,X1) be the associated rational boundary component. The Hodge structureon V defined by an element x ∈ H2g is pure of the type (−1, 0), (0,−1), and F 0VC isa maximal totally Ψ-isotropic subspace of VC. Let x1 ∈ X1 be the image of x, then therational mixed Hodge on V that is defined by x1 has the weight filtration

WnV =

0 if n < −3,V ′ if n = −2,(V ′)⊥ if n = −1,V if n ≥ 0.

and by 4.12 the same Hodge filtration as the pure Hodge structure. The decompositionof VC into eigenspaces under hx is VC = F 0VC ⊕ F 0VC. By 1.2 the decomposition intoeigenspaces under hx1 turns out to be V = V −1,−1 ⊕ V −1,0 ⊕ V 0,−1 ⊕ V 0,0, where

V −1,−1= V ′C,V 0,−1 = (V ′)⊥C ∩ F 0VC,V −1,0 = V 0,−1, andV 0,0 = F 0VC ∩ (F 0VC + V ′C).

This impliesP1 = q ∈ Q | q acts trivially on V/(V ′)⊥, and

U1 = q ∈ Q | q acts trivially on V/V ′ and on (V ′)⊥.

The map u1 7→ Ψ(., (id−u1)(.)) identifies U1 with the vector space of all symmetric bilinearforms on V/(V ′)⊥. Clearly this isomorphism is Q-equivariant under the natural operationof Q on U1 and on V/(V ′)⊥. It is easy to check that Q maps onto GL(V/(V ′)⊥).

In the case dim(V ′) = 1 it turns out that (P1,X1) is isomorphic to the mixed Shimuradata (P2(g−1),X2(g−1)) defined in 2.25. At the other extreme, where V ′ is a maximal totally

isotropic subspaces of V , (P1,X1) is isomorphic to a g(g+1)2 -fold fibre product of (P0,X0)

with itself over (Gm,Q,H0).

4.26. Continuation of 4.25: The cone: Let H0 be the connected component ofH2g that contains x. Let λ : Q ∼−−→ Q(1) be the isomorphism defined by H0 according to2.8, and let C := C(H0, P1) and C∗(H0, P1). Then λ(C) is an open cone in U1(R).

Claim: (a) λ(C) is the cone of all positive definite symmetric biliniear forms on(V/(V ′)⊥)R.

(b) λ(C∗) is the cone of all positive semi-definite symmetric biliniear forms on (V/(V ′)⊥)R,whose degenerate subspace is defined over Q.

Consequence: In general the cone in U1(R)(−1) does depend on the connected com-ponent of X . Indeed in the case under consideration, λ depends on H0 but by λ(C) doesnot.

70

Proof. Fix a choice for√−1 such that λ(1) = 2π

√−1. By 1.11 and 2.8, the pairing

Ψ(., hx(√−1)(.)) on VR is symmetric and positive definite. Hence the associated hermitian

form (v, v′) 7→ Ψ(v, hx(√−1)(v′)) on VC is also positive definite. By 1.3, and since hx(

√−1)

is defined over R, we have hx(√−1)(v) =

√−1 · v for every v ∈ F 0VC. Now let u1 ∈ C be

the imaginary part of x1. By 4.12, it is characterized by the fact that u−11 V 0,0 is defined

over R. Let B be the symmetric form associated to u1 by the identification in 4.25. Forany 0 6= v ∈ VR such that u1v ∈ V 0,0, we get

0 < Ψ(u1v, hx(√−1)(u1v)) = Ψ(u1v,

√−1 · u−1

1 v)

= Ψ(v,√−1 · u−2

1 v)

= Ψ(v,√−1 · (u−2

1 − id)v)

= 2 ·B([v],√−1 · [v])

=1

π· λ B([v], [v]),

where we used u1 = u−11 , and the fact that Ψ is alternating. This shows that B is pos-

itive definite, as desired. Now GL(V/(V ′)⊥)(R) transitively permutes all positive definitesymmetric bilinear forms, whence (a). The assertion (b) follows from the fact that theboundary components between (P,X ) and (P1,X1) correspond bijectively to the subspacesV ′′ ⊂ V ′ defined over Q. q.e.d.

4.27. Example: In the case W/U 6= 1 the situation is somewhat more complicated.Consider the following special case. Let (P,X ) be the mixed Shimura data (P2g,X2g) definedin 2.25. for g = 1. Let (P1,X1) be a proper rational boundary component of (P,X ). By4.25 we can view (P,X ) itself as rational boundary component of (CSp2g,Q,H2g) for g = 2.Thus, for a suitable choice of coordinates, U1 is isomorphic to the space of all symmetic2 × 2-matrices, and Q acts on this space by A 7→ B · A ·t B for B ∈

(∗ ∗0 ∗). The image of

U1 ∩W is the subspace of all matrices of the form(∗ ∗∗ 0

), the image of U the subspace of all

matrices of the form(∗ 0

0 0

). In other coordinates, U1 corresponds to Q3, and Q acts on this

through all matrices of the form st 0 00 s 00 0 s/t

· 1 2a a2

0 1 a0 0 1

for s, t ∈ Gm,Q and a ∈ Ga,Q. We see in particular that the unipotent radical of Q may actthrough quadratic terms on U1.

The cone can be determined as follows. In 4.26 the cone λ(C) was the set of allpositive definite real symmetric 2× 2-matrices. Thus 4.21 (a) implies that in our case thecone λ(C(X 0, P1)) is the set of all matrices of the form

(a bb c

)with a, b ∈ R and c ∈ R>0.

Moreover by 4.21, λ(C∗(X 0, P1)) is the union of λ(C(X 0, P1)) with the set of all matricesof the form

(a 00 0

)with a ∈ R.

71

Chapter 5

Torus embeddings

This chapter contains some results about torus embeddings that will be needed later. In 5.1–5 we begin by recalling standard facts, most of which can be found for instance in [KKMS]ch.I or in [AMRT] ch.I. In 5.7 we derive a density result of concerning the formal completionalong a stratum. In 5.9 we describe the topology of a torus embedding of complex spaces.Starting from 5.10 we study equivariant line bundles on relative torus embeddings. Inparticular we show how they can themselves be described as torus embeddings. The mainresult is the ampleness criterion 5.19, which generalizes [KKMS] ch.I §3 thm. 13 to therelative case. In 5.20–25 we deal with the construction of cone decomposition that satisfythe ampleness condition. The chapter ends with a study of canonical sheaves on torusembeddings in 5.26–27.

5.1. Cone decompositions: Let M be a finite dimensional Q-vector space. A subsetσ ⊂ MR is called a convex rational polyhedral cone if it can be described in one of thefollowing two equivalent ways:

(i) There exists finitely many `i ∈ M∨, i ∈ I (note that the `i are defined over Q,) suchthat σ = x ∈MR | ∀i ∈ I 〈`i, x〉 ≥ 0.(ii) There exists finitely many xi ∈ M , i ∈ I (note that the xi are defined over Q), suchthat σ =

∑i∈I λi · xi | ∀i ∈ I λi ∈ R≥0.

Let σ = x ∈MR | ∀i ∈ I 〈`i, x〉 ≥ 0 be a convex rational polyhedral cone. A face of σis a subset of the form x ∈ σ | ∀i ∈ J 〈`i, x〉 = 0 for some subset J ⊂ I. A face is again aconvex rational polyhedral cone. The interior of σ is the subset σ0 of all points that do lie ina proper face of σ. This is the topological interior of σ within the R-subspace generated byσ, but not in general in MR. These definitions are independant of the chosen representationof σ. Any convex rational polyhedral cone is the disjoint union of the interiors af all itsfaces. Observe that the intersection of any two convex rational polyhedral cones in MR isagain a convex rational polyhedral cone.

For any convex rational polyhedral cone σ in MR let σ := ` ∈M∨R | ∀x ∈ σ〈`, x〉 ≥ 0.This is a convex rational polyhedral cone in M∨R , called the dual cone. Clearly ˇσ = σ. Ifτ ⊂ σ, then σ ⊂ τ . The dual cone σ contains a basis of M∨ if and only if σ does not containa nontrivial linear subspace. Every such convex rational polyhedral cone σ possesses thecone 0 as one of its faces. Note that 0∨ = M∨R .

72

A rational partial polyhedral decomposition of MR is a collection Σ of a convex rationalpolyhedral cones, such that the following conditions hold:

(i) Any face of any σ ∈ Σ is also contained in Σ.

(ii) The intersection of any two σ, τ ∈ Σ is a face both of σ and of τ .

(iii) 0 ∈ Σ.

The conditions (ii) and (iii) imply that 0 is a face of every σ ∈ Σ, so by (i) σ contains anontrivial linear subspace. Conversely if Σ satisfies (i) and (ii), and some σ ∈ Σ possesses nonontrivial linear subspace, then (iii) follows. Let |Σ| :=

⋂σ∈Σ σ. If C ⊂MR is a subset which

contains |Σ|, then we call Σ a rational partial polyhedral decomposition of C. If moreoverC = |Σ|, then we call Σ a (complete) rational polyhedral decomposition of C. If Σ1 and Σ2

are rational polyhedral decompositions, then so is Σ3 := σ1 ∩ σ2 | σ1 ∈ Σ1, σ2 ∈ Σ2. Ifboth Σ1 and Σ2 are finite or complete rational polyhedral decompositions of C, then Σ3 hasthe same property. If |Σ1| = |Σ2|, and every σ ∈ Σ1 is contained in some τ ∈ Σ2, then Σ1

is called refinement of Σ2. Equivalently Σ1 is a refinement of Σ2 if and only if |Σ1| = |Σ2|and every τ ∈ Σ2 is the union of all σ ∈ Σ that are contained in τ .

5.2. Affine torus embeddings: Let k be a field of characteristic zero, and T asplit algebraic torus over k, that is T ∼= (Gm,k)dim(T ). Let X∗(T ) := Hom(T,Gm,k) be itscharacter group and Y∗(T ) := Hom(Gm,k, T ) its cocharacter group. The bilinear map

X∗(T )× Y∗(T )→ Hom(Gm,k,Gm,k) = Z, (χ, λ) 7→ deg(χ λ)

is a perfect pairing. The affine coordinate ring of T is canonically isomorphic to the groupring k[X∗(T )]. Let σ be a convex rational polyhedral cone in Y∗(T )R that does not containa nontrivial linear subspace. Define Tσ := Spec k[σ∩X∗(T )], this is a normal affine varietyover k. Since σ∩X∗(T ) contains a Q-basis of X∗(T )Q, it can easily be shown that it containsa Z-basis of X∗(T ), so Tσ contains T as an open dense subvariety. Moreover the action ofT by multiplication on itself extends uniquely to an action on Tσ.

Call σ smooth (with respect to the lattice Y∗(T ) ⊂ Y∗(T )Q) if the semigroup σ ∩ Y∗(T )can be generated by a subset of a Z-basis of Y∗(T ). The variety Tσ is smooth if and onlyif σ is smooth. As an example, take T = Gm,k and σ = R≥0 ⊂ R = Y∗(Gm,k)R, then oneeasily sees that Tσ ∼= A1

k with the obvious embedding of Gm,k. In general the embeddingT → Tσ is smooth if and only if it is isomorphic to the embedding (Gm,k)dim(T ) → (A1

k)r ×

(Gm,k)dim(T )−r for some integer r. Then the complement Tσ r T is a union of smoothdivisors with only normal crossings.

There is a unique closed T -orbit Tσ in Tσ. Its ideal in k[σ ∩ X∗(T )] is generated byχ ∈ X∗(T ) | ∀λ ∈ σ0 〈λ, χ〉 > 0. thus it is canonically isomorphic to the torus withcharacter group X∗(T ) ∩ (R · σ)∨ = χ ∈ X∗(T ) | ∀λ ∈ σ 〈λ, χ〉 = 0 and cocharactergroup (Y∗(T ) + R · σ)/R · σ. The inclusion of semigroups X∗(T ) ∩ (R · σ)∨ → X∗(T ) ∩ σdefines a morphism πσ : Tσ → Tσ, the canonical projection. In particular Tσ is canonicallyisomorphic to a quotient of T .

If τ is a face of σ, then Tτ is an open subvariety of Tσ. The T -orbits in Tσ are justthe Tτ for the faces τ of σ, and the unique open orbit T ⊂ Tσ equals T0. The closure of

Tτ in Tσ is the union of all Tρ such that τ is a face of ρ and ρ a face of σ. By the above

73

Tτ is the torus with cocharacter group (Y∗(T ) + R · τ)/R · τ , so let us consider the convexrational polyhedral cone στ := σmodR · τ . It defines a torus embedding (Tτ )στ of Tτ , andone easily verifies that Tτ,στ is isomorphic to the unique reduced closure subscheme of Tσwith support the closure of Tτ in Tσ. Moreover the canonical projection Tτ → Tτ extendsuniquely to a T -equivariant morphism Tσ → (Tτ )στ .

5.3.General torus embeddings: Let Σ be a rational polyhedral decomposition ofY∗(T )R. Then the Tσ for σ ∈ Σ can be glued together canonically to a normal scheme TΣ.This contains every Tσ as an open dense subscheme. It is of finite type over k if and onlyif Σ is finite. It is proper if and only if Σ is finite and |Σ| = Y∗(T )R. It is smooth andthe complement is a union of smooth divisors with only normal crossings, if and only if allσ ∈ Σ are smooth. Then we call Σ smooth (with respect to the lattice Y∗(T ) ⊂ Y∗(T )Q).

The action of T extends to TΣ, and the T -orbits are in one-to-one correspondence withthe σ ∈ Σ. Thus we have a bijection between all T -variant subsets S ⊂ TΣ and all subsetsΣ1 of Σ. Clearly S is open in TΣ if and only if every face of a cone in Σ1 is also in Σ1, orequivalently in

⋃σ∈Σ1

σ0 is relatively closed in |Σ|. Dually it follows that S is closed in TΣ

if and only if⋃σ∈Σ1

σ0 is relatively open in |Σ|.For any σ ∈ Σ the closure of Tσ in TΣ is the union of all Tτ such that τ ∈ Σ and σ is a

face of τ . Fix σ, then

Σσ := τ modR · σ | τ ∈ Σ such that σ is a face of τ.

is a rational partial polyhedral decomposition of Y∗(Tσ)R = (Y∗(T )+R ·σ)/R ·σ. The uniquereduced closed subscheme of TΣ with support the closure of Tσ is canonically isomorphicto (Tσ)Σσ . Let

Σ1 := ρ ∈ Σ | ∃τ ∈ Σ such that σ and ρ are faces of τ.

Then the canonical projection extends to a morphism πσ : TΣ1 → (Tσ)Σσ.

5.4. Functoriality and isogenies: Let φ : T1 → T2 be a homomorphism of tori.By λ 7→ φ λ it induces a linear map φ∗ : Y∗(T1) → Y∗(T2). Let Σ1 and Σ2 be rationalpartial polyhedral decompositions of Y∗(T1)R, resp. Y∗(T2)R. Then φ extends to a morphismT1,Σ1 → T2,Σ2 if and only if for every σ1 ∈ Σ1 there exists σ2 ∈ Σ2 such that φ∗(σ1) ⊂ σ2.

Let T be a torus and Γ a discrete group acting from the left hand side on T by groupautomorphisms. Then Γ acts from the left hand side on Y∗(T ), and we call a rational partialpolyhedral decomposition Γ-invariant if σ ∈ Σ and γ ∈ Γ implies γ∗(σ) ∈ Σ. The action ofΓ on T extends to TΣ if and only if Σ is Γ-invariant.

If φ : T1 → T2 is an isogeny, then φ∗ induces an isomorphism Y∗(T1)R → Y∗(T2)R.Let Σ be a rational partial polyhedral decomposition of this vector space, then we get amorphism T1,Σ → T2,Σ. The kernel of φ acts by translations on T1, and T2 is just thequotient of T1 by this action. Since T1 → T1,Σ is an equivariant embedding, the action ofker(φ) extends to T1,Σ, and one easily verifies (on the affine rings of all Tσ) that T2,Σ is thescheme-theoretic quotient of T1,Σ by ker(φ). This shows that torus embeddings commutewith isogenies.

74

5.5. The relative case: Let T be as before, S a scheme over k, and X → S aT -torsor. Let Σ be a rational partial polyhedral decomposition of Y∗(T )R. Locally for theZariski topology on S there exist sections in X. Every local section Σ defines a trivializationof X, hence a T -equivariant embedding X → XΣ over S. Since the difference of anytwo local sections is a unique morphism to T , and the embeddings are T -equivariant,we get a canonical isomorphism of the two respective embeddings. Thus there exists upto isomorphism a unique global T -equivariant embedding X → XΣ that is locally on Sisomorphic to TΣ × S.

All of the above notions are constructions, in particular the canonical projection, extendto the relative case. We have functoriality even in the following general situation. LetXi → Si be a Ti-torsor and Σi a rational partial polyhedral decomposition of Y∗(Ti)R fori = 1, 2. Let φ : T1 → T2 be a homomorphism, and suppose we are given an equivariantcommutative diagram of morphisms

X1 −→ X2

↓ ↓S1 −→ S2

Suppose furthermore that for every σ1 ∈ Σ1 there exists σ2 ∈ Σ2 such that φ∗(σ1) ⊂ σ2.Then the morphism X1 → X2 extends uniquely to a morphism X1,Σ1 → X2,Σ2 . In fact,this is clear from the construction if S1 = S2 or if T1 = T2. In general consider the T2-torsor X2 ×S2 S1 → S1, then the composite X1,Σ1 → (X2 ×S2 S1)Σ2 → X2,Σ2 is the desiredmorphism.

The same constructions go through if T itself is only given as a locally constant familyof k-split tori over S, and the rational partial polyhedral decomposition is given fibrewiseand locally constant over S. Let us also mention another generalization, where T is nota split torus. In this case Y∗(T ) := Hom(Gm,k, Tk), where k is the algebraic closure of k.

The Galois group Gal(k/k) operates on this through a finite quotient, and by functorialityall construtions descend to k if and only if the rational partial polyhedral decomposition isinvariant under Gal(k/k).

5.6. Completion along an orbit: For simplicity assume that σ is of top dimensionin 5.2, i.e. that R · σ = Y∗(T )R. Writing Λ+ := σ ∩ X∗(T ), the coordinate ring of Tσ isk[Λ+], and the ideal of the unique closed orbit is generated by all nonzero elements of Λ+.Thus the completion of k[Λ+] with respect to this ideal can be described as

k[[Λ+]] = lim←−

k[Λ+]/A · k[Λ+],

where A runs through all subsets A ⊂ Λ+ such that A+Λ+ = Λ+ and Λ+rA is finite. Theformal completion Tσ of Tσ along the closed orbit is just the formal spectrum of k[[Λ+]].For relative torus embeddings, and hence in particular if σ is not of top dimension, we getsimilar formulas.

Let T ′ → T ′σ′ be another affine torus embedding, with σ′ of top dimension. By 5.4, ahomomorphism φ : T ′ → T extends to a morphism T ′σ′ → Tσ if and only if φ∗(σ

′) ⊂ σ.Clearly, the closed orbit T ′σ′ is mapped to the closed orbit Tσ if and only if φ∗(σ

′0) ⊂ σ0.

75

In this case we get a morphism of formal schemes T ′σ′ → Tσ, corresponding to a continuoushomomorphism k[[Λ+]]→ k[[Λ′+]].

By [EGA-I] 5.4.2, a morphism of schemes f : X → Y is called scheme-theoreticallydominant, if the associated homomorphism OY → f∗(OX) is injective. We extend thisnotion to formal schemes in the obvious way: namely, a morphism f : X → Y of formalschemes is scheme-theoretically dominant if the associated homomorphism OY → f∗(OX )is injective.

5.7. Proposition: Let T → Tσ be an affine torus embedding, with σ of top dimension.Let X → S be a T -torsor, and denote by Xσ the formal completion of Xσ along theclosed orbit. For every i in some index set I consider the same situation again, i.e. aTi-torsor Xi → Si, σi of top dimension, and the formal completion Xi,σi . For every i ∈ I,let φi : Ti → T be a homomorphism, and (Ψi, ψi : (Xi, Si) → (X,S) a Tiequivariantmorphism. We assume:

(i) φi,∗(σ0i ) ⊂ σ0 for every i ∈ I. In particular, by 5.6 we get induced morphisms Ψi :

Xi,σi → Xσ.

(ii)⋃i∈I φi,∗(σ

0i ) is dense in σ0 (in the usual topology).

(iii) Every ψi : Si → S is scheme-theoretically dominant.

Then the morphism ∐i∈I

Ψi :∐i∈I

Xi,σi −→ Xσ

is scheme-theoretically dominant.

Proof. Since the assertion is local in S, we may assume that S is affine, say S =Spec(R), and that X = T × S = Spec(R[X∗(T )]). We may also replace each Xi bya scheme over Xi. In particular, we may replace Si by open affine coverings, so thatXi∼= Ti × Si = Spec(Ri[X

∗(Ti)]). By assumption, ψi corresponds to an injection ψ∗i :R → Ri, and by equivariance, Ψ∗i : R[X∗(T )] → Ri[X

∗(Ti)] extends ψ∗i and is givenby X∗(T ) 3 χ 7→ φ∗i · ξi(χ) for some group homomorphism ξi : X∗(T ) → R∗i . WritingΛ+ := σi ∩X∗(T ) as in 5.6, and Λi,+ = σi ∩X∗(Ti), we have to prove the injectivity of thehomomorphism ∏

i∈IΨ∗i : R[[Λ+]] −→

∏i∈I

Ri[[Λi,+]].

For every i ∈ I denote by σi,j all 1-dimensional convex rational polyhedral cones in Y∗(Ti)Rwith σ0

i,j ⊂ σ0i . For all (i, j), let Ti,j ⊂ Ti be the unique subtorus with Y∗(Ti,j)R = R · σi,j .

Since the union⋃j σ

0i,j is dense in σ0

i , the pairs (Ti,j , σi,j) together with the morphisms(Ψi|Ti,j × Si, ψi) satisfy the same assumptions. Thus we may without loss of generalityassume that all Ti have dimension 1, or even that all Ti = Gm,k and σi = R≥0, so Λi,+ = Z≥0.In the latter case the set of all φ∗i can be viewed as a subset Φ ⊂ σ0 ∩ Y∗(T ) such thatR>0 · Φ is dense in σ0.

By 5.6, R[[Λ+]] = lim←−

R[Λ+]/A ·R[Λ+], where A runs through all subsets A ⊂ Λ+ suchthat A+ Λ+ = Λ+ and Λ+ r A is finite. Let A be the collection of all such subsets A, forwhich the kernel of the above homomorphism is contained in the ideal A ·R[[Λ+]]. We haveto prove that A is cofinal, i.e. that

⋂A∈AA = ∅. Fix any i0 ∈ I, then for any x > 0 there

76

are only finitely many λ ∈ Λ+ with φ∗i0 ≤ x. Thus it suffices to prove: For every A ∈ Athere exists a0 ∈ A with φ∗i0(a0) ≤ φ∗i0(a) for all a ∈ A, and such that A′ := A r a0 ∈ A.Note that here the assumption A ∈ A means that the kernel of the homomorphism

R[[Λ+]]/A′ ·R[[Λ+]] −→∏i∈I

Ri[[Z≥0]]/φ∗i (A′) ·Ri[[Z≥0]]

is contained in A · R[[Λ+]]/A′ · R[[Λ+]] ∼= [a0] · R. Since Ψ∗i (a0) = φ∗i (a0) · ξi(a0) withξi(a0) ∈ R×i , we certainly have A′ ∈ A if there exists i ∈ I such that φ∗i (a0) < φ∗i (a) for alla0 6= a ∈ A. In other words, it suffices to prove: For all nonempty subsets A ⊂ Λ+ thereexists a0 ∈ A and φ∗i ∈ Φ, such that φ∗i (a0) ≤ φ∗i0(a) and φ∗i (a0) < φ∗i (a) for all a0 6= a ∈ A.

To show this, let x := min(φ∗i0(A)) and K := λ ∈ σ | φ∗i0(λ) = x, then A ∩ K is anonempty finite set. We take a0 to be any extreme point of the convex closure of A ∩K,this is necessarily a point in A. By construction, σ = R≥0 · K, and there exists ε ∈ Q>0

so that A r K ⊂ R≥1+ε · K. Thus a0 is also an extreme point of the convex closure ofA ∪ R≥1+ε ·K. If we let τ ⊂ X∗(T )R the convex rational polyhedral cone generated by

λ− a0 | λ ∈ A ∪ (1 + ε) ·K,

this means that τ does not contain a nontrivial linear subspace. Now by constructuionσ ⊂ τ , whence τ ⊂ σ. But τ0 is open in Y∗(T )R, so by assumption Φ ∩ τ0 is non-empty.Any φ∗i in this set satisfies the desired condition. q.e.d.

5.8. Torus embeddings of complex spaces: Let T be a torus over C, S a complexspace (see [BPV] ch.I §8), and X → S an analytic T (C)-torsor. Again X → S is locallytrivial, so for every rational partial polyhedral decomposition of Y∗(T )R the constructionof 5.5 yields a canonical T (C)-equivariant embedding X → XΣ of complex spaces, that islocally on S isomorphic to T (C) × S → TΣ(C) × S. If S is normal, then XΣ is normal.Again all the other notions and constructions make sense in this case.

Note the obvious canonical isomorphism T (C) ∼= Y∗(T ) ⊗ C×. The continious homo-morphism

ord = ordT : T (C) = Y∗(T )⊗ C× → Y∗(T )⊗ R, λ⊗ z 7→ λ⊗ (− log |z|)

(note the minus-sign) identifies Y∗(T )R with the quotient of T (C) by its unique maximalcompact subgroup. Clearly ordT is functorial in T . More generally let X → S be ananalytic T (C)-torsor over a complex space S. Then the quotient of X by the maximalcompact subgroup of T (C) is a real analytic Y∗(T )R-torsor on S. It can be C∞-trivialized(using a partition of 1), and any trivialization induces a projection X → Y∗(T )R that weagain denote by ord = ordX . Since any two trivializations differ by a C∞-section in Y∗(T )R,which can be lifted to T (C), this map ordX is unique up to translation by C∞-sections inT (C).

Let Σ be a rational partial polyhedral decomposition of Y∗(T )R. The following propo-sition determines the topology on XΣ.

77

5.9. Proposition: For every σ ∈ Σ, and all subsets V ⊂ Y∗(T )R and U ⊂ Xσ considerthe following subset of XΣ: ∐

τ face of σ

πτ (ord−1(V + σ) ∩ π−1σ (U)).

(a) For V and U open, this set is open. Moreover these sets form a base for the topologyon XΣ.

(b) For V and U compact, this set is compact.

Proof. For fixed σ, apart from the openness, resp. the compactness, we have to showthat the sets in (a) contain a base for the neighborhoods in XΣ of every point in Xσ.First we reduce the assertions to a special case. The set in question is contained in Xσ,which is open in XΣ, so whithout loss of generality we may assume XΣ = Xσ. Let T ′

be the subtorus of T whose cocharacter group is R · σ ∩ Y∗(T ). Then via the canonicalprojection πσ : Xσ → Tσ the embedding X → Xσ can be viewed as the relative torusembedding associated to T ′ → T ′σ. After replacing T by T ′ we may therefore assume thatR · σ = Y∗(T )R. Now all assertions are local in S, so we may also assume that the T -torsorX → S is trivial. Moreover the topology on XΣ is invariant under translation by C∞-sections S → T (C), so we may assume that the map ordX comes from the correspondingmap on T (C). Then the set in question is the product of a subset of Tσ(C) with the subsetU ⊂ Xσ = S. Since Xσ

∼= Tσ(C)×S carries the product topology, this shows that it sufficesto treat the case X = T . Then by R · σ = Y∗(T )R the stratum Tσ(C) is just a point, so wehave to show that the subset ∐

τ face of σ

πτ (ord−1(V + σ)).

of Tσ(C) is open if V is open, compact if V is compact, and for V open they form a basefor the neighborhoods of Tσ(C) in Tσ(C).

For V open we clearly have V +σ = V +σ0 =⋃λ∈V λ+σ0, hence it suffices to prove the

assertions for V = V + σ = λ+ σ0 for all λ ∈ Y∗(T )R. If V is compact, the T -equivarianceof the torus embedding yields∐

τ face of σ

πτ (ord−1(V + σ)) = ord−1(V ) ·∐

τ face of σ

πτ (ord−1(σ)),

where ord−1(V ) is a compact subset of T (C). Thus for the compactness we may assumeV = V + σ = λ+ σ for some λ ∈ Y∗(T )R.

Choose nonzero generators X1, . . . , Xn of the semigroup σ ∩ X∗(T ). They define aclosed embedding Tσ = SpecC[σ ∩X∗(T )] → AnC that maps Tσ to 0. The relation ˇσ = σimplies that for all λ ∈ Y∗(T )R

λ+ σ = y ∈ Y∗(T )R | ∀i = 1, . . . , n 〈Xi, y〉 ≥ 〈Xi, λ〉.

Henceord−1(λ+ σ) = t ∈ T (C) | ∀i = 1, . . . , n |Xi(t)| ≤ e−〈Xi,λ〉,

78

and for every face τ of σ

πτ (ord−1(λ+ σ)) = t ∈ Tσ(C) | ∀i = 1, . . . , n |Xi(t)| ≤ e−〈Xi,λ〉,

(note that in the last formula some Xi(t) may be zero). Thus the set in question is theintersection of Tσ(C) with the compact subset

ξ ∈ An(C) | ∀i = 1, . . . , n |ξi| ≤ e−〈Xi,λ〉

of An(C) hence itself compact. This shows (b). For λ + σ0 instead of λ + σ above wehave strict in equalities, so by the same argument the set is open. Finally since σ does notcontain a nontrivial linear subspace, λ can be chosen to make all 〈Xi, λ〉 arbitrarily large.Therefore the sets form a base of neighborhoods of 0 in An(C), whence the correspondingassertion for Tσ(C). q.e.d.

5.10. Line bundles as torus embeddings: Let S be a scheme of finite type overk, and π : X → S a Gm-torsor. The cone σ0 := R≥0 ⊂ Y∗(Gm)R defines a relative torusembeddding X → Xσ0 over S. Clearly Xσ0 → S is just the line bundle associated to thetorsor X → S. Denote Xσ0 by L and the projection L → S again by π. Let L be theinvertible sheaf on S of all local sections in L. Then π∗(OL) is canonically isomorphic tothe sheaf of OS-algebras

⊕n≥0 L⊗(−n). Indeed, for any open U ⊂ S we have canonically

HomOU−algebra(π∗(OL),OU ) ∼= MorS(U,L) ∼= Γ(U,L) ∼=∼= HomOU (L⊗−1,OU ) ∼= HomOU−algebra(

⊕n≥0

L⊗(−n),OU ).

For i = 1, . . . , n let ai ∈ Z and Xi → S be a Gm-torsor. Consider the homomorphism

φ : Gnm → Gm, (t1, . . . , tn) 7→ ta11 · . . . · t

ann

The action of Gnm on(X1 ×S . . .×S Xn)/ ker(φ)

factors through φ, and endows this quotient with the structure of a Gm-torsor. It is easy tosee that this Gm-torsor is canonically isomorphic to the tensor product X⊗a1

1 ⊗ . . .⊗X⊗ann .In fact, let xi be local sections of Xi, then the map

Xi ×S . . .×S Xn → X⊗a11 ⊗ . . .⊗X⊗ann , (x1, . . . , xn) 7→ xa1

1 ⊗ . . .⊗ xann

is φ-equivariant and factors through (X1×S . . .×SXn)/ ker(φ). This yields a convenient wayto describe tensor products of Gm-torsors. In particular the inverse of a Gm-torsor is justthe same scheme X → S with the opposite Gm-action. For any integer n ≥ 1 X⊗n ∼= X/µnwith the Gm-action through the quotient Gm ı Gm/µn ∼−−→ Gm, t 7→ tn. Clearly the sameresults hold if S is a complex space, T a torus over C and X → S a T (C)-torsor.

5.11. Equivariant line bundles on torus embeddings: Let 1 → Gm → T ′ π−−→T → 1 be a short exact sequence of split tori over k. Then T ′ is a Gm-torsor on T ,

79

equivariant under the action of T ′. Note that an equivariant trivialization of this torsoris equivalent to a splitting of the above short exact sequence. Let Σ be a rational partialpolyhedral decomposition of Y∗(T )R. We want to study how the Gm-torsor T ′ → T can beextended T ′-equivariantly to TΣ.

Since Tσ is affine for every σ ∈ Σ, the extension must be trivializable over Tσ. Thusthere exists a splitting eσ : T → T ′ such that the extension of the Gm-torsor to Tσ isisomorphic to the torus embedding T ′eσ∗ (σ). Note that eσ is not uniquely determined by

eσ∗(σ) unless R · σ = Y∗(T )R. Globally the extension must be the torus embedding T ′Σ+ forsome rational partial polyhedral decomposition Σ+ of Y∗(T

′)R. The necessary and sufficientcondition on Σ+ is that π∗ induces a bijection Σ+ → Σ, σ′ 7→ π∗(σ

′), such that for everyσ′ ∈ Σ+ there exists a splitting eσ : T → T ′ such that σ′ = eσ∗ π∗(σ′).

Let Σ+ be such a decomposition. Consider the convex rational polyhedral cone σ0 :=R≥0 ⊂ Y∗(Gm)R → Y∗(T

′)R, and define Σ′ := σ′, σ′ + σ0 | σ′ ∈ Σ+. One easily verifieslocally that T ′Σ+ → T ′Σ′ is the relative torus embedding over TΣ associated to the torus Gmand the cone σ0. Thus by 5.10 T ′Σ′ is the line bundle on TΣ associated to the Gm-torsorT ′Σ∗ .

Let S be a scheme of finite type over k, X ′ → S a T ′-torsor and X : +X ′/Gm theassociated T -torsor on S. Let X ′Σ′ → XΣ be the associated relative torus embeddings, thisis a line bundle. Any splitting e : T → T ′ determines an isomorphism Gm ∼= T ′/e(T ),through which X ′/e(T ) becomes a Gm-torsor. Denote by L the invertible sheaf on XΣ

associated to X ′Σ′ . We keep these notations for the rest of this chapter. The followingproposition determines the direct image of L under the projection f : XΣ → S.

5.12. Proposition: (a) In the situation of 5.11 f∗L is canonically isomorphic to thedirect sum of the invertible sheaves associated to the line bundles

(X ′σ0)/e(T )→ S

for all splittings e : T → T ′ that extend to a morphism TΣ → T ′Σ.

(b) The same assertion also holds if T is a torus over C, X → S an analytic T (C)-torsorover a complex space S, and Σ a finite and complete rational polyhedral decomposition ofY∗(T )R.

Proof. For any splitting e : TΣ → T ′Σ′ let T ′ → T × (T ′/e(T )) be the canonical mapt′ 7→ (π(t′), [t′]). It is clearly an isomorphism, and its inverse extends to a morphism

T × (T ′/e(T )) ∼←−− T ′∩↓

∩↓

TΣ × (T ′σ0/e(T )) −→ T ′Σ′

Indeed, if we decompose T ′ ∼= T × Gm via the splitting e, the torus embedding on thebottom left is covered by affines Tσ ×A1 = T ′σ+σ0

. Since by assumption σ ⊂ eσ(σ) + σ0, wealso have σ + σ0 ⊂ eσ(σ) + σ0, hence the morphism extends to T ′σ+σ0

→ T ′eσ(σ)+σ0⊂ T ′Σ′ ,

as desired. This implies that the analogous morphism of the torsors also extends:

X × (X ′/e(T )) ∼←−− X ′∩↓

∩↓

XΣ × (X ′σ0/e(T )) −→ X ′Σ′

80

since it does so locally. Thus for any open U ⊂ S every local section y : U → (X ′σ0)/e(T )

yields a sectionXΣ ×S U id×y−−−−→ XΣ × (X ′σ0

/e(T )) −→ X ′Σ′ ,

i.e. an element of Γ(XΣ ×S U,L) = Γ(U, f∗L). This defines a canonical homomorphismfrom the invertible sheaf associated to (X ′σ0

/e(T )) to f∗L. Taking the direct sum over allsplittings e : TΣ → T ′Σ′ we get the desired canonical homomorphism.

It remains to show that in both cases (a) and (b) this is an isomorphism. Since thisis a local property, we may assume that X ′ is a trivial torsor, which then reduces to thecase X ′ = T ′, resp. X ′ = T ′(C). In case (b) TΣ(C) is compact by assumption, hence theholomorphic sections of L coincide with the algebraic sections. Thus for the rest of theproof we may stay in the algebraic category.

Now Γ(TΣ,L) is just the intersection of all Γ(Tσ,L) inside Γ(T,L). Since likewise anysplitting T → T ′ extends to TΣ → T ′Σ′ if and only if it extends to Tσ → T ′eσ(σ)+σ0

forevery σ ∈ Σ, we are reduced to the case TΣ = Tσ. After using the splitting eσ to writeT ′ ∼= Gm × T , we may assume that T ′ = Gm × T and the embedding is T ′σ0×σ = A1 × Tσ.Then we have canonical isomorphisms

Γ(Tσ,L) ∼= MorTσ(Tσ,A1 × Tσ ∼= Mor(Tσ,A1) ∼= Γ(Tσ,OTσ) ∼= k[X∗(T ) ∩ σ].

Any splitting e : T → Gm × T is of the form t 7→ (χ(t), t) for some χ ∈ X∗(T ), soe : Y∗(T )→ Z⊕Y∗(T ) is given by λ 7→ (〈χ, λ〉, λ). It extends to a morphism Tσ → A1×Tσif and only if e∗(σ) ⊂ σ0 × σ, i.e. if 〈χ, λ〉 ≥ 0 for all λ ∈ σ, which is equivalent to χ ∈ σ.Hence the map under consideration goes from

⊕χ∈X∗(T )∩σ k · χ = k[X∗(T ) ∩ σ] to itself,

and it suffices to show that it is the identity map.

The section associated to e is determined by the following diagram

T (id,1)−−−−−→ T × (Gm × T )/e(T ) ∼←−− Gm × T pr1−−−→ Gm

∩↓

∩↓

∩↓

∩↓

Tσ (id,1)−−−−−→ Tσ × (A1 × T )/e(T ) −→ A1 × Tσ pr1−−−→ A1

On the level of cocharacter groups this diagram reads

Y∗(T ) (id,0)−−−−−→ Y∗(T )⊕ (Z⊕ Y∗(T ))/e∗(Y∗(T )) ∼←−− Z⊕ Y∗(T )

pr1−−−→ Zλ 7−→ (λ, [(0, 0)]) = (λ, [(〈χ, λ〉, λ)]) (〈χ, λ〉, λ) 7−→ 〈χ, λ〉

This shows that the image of the section associated to e is indeed the cocharacter χ, asdesired. q.e.d.

5.13. Example: In the situation of 5.11 assume that T ′ = Gm × T and Σ′ =0 × σ, σ0 × σ | σ ∈ Σ, in which case L is just the structure sheaf of XΣ. Then f∗L isthe direct sum of the invertible sheaves associated to the Gm-torsors X/ ker(χ) → S, forall χ ∈ Hom(T,Gm) = X∗(T ) that take nonpositive values on |Σ|. In fact, any splittinge : T → T ′ is of the form e(t) = (χ−1(t), t) for χ ∈ X∗(T ), and it extends to TΣ → T ′Σ′ =A1 × TΣ if and only if χ−1 is nonnegative on |Σ|. By 5.12 (a), f∗OX is the direct sum ofthe invertible sheaves associated to the line bundles (X ′σ0

/e(T ) → S for all such χ. But

81

(X ′σ0/e(T ) = (A1×X)/e(T ) ∼= X/ ker(χ), with the Gm-structure given by the commutative

diagramGm × T −→ (Gm × T )/e(T )

↑∪

o‖T

χ−−→ GmIn particular, if TΣ = Tσ is an affine torus embedding, then the χ in question are preciselythose with −χ ∈ σ. Assume moreover that σ is of top dimension, and denote by f : Xσ → Sthe formal completion of Xσ along the closed orbit, as in 5.7. By the description in 5.6,f∗OXσ is canonically isomorphic to the direct product of the above invertible sheaves.

5.14. Proposition: In the situation of 5.12 (a) or (b) assume that Σ is a finite andcomplete rational polyhedral decompostion of Y∗(T )R, and that Y∗(T )Rr |Σ′| is convex. Forevery σ ∈ Σ with R · σ = Y∗(T )R let eσ : T → T ′ be the splitting defined in 5.11, and Mσ

the invertible sheaf on S associated to the line bundle (X ′σ0/eσ(T )→ S.

(a) For every such σ Mσ is a direct summand of f∗L as in 5.12.

(b) For every n ≥ 1 and every direct summandM of f∗(L⊗n) according to 5.12 there existintegers b ≥ 1 and aσ ≥ 0, for all σ as above, such that b · n =

∑σ aσ and

M⊗b ∼=⊗M⊗aσσ .

Remark. In general b cannot be chosen equal to 1, although it can be bounded in-dependently of M and n. This is related to the fact that on an arbitrary (non-smooth)complete torus embedding there exist ample invertible sheaves that are not very ample.

Proof. After choosing a splitting we may assume that T ′ = Gm × T . For every σof top dimension write eσ = (χσ, id) for χσ ∈ X∗(T ). Recall (5.11) that for such σ χσis uniquely determined by σ and Σ′. Consider arbitrary aσ ∈ Z. By 5.10

⊗σM⊗aσσ is

the invertible sheaf associated to the Gm-torsor (X ′ ×S . . . ×S X ′)/ ker(φ), where φ is thecomposite homomorphism

ΠσT′ ı ΠσT

′/eσ(T ) ∼←−− ΠσGm ı Gm,

((xσ, tσ)) 7→ ([(xσ, tσ)]), ([(xσ, 1)])←p (xσ) 7→ Πxaσσ .

An easy calculation shows that the composite homomorphism

Gm × T = T ′ diag−−−−→ ΠσT

′ φı Gm

is given by ψ : (x, t) 7→ xa · χ∗(t)−1, where χ∗ :=∑

σ aσ · χσ and a :=∑

σ aσ. This showsthat the composite map

X ′ diag−−−−→ X ′ ×S . . .×S X ′ ı (X ′ ×S . . .×s X ′)/ ker(φ)

induces a T ′-equivariant isomorphism

X ′/ ker(ψ) ∼−−→ (X ′ ×S . . .×S X ′)/ ker(φ).

Thus⊗

σM⊗aσσ is the invertible sheaf associated to the Gm-torsor X ′/ ker(ψ)→ S.

82

By 5.10 L⊗n is the invertible sheaf associated to the line bundle X ′Σ′/µn → XΣ. Sincecanonically Y∗(T

′/µn) = Y∗((Gm/µn)× T ) = 1n · Z⊕ Y∗(T ), every splitting e : T → T ′/µn

is on the cocharacter group given as (χ, id) for some χ ∈ 1n · X

∗(T ). An easy calculationshows that the composite homomorphism

T ′ = Gm × T ı ((Gm/µn)× T )/e(T ) ∼←−− Gm

is given by ξ : (x, t) 7→ xn · (χn)(t)−1. Thus M is the invertible sheaf associated to theGm-torsor X ′/ ker(ξ) → S for some χ ∈ 1

n ·X∗(T ). For any integer b it follows that M⊗b

is the invertible sheaf associated to the Gm-torsor X ′/ ker(ξb) → S. Thus for (b) we haveto prove that for any χ ∈ 1

n ·X∗(T ) such that the section (χ, id) : T → T ′/µn extends to a

morphism T − Σ→ T ′Σ′/µn, there exist integers b ≥ 1 and aσ ≥ 0 such that b · n =∑

σ aσand ψ = ξb. This last equality is equivalent to b · n · χ =

∑aσ · χσ. Let ασ := aσ

b·n for allσ, then these must be nonnegative rational numbers with

∑ασ = 1 and χ =

∑ασ · χσ.

Since we can make b arbitrarily large, it suffices to find such ασ ∈ Q≥0. An easy argumentshows that if such elements exist in R≥0, then one can also find them in Q≥0. Thus we arereduced to the assertion: χ lies in the convex closure of the χσ.

Let us analyze the condition that a splitting extends to TΣ. Let f : Y∗(T )R → R be theunique function such that for all σ ∈ Σ and all λ ∈ σ eσ∗(λ) = (f(λ), λ). This function iscontinious, coincides with χσ for each σ of top dimension, and has integer values on Y∗(T ).Moreover by definition

|Σ′| = (f(λ) + α, λ) | λ ∈ Y∗(T )R, α ∈ R≥0,

hence the convexity of Y∗(T )R r |Σ′| is equivalent to the convexity of f (in the sense of[KKMS] footnote p.27), i.e. for all λ, λ′ ∈ Y∗(T )R and α ∈ [0, 1]f(αλ+ (1−α)λ′ ≥ αf(λ) =(1− α)f(λ′). Consider the set

K := x ∈ X∗(T )R | ∀λ ∈ Y∗(T )R〈x, λ〉 ≥ f(λ).

Let χ ∈ 1n ·X

∗(T ), then by 5.4 the splitting (χ, id) : T → T ′/µn extends to a morphismTΣ → T ′Σ′/µn if and only if for all σ ∈ Σ (χ, id)(σ) ⊂ eσ∗(σ) + σ0. But this is equivalentto ∀λ ∈ Y∗(T )R 〈χ, λ〉 ≥ f(λ), so the splitting extends if and only if χ ∈ K. Thus theassertion (a) is equivalent to χσ ∈ K for all σ ∈ Σ of top dimension, and (b) is equivalentto the statement: Every χ ∈ K ∩ 1

n · X∗(T ) lies in the convex closure of these χσ. Both

are implied by the following lemma. Observe that up to now we have not used any of theassumptions on Σ′. q.e.d.

5.15. Lemma: Let K ⊂ X∗(T )R and χσ ∈ X∗(T ) for all σ ∈ Σ with R · σ = Y∗(T )Ras in the proof of 5.14. Then K is the convex closure of these χσ.

Proof. Let us first check that the χσ lie in K. For any λ ∈ Y∗(T )R we have to showthat 〈χσ, λ〉 ≥ f(λ). Choose any λ′ ∈ σ0, then σ0 is open in Y∗(T )R, hence there exists0 < α < 1 such that αλ+ (1− α)λ′ ∈ σ0. The convexity of f implies f(αλ+ (1− α)λ′) ≥α ·f(λ)+(1−α) ·f(λ′). But f coincides with 〈χσ〉 on σ, so it follows 〈χσ, αλ+(1−α)λ′〉 ≥α · f(λ) + (1− α) · 〈χσ, λ′〉. This implies 〈χσ, λ〉 ≥ f(λ), as desired.

83

Next K is an intersection of closed half spaces, hence it is closed and convex. We claimthat K is compact. To prove this it suffices to show that it is bounded in any fixed euclideanmetric on X∗(T )R. Assume it is not bounded, and fix x ∈ K. For every α ≥ 0 let Bα(x) bethe open ball of radius α around x. Let S := ∂B1(x), then for every α ≥ 1 the intersectionof S with the convex closure of x ∪ (K r Bα(x)) is a nonempty closed subset, whichdecreases as α increases. By the compactness of S their intersection is nonempty, hence Kcontains some half-line x + R≥0 · x′ for x′ ∈ S. Let λ ∈ Y∗(T )R be such that 〈x′, λ〉 < 0,then it follows that 〈x + αx′, λ〉 ≥ f(λ) for all α ∈ R≥0, which is clearly a contradiction.Thus K is compact.

Recall that an element x of a closed convex subset A of a finite dimensional R-vectorspace is called an extreme point of A if it cannot be written in the form 1

2 · (x′ + x′′) with

x′, x′′ ∈ A and x′ 6= x 6= x′′. By the Krein-Milman theorem every compact convex set Ais the convex closure of its extreme points. Thus in our case it suffices to prove that theextreme points of K are just the χσ.

Fix an extreme point x0 ∈ K and let

A := λ ∈ Y∗(T )R | 〈x0, λ〉 = f(λ).

This is a convex subset. Indeed, let λ, λ′ ∈ A, 0 ≤ α ≤ 1 and λ′′ := αλ + (1 − α)λ′.Then the sequence of inequalities f(λ′′) ≤ 〈x0, λ

′′〉 = α · 〈x0, λ〉 + (1 − α) · 〈x0, λ′〉 =

α · f(λ) + (1 − α) · f(λ′) ≤ f(λ′′) shows that f(λ′′) = 〈x0, λ′′〉, hence λ′′ ∈ A, as desired.

Clearly A∩ σ is a convex rational polyhedral cone for every σ ∈ Σ. Since A is convex, it isitself a convex rational polyhedral cone.

We want to show that R · A = Y∗(T )R. Let τi ⊂ Σ be the finite subset of all τ withdim(R · τ) = 1. Choose λi ∈ τ0

i for every i, then we have K = x ∈ X∗(T )R | ∀i〈x, λi〉 ≥f(λi). Indeed, every λ ∈ Y∗(T )R lies in some σ ∈ Σ, hence is a nonnegative linearcombination of the λi such that τi is the face of σ, and f is linear on σ. Let I be the set ofall i such that 〈x0, λi〉 = f(λi), then by definition L := x ∈ X∗(T )R | ∀i 6∈ I〈x, λi〉 ≥ f(λi)is a neighborhood of x0 in X∗(T )R. Now K contains the subset

L ∩ x ∈ X∗(T )R | ∀i ∈ I〈x, λi〉 = f(λi) = L ∩ (x0 + x ∈ X∗(T )R | ∀i ∈ I〈x, λi〉 = 0),

so since x0 is an extreme point of K, the linear subspace x ∈ X∗(T )R | ∀i ∈ I〈x, λi〉 = 0 isreduced to a point. In other words the set λi | i ∈ I generates Y∗(T )R. But by definitionit is a subset of A, whence R ·A = Y∗(T )R.

Finally it now follows that there exists σ ∈ Σ with R ·σ = Y∗(T )R and σ0∩A0 6= ∅. Fixλ ∈ σ0 ∩ A0, then f coincides with both χσ and x0 in a neighborhood of λ. This impliesx0 = χσ, as desired. q.e.d.

Next we study ampleness criteria in our setting. It turns out that the property “anti-ample” has a more natural geometric meaning.

5.16. Ample line bundles: In the situation of 5.12 assume that |Σ| is convex, forinstance |Σ| = Y∗(T )R. We call Σ′ strictly convex if

(i) |Σ′| is convex and

84

(ii) Σ+ consists of all cones |Σ′|∩ker(〈x, 〉) for all x ∈ X∗(T )Q such that 〈x, 〉 is nonnegativeon |Σ′|, and restricts to the identity on R · σ0.

Note that if these conditions are satisfied, then Σ+, hence also Σ′ and Σ, are uniquelydetermined by |Σ′|. If Σ is finite, then condition (i) means that |Σ′| is a convex rationalpolyhedral cone, and (ii) means that Σ+ consists of all faces of |Σ′| that do not contain σ0.

Assume now that Σ is finite, and |Σ| convex. Then the invertible sheaf L⊗(−1) isrelatively ample with respect to f : XΣ → S if and only if Σ′ is strictly convex. Indeedthis is a reformulation of [KKMS] ch.I §3 thm.13 under slightly stronger assumptions. Butlet us briefly indicate another, somewhat amusing proof, which exhibits the ralationshipbetween ampleness and convexity within the setting of torus embeddings. Without loss ofgenerality let X = T . If Σ′ is strictly convex, then we have a torus embedding T ′ → T ′|Σ|,hence a commutaive diagram

TΣ zero section−−−−−−−−−−−→ T ′Σ′open←−−−− T ′Σ+

↓↓ ↓↓ ‖S ∼= T ′|Σ′| closed orbit−−−−−−−−→ T ′|Σ′|

open←−−−− T ′Σ+

Observe that the morphism T ′Σ+ → T ′|Σ′|, which exists if |Σ′| is convex, is an open embedding

if and only if Σ′ is strictly convex. Grauert’s ampleness criterion ([EGA–II] thm. 8.9.1) nowimplies that L⊗(−1) is ample on TΣ (note that the convention of [EGA–II] 1.7.8 concerningthe relation between T ′Σ′ and L is the opposite of ours). Conversely, if L⊗(−1) is ampleon TΣ, we have the same diagram with T ′|Σ′| replaced by the affine cone associated to Lby [EGA–II] 8.8.2. But by definition the T ′-action extends to this cone, so by [EGA–II]8.8.6 (ii) and [KKMS] ch.I thm. 1’ this cone is isomorphic to T ′τ for some convex rationalpolyhedral cone τ ⊂ Y∗(T

′)R. One easily verifies that τ = |Σ′|, and the strict convexity ofΣ′ follows.

For the proof of theorem 5.19 we shall need the following lemmata.

5.17. Lemma: Let f : X → Y be the proper morphism of schemes of finite typeover k. Let L be an invertible sheaf on X, Li for i = 1, . . . , n invertible sheaves on Y suchthat

(i) L is very ample with respect to f (in the sense of [EGA–II] 4.4.2).

(ii) For all i Li is ample on Y .

(iii) f∗L ∼=⊕n

i=1 Li.Then L is ample on X.

Proof. By [EGA–II] 4.6.11 and 4.4.10 there exists a nonnegative integer k such thatL ⊗ f∗L⊗k1 is very ample. By [EGA–II] 4.5.10 it suffices to prove that L⊗(M+1) is veryample for some large positive integer M , to be chosen below. Since

L⊗(M+1) ∼= (L ⊗ f∗L⊗k1 )⊗ (L⊗M ⊗ f∗L⊗(−k)1 ),

by [EGA–II] 4.4.8 it suffices to choose M such that the sheaf L⊗M⊗f∗L⊗(−k)1 is generated by

global sections. Since L is relatively very ample, the canonical homomorphism f∗f∗L → Lis surjective. Thus there is a surjective homomorphism f∗(f∗L)⊗M → L⊗M , so it suffices

85

to have (f∗L)⊗M ⊗ L⊗(−k)1 generated by global sections on Y . But (f∗L)⊗M is a direct

sum of tensor products of M copies of different Li, so M must satisfy: For all integers mi

such that m1 ≥ −k, mi ≥ 0 for i = 2, . . . , n, and M =∑n

i=1mi, the sheaf⊗n

i=1 L⊗mii is

generated by global sections.

To produce such an M note that there exists a positive integer m0 such that for all

m ≥ m0 and all i, j ∈ 1, . . . , n the sheaves Li⊗L⊗mj , L⊗(−1)1 ⊗L⊗mj , L⊗mj are generated

by global sections ([EGA–II] 4.5.5). Any tensor product of such sheaves is also generatedby global sections. Now let m = (m1, . . . ,mn) ∈ Zn such that m1 ≥ −k, ∀i = 2, . . . , nmi ≥ 0, and M =

∑ni=1mi. Write m = m0 · a + b with a, b ∈ Zn such that ∀i = 1, . . . , n

ai ≥ 0, −k ≤ b1 < m0, and ∀i = 2, . . . , n 0 ≤ bi < m0. If∑n

i=1 ai ≥∑n

i=1 |bi|, then we canwrite m as a sum of n-tuples of the form m0 · ei, m0 · ei + ej and m0 · ei− e1, where ei is the

ith standard basis vector. So in that case the sheaf is generated by global sections. Notethat

N := (n− 1) · (m0 − 1) = maxk,m0 − 1 ≥n∑i=1

|bi|.

Thus for M := (m0 + 1) ·N we get

n∑i=1

ai =1

m0· (M −

n∑i=1

bi) ≥1

m0· (M −N) = N ≥

n∑i=1

|bi|,

as desired. q.e.d.

5.18. Lemma: Let f : X → Y be a holomoprphic map of compact normal complexspaces, and L and M invertible sheaves on X, Y respectively, such that M is ample, andL is ample on every fibre of f . Then there exists a positive integer n such that L⊗ f∗M⊗nis ample. In particular X is projective.

Proof. By the direct image theorem (see [GR] ch.10 §4.6) f∗L is a coherent sheaf.By the ampleness of M there exists a positive integer n such that (f∗L) ⊗ M⊗(n−1) isgenerated by global sections. We shall prove the ampleness of L ⊗ f∗M⊗n for every suchn. By Grauert’s ampleness criterion (see [BPV] ch.I thm. 19.3) we have to show that forevery irreducible closed analytic subset Z ⊂ X of strictly positive dimension there exists aninteger m ≥ 1 such that (L ⊗ f∗M⊗n)⊗m|Z possesses a nonzero section which has at leastone zero. If Z is contained in a fibre of f , then (L⊗f∗M⊗n)|Z is isomorphic to L|Z , so thisfollows from the ampleness of L on the fibres. Otherwise by the proper mapping theorem(see [GR] ch.10 §6.1) f(Z) is a closed analytic subset of Y , of strictly positive dimension.Thus by the ampleness of M there exists an integer m ≥ 1 and a nonzero section s ofM⊗m|f(Z) that has at least one zero. Let t be a section of (f∗L) ⊗M⊗(n−1) that doesnot vanish identically on f(Z). Denoting the pullback to Z of these sections by the sameletters, it follows that t⊗m ⊗ s is a nonzero section of (L ⊗ f∗M⊗n)|Z , whose zero locus isnon-empty since it contains that of s. This is the desired section. q.e.d.

5.19.Theorem: In the situation of 5.12 (a) or (b) assume that Σ is a finite andcomplete rational polyhedral decomposition of Y∗(T )R. In the case (b) suppose in addition

86

that S is compact. For every section e : T → T ′ let Me be the invertible sheaf on Sassociated to the line bundle (X ′σ0

)/e(T )→ S. Then the following assertions are equivalent:

(a) L⊗(−1) is ample on XΣ.

(b) Σ′ is strictly convex, and M⊗(−1)eσ is ample on S for all σ ∈ Σ with R · σ = Y∗(T )R.

Proof. Let us first reduce the analytic case to the algebraic case. For this, if S is acompact space, it suffices to prove that S and XΣ are projective whenever (a) or (b) holds.If (b) holds, then there exists an ample line bundle on S, and a line bundle on XΣ thatis ample in every fibre over S. By 5.18 there exists also an absolutely ample line bundleon XΣ, so both S and XΣ are projective. If (a) holds, then XΣ is projective. Choose anyσ ∈ Σ with R · σ = Y∗(T )R, then Xσ = S is also projective, as desired. For the rest of theproof S is a scheme over k.

Assume (b), then for every n ≥ 1 f∗(L⊗(−n)) is a direct sum of ample invertible sheaves

on S . In fact, this is a consequence of 5.14 (b), the ampleness of the M⊗(−1)eσ , and the

fact that a tensor product of ample invertible sheaves is again ample ([EGA-II] 4.5.7).Thus if we choose n so that L⊗(−n) is relatively very ample for f : XΣ → S, then by5.17 it is already absolutely ample, whence (a). Conversely assume (a). Then the strictconvexity of Σ′ follows from 5.16. For any σ ∈ Σ with R · σ = Y∗(T )R we have a sectioniσ : S ∼= Xσ → Xσ, and it follows that i ∗σ L⊗(−1) is ample. Now L|Xσ is the invertible sheaf

associated to the Gm-torsor X ′/eσ(T ) on Xσ∼= X/T , hence i ∗σ L is isomorphic to Meσ .

Thus M⊗(−1)eσ is ample, which proves (b). q.e.d.

5.20. Construction of projective refinements: We now want to deal with theconstruction of rational polyhedral cone decompositions that are strictly convex in the senseof 5.16. Our main point of interest is to construct refinements of given cone decompositions,subject to strict convexity and/or smoothness conditions.

Consider a short exact sequence of finite dimensional Q-vector spaces 0→ Q→M ′ π−−→M → 0, and let σ0 := R≥0 × 0. Let σ ⊂ MR be a convex rational polyhedral cone, andconsider finite rational partial polyhedral decompositions Σ, Σ+, Σ′ as in 5.11, such that|Σ| = σ. If Σ′ is strictly convex, then the definition 5.16 shows that Σ+, hence also Σ andΣ′ are uniquely determined by |Σ′|. Conversely let σ′ be a convex rational polyhedral conein M ′R such that σ′ ∩R · σ0 = σ0 and π(σ′) = σ. Let Σ+ be the set of all faces of σ′ that donot contain σ0, and Σ := π(σ+) | σ+ ∈ Σ+ and Σ′ := σ+, σ+ + σ0 | σ+ ∈ Σ+, then itis easy to see that Σ′ strictly convex with |Σ′| = σ′ and |Σ| = σ.

Next we fix a splitting M ′ = Q×M . For any∑

as above there exists a unique functionf : σ → R such that ∑

= (t, x) ∈ R×MR | t+ f(x) ≥ 0.

In fact, if e = (χ, id) is the splitting M → M ′ associated to some τ ∈ Σ, then f coincideswith −χ on τ (This function is the negative of the function considered in the proof of5.14). Clearly f is continuous, linear on every τ ∈ Σ, takes rational values on σ ∩ M ,and is convex in the sense that f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y) for all x, y ∈ MRand α ∈ [0, 1]. Conversely suppose that we are given a convex continuous piecewise linearfunction f : σ → R, such that f(α · x) = α · f(x) for all x ∈ σ and α ∈ R≥0, and which

87

takes rational values on σ ∩M . We call such a function a piecewise linear convex rationalfunction. Then the above formular defines a convex rational polyhedral cone

∑with the

described properties. Denote by Σ(f) the corresponding rational polyhedral decompositionof σ, it consists af all maximal subcones of σ on which f is linear.

In the following propositions we shall use this description to construct certain refine-ments of cone decompositions. The general idea in these constructions is the inductionover the number and the dimension of the cones in the given decomposition. Thus in theinduction step we assume that the construction is already carried out on every proper faceof a convex rational polyhedral cone σ, and we extend it to all of σ.

5.21. Proposition: Let σ ⊂MR be a convex rational polyhedral cone that does notcontain a nontrivial linear subspace, and Σ a finite rational polyhedral decomposition with|Σ| = σ. Let fσ : σ r σ0 → R be a function such that for every proper face ρ of σ, therestriction f0|ρ is a piecewise linear convex rational function, and Σ(f0|ρ) is a refinement ofπ ∈ Σ | π ⊂ ρ. Then there exists a piecewise linear convex rational function f : σ → R,which coincides with f0 on σ r σ0, such that Σ(f) is a refinement of Σ.

Remark. The assumption on σ is not necessary. By induction the proposition impliesthat every finite rational polyhedral decomposition Σ, such that |Σ| is convex, possesses arefinement which admits a strictly convex Σ′ as in 5.16.

Proof. Let f be any piecewise linear convex rational function whose restriction toσ r σ0 is f0. That there exists such a function is obvious if dim(R · σ) ≤ 1. Otherwise σis the convex closure of σ r σ0, so there exists a unique smallest convex function which is≥ f0 on σ r σ0, and it is easy to see that its restriction to σ r σ0 is f0. After replacing Σby its refinement τ ∩ π | τ ∈ Σ, π ∈ Σ(f) we may assume that Σ(fo|ρ) = τ ∈ Σ | τ ⊂ ρfor every proper face ρ of σ.

Next let x ∈ σ0 and T be the set of all faces of ρ+ R≥0 · x for all proper faces ρ of σ.This is a rational polyhedral decomposition of σ that contains every proper face of σ, andevery τ ∈ T is contained in ρ ∪ σ0 for some proper face ρ of σ. After replacing Σ by therefinement τ ∩ π | π ∈ Σ, τ ∈ T we may assume the same property for Σ.

Since T contains every proper face of σ, we still have Σ(fσ|ρ) = τ ∈ Σ | τ ⊂ ρ forevery proper face ρ of σ.

Now fix τ ∈ Σ. We want to find a piecewise linear convex rational function fτ : σ → R,which coincides with f0 on σ r σ0, such that τ ∈ Σ(fτ ). Let ρ be a proper face of σ suchthat τ ⊂ ρ ∪ σ0. By assumption ρ ∩ τ ∈ Σ(f0|ρ), so there exists a linear form: ` : M → Qsuch that ` = f0 on ρ ∩ τ , and ` > f0 on ρ r τ . Let m : M → Q be a linear form whichvanishes on ρ, and is strictly positive on σ r ρ. Since we may replace ` by ` + α ·m forarbitrarily large α ∈ Q>0, we can choose ` such that ` > f0 on σr (ρ∪σ0). Let fτ : σ → Rbe the unique smallest convex function, which is ≥ f0 on σ r σ0, and ≥ ` on τ . Since byassumption τ ⊂ ρ∪σ0, we have f0 = ` on (σrσ0)∩ τ . Thus the restriction of fτ for σrσ0

is f0, and fτ > ` on σ r τ . Since fτ = ` on τ , we have τ ∈ Σ(fτ ), as desired.

Finally let f = card(Σ)−1 ·∑

τ∈Σ fτ be the average of all fτ . This is again a piecewiselinear convex rational function and coincides with f0 on σrσ0. Moreover it is easy to checkthat Σ(f) consists of all intersections of elements of the different Σ(fτ ). In particular Σ(f)

88

is a refinement of every Σ(fτ ), hence every τ is the union of cones in Σ(f). But this meansthat Σ(f) is a refinement of Σ, as desired. q.e.d.

5.22. Lemma: Let σ ⊂ MR be a convex rational polyhedral cone, f : σ → Ra piecewise linear convex rational function, and g : σ → R a function whose restrictionto every ρ ∈ Σ(f) is piecewise linear convex rational. Then for every sufficiently smallε ∈ Q>0 the function f + ε · g is again piecewise linear convex rational, and Σ(f + ε · g) =⋃ρ∈Σ(f) Σ(g|ρ).

Proof. Let R≥0 · xi be the finitely many one dimensional cones in⋃ρ∈Σ(f) Σ(g|ρ). Let

ρ ∈ Σ(f) with R ·ρ = R ·σ, and π ∈ Σ(g|ρ) with R ·π = R ·ρ. Let ` and m be linear functionson M with `|ρ = f |ρ and m|π = g|π. Then (`+ ε ·m)(xi) = (f + ε · g)(xi) for every xi ∈ π.But for every xi 6∈ π we have either `(xi) > f(xi), or `(xi) = f(xi) and m(xi) > g(xi).Thus for any sufficiently small ε ∈ Q>0 we have (`+ε ·m)(xi) > (f+ε ·g)(xi) for all xi 6∈ π.If we have this for every π, then f + ε · g is convex, it is automatically a piecewise linearconvex rational function, and π is the subset of σ on which `+ ε ·m coincides with f + ε · g.Since every cone in

⋃ρ∈Σ(f) Σ(g|ρ) is a face such a π, the assertion follows. q.e.d.

5.23. Proposition: Consider the situation of 5.21. Assume that in addition we aregiven a lattice MZ ⊂M , and that for every proper face τ on σ, every π ∈ Σ(f0|τ ) is smoothin the sense of 5.2. Then the function f in 5.21 can be chosen such that every π ∈ Σ(f) issmooth.

Remark. By induction the proposition implies that every finite rational polyhedraldecomposition Σ, such that |Σ| is convex, possesses a smooth refinement which admits astrictly convex Σ′ as in 5.16.

Proof. Suppose first that Σ consists of the faces of σ. In this case we just have afunction f0 : σrσ0 → R, whose restriction to every proper face τ of σ is a piecewise linearconvex rational function, such that every π ∈ Σ(f0|τ ) is smooth. The assertion is that thereexists a piecewise linear convex rational function f : σ → R, which coincides with f0 onσ r σ0, such that every π ∈ Σ(f) is smooth. This is precisely what is shown in the proof of[KKMS] ch.I §2 thm. 11 on pp.33–35.

In general let f be as in 5.21. By 5.22 it suffices to construct a function g : σ → R,which is zero on σ r σ0, and piecewise linear convex rational on every ρ ∈ Σ(f), such thatevery π ∈ Σ(g|ρ) is smooth. Let T be a rational partial polyhedral decomposition containedin Σ(f), which contains every ρ ∈ Σ(f) that lies in a proper face of σ. We shall define g on|T | by induction over T . First let T consist of all ρ ∈ Σ(f) that lie in a proper face of σ,and define g to be zero on σ r σ0. By assumption this function satisfies the requirements.For the induction step choose ρ ∈ Σ(f)rT of minimal dimension. Then g is already definedon ρr ρ0, and the function g|ρrρ0 satisfies the assumptions of the special case above. Thuswe can extend g to a function on |T | ∪ ρ with the desired properties. q.e.d.

5.24. The barycentric subdivision: Let MZ be a lattice in M , and σ ⊂ MR aconvex rational polyhedral cone that is smooth with respect to MZ. Then by definitionσ =

∑di=1 R

≥0 · xi for certain xi ∈ MZ which are part of some basis of MZ. The R≥0 · xi

89

are the one dimensional faces of σ, and the xi are uniquely determined up to permutation.The barycentric subdivision of σ is then defined as follows.

For every nonempty subset S ⊂ 1, . . . , d let xS :=∑

i∈S xi. For every 0 ≤ r ≤ d

consider a sequence S of subsets ∅ ⊂6= S1⊂6= . . . ⊂6= Sr ⊂ 1, . . . , d, and let σS :=

∑ri=1 R

≥0 ·xSi . It is easy to check that the set Σ of all σS is a rational polyhedral decomposition ofσ (compare [Sp] §3.3 thm. 9). This is the barycentric subdivision of σ. By construction itdepends only on σ and the lattice MZ. Moreover every σS is smooth, and for any face τ ofσ the decomposition σS ∈ Σ | σS ⊂ τ is the barycentric subdivision of τ .

The barycentric subdivision has the following property, which is why we are interestedin it. Consider σS ∈ Σ. Every face of σS is of the form σS′ for a subsequence S′ of S. Wesaw above that every automorphism of MZ ∩ R · σ that fixes σ comes from a permutationof the xi. This implies that no two distinct faces of the same σS are conjugate under suchan automorphism. More generally let τ1, τ2 be the two faces of σ, and φ an isomorphismMZ ∩ R · τ2 →MZ ∩ R · τ2 that identifies τ1 with τ2. If σS′ is a face of σS that is containedin τ1, and φ(σS′) is a face of σS , then again we must have φ(σS′) = σS′ .

5.25. Lemma: In 5.24 suppose that we are given a function f0 : σ r σ0 → R,whose restriction to every proper face ρ of σ is a piecewise linear convex function, suchthat Σ(f0|ρ) is the barycentric subdivision of ρ. Then there exists a piecewise linear convexrational function f : σ → R, which coincides with f0 on σ r σ0, such that Σ(f) is thebarycentric subdivision of σ.

Proof. let f be the smallest convex function on σ with f(x1 + . . . + xd) = C, andwhich is ≥ f0 on σ r σ0. If C is a sufficiently large positive rational number, then f hasthe desired properties. q.e.d.

5.26. Canonical sheaves on torus embeddings: Let π : X → S be a torsorunder a torus T of dimension d, and j : X → XΣ a relative torus embedding. What weshall do will be equally valid in the algebraic and the analytic category. Let ωXΣ/S [dlog]be the sheaf of meromorphic relative differential d-forms on XΣ over S with at most simplepoles along XΣ rX. It is a coherent subsheaf of j∗ωX/S , called the relative canonical sheafwith logarithmic poles along XΣ r X. Similarly if S is smooth and equidimensional ofdimension d′ over the filed k, let ωXΣ

[dlog] be the sheaf of meromorphic relative differential(d + d′)-forms on XΣ over k with at most simple poles along XΣ r X. It is a coherentsubsheaf of j∗ωX := j∗ωX/k, called the canonical sheaf with logarithmic poles along XΣrX.Observe that these definitions depend only on the embedding X → XΣ and the morphismπ : XΣ → S and not on either T or the fact that X → XΣ is given as a torus embedding.The second definition does not even depend on S. Note also the canonical isomorphism

ωXΣ[dlog] ∼= ωXΣ/S [dlog]⊗ π∗ωS .

We are also interested in the T -invariant differentials on X. By definition the sheaf of T -invariant relative differential d-firms ωTX/S is the difference kernel of the two homomorphismsof OS-sheaves

pr ∗2 , µ∗ : π∗ωX/S → (π pr2)∗ωT×X/T×S ,

90

where µ : T × X → X denote the T -action. It is an invertible coherent sheaf on S,and the canonical map π∗π∗ωX/S → ωX/S induces an isomorphism π∗ωTX/S → ωX/S . Let

i : ωTX/S → π∗ωX/S be the inclusion. The invariance property implies that for any openU ⊂ S and any morphism t : U → T the following diagram is commutative

ωTX/Si−−→ π∗ωX/S

i ↓ [t· ]∗

π∗ωX/S ,

i.e. [t· ]∗ induces the identity on ωTX/S

5.27. Proposition: Let [1] : S → T be the constant morphism s 7→ 1. There existcanonical isomorphisms

ωXΣ/S [dlog] ∼= π∗ωTX/S , and

ωTX/S∼= [1]∗ωT .

If Σ is finite and complete, they induce a canonical isomorphism

π∗ωXΣ/S [dlog] ∼= [1]∗ωT .

Proof. For the first isomorphism note that j∗j∗π∗ωTX/S

∼= j∗π∗ωTX/S

∼= j∗ωX/S , so

π∗ωTX/S is like ωXΣ/S [dlog] canonically isomorphic to a subsheaf of j∗ωX/S . Thus we haveto show that their images are equal. Since this is a local question, we may assume thatX = T × S, which then reduces to the case X = T . Let ω be a nonzero T -invariant d-formon T , we then have to show that ωTΣ

[dlog] is generated by ω. Now it is well-known that ωhas precisely a simple pole at every stratum Tσ of codimension 1. Indeed, for any such σthe embedding T → Tσ is isomorphic to the enbedding Gdm → Gd−1

m ×A1, and if T1, . . . , Tdare the d coordinate functions, then ω corresponds to the differential dT1

T1∧ . . .∧ dTd

Td, which

clearly has a pole of order 1 along Gd−1m × 0. This shows that ωTΣ

[dlog] = ω · OTΣ, as

desired.

To construct a canonical isomorphism ωTX/S∼= [1]∗ωT we begin with the case X = T×S.

In that case we define it as the composite homomorphism

ωTX×S/Si−−→ π∗ωT×S/S −→ π∗[1]∗[1]∗ωT = [1]∗ωT ,

which is well-known to be an isomorphism. For arbitrary X we shall show that there existsa unique homomorphism ωTX/S → [1]∗ωT which for every local trivialization over an openU ⊂ S becomes equal to th homomorphism above. Then it is clearly an isomorphism, asdesired. Now such a homomorphism exists if and only if for every open U ⊂ S and everymorphism t : U → T the diagram

ωTT×U/U

[t· ]∗ ↓

ωTT×U/U −→ [1]∗ωT |U

91

commutes. But by the last remark in 5.16 [t· ]∗ induces the identity on ωTT×U/U , as desired.

Together these maps yields a canonical isomorphism ωXΣ/S [dlog] ∼= π∗[1]∗ωT , whencean isomorphism π∗ωXΣ/S [dlog] ∼= π∗π

∗[1]∗ωT . But for Σ finite and complete, XΣ → S is alocally trivial bundle with fibre TΣ which is proper, connected and reduced so the canonicalmap [1]∗ωT → π∗π

∗[1]∗ωT is an isomorphism. This gives the last isomorphism. q.e.d.

92

Chapter 6

Toroidal compactification

In this chapter we construct (partial or complete) toroidal compactifications of mixedShimura varieties. For our purposes it is best to carry out this construction within theframework of the objects MKf (P,X )(C), instead of considering just one connected compo-nent at a time. This, and the fact that we consider mixed Shimura varieties as opposed topure ones,makes it necessary to redo as a whole the construction of the toroidal compact-ification in [AMRT]. The general line of the proof is the same as there, and of course wetake resort to the reduction theoretic results [AMRT] pp.116, 157 and 258–9, see 6.2, 6.19and 6.22 below.

We begin in 6.1–2 by describing the Baily-Borel compactification within our framework.For this we use an explicit description of the Satake-topology that is more or less containedin [AMRT] ch.III §6 (see 6.2 below). It is striking that the Satake-topology can be describedin a way completely analogous to the topology on a torus embedding, as described in 5.9.

In 6.4–5 we state the requirements for the systems of cone decompositions, on which ourtoroidal compactification will depend. Then, in 6.6–9, we study torus embeddings “alongthe unipotnet fibre”: these are just the torus embeddings with respect to a torsor structureas described in 3.12. To define the general toroidal compactification of MKf (P,X )(C), webegin by considering a certain “big” infinite disjoint union U of open subsets of certainother mixed Shimura varietes, associated to all rational boundary components of (P,X ).In any particular case, it would suffice to consider only finitely many components of U , butwith our choice, U has a much nicer functorial behavior. We show how MKf (P,X )(C) canbe identified with the quotient of U by an explicit equivalence relation ∼ (6.10–12). Thenwe use the torus embeddings along the unipotent fibre to define an open embedding U → U(6.13). In 6.14–17 we extend ∼ to an equivalence relation on U . The study of the quotientU/ ∼, which will be the desired toroidal compactification, is then carried out in two steps.

First we sonsider the action of a certain arithmetic subgroups on some subset of U . Themain result 6.20 is that this action is properly discontinuous. Then in 6.21 we relate U to theBaily-Borel compactification of Mπ(Kf )((P,X )/W )(C), and in 6.22 we prove that this actionis in some sense locally equivalent to the whole equivalence relation. As in consequence thegraph ∼ is closed (6.23), so U/ ∼ is a Hausdorff space, and locally isomorphic to quotientsof subsets of U by finite groups. It is therefore a normal complex space. This is the desiredtoroidal compactification (6.24). We finish the chapter by describing the functorial behavior

93

of this compactification (6.25), and by giving smoothness (6.26), resp. compactness (6.27)criterion.

6.1. Cores: Let (P,X ) be mixed Shimura data, Q ⊂ P an admissible Q-parabolicsubgroup, and (P1,X1) a corresponding rational boundary component of (P,X ). Let X 0 ⊂X+ be a connected component. Choose a Z-lattice U1(Z)(−1) in U1(Q)(−1), and let D0

be the convex closure of U1(Z)(−1) ∩ C(X 0, P1). A subset D ⊂ C(X 0, P1) is called a coreif there exists numbers Λ, µ ∈ R>0 such that Λ ·D0 ⊂ D ⊂ µ ·D0. Clearly this definitionis independant of the choice of the lattice (Compare [AMRT] ch.II §5.1). If D is a core,then so are the following sets, as is easy to see: its closure D, its interior D0, and the setsR≥1 ·D and D+U1(R)(−1). Moreover by choosing a small enough lattice it becomes clearthat the set D + C(X 0, P1) is also a core, and that C(X 0, P1) = R>0 ·D.

6.2. The Baily-Borel compactification: Let (P,X ) be pure Shimura data, and Kf

an open compact subgroup of P (Af ). Since every connected component of MKf (P,X )(C)is a qoutient of a hermitian symmetric domain by an arithmetic group, it possesses acanonical minimal compactification as a normal complex space, the (Satake-)Baily-Borelcompactification (see [BB] or [AMRT]). We want to reformulate the main points of itsconstruction in the language developed in the previous chapters.

First one definesX ∗ :=

∐X1/W1,

where the disjoint union is extended over all rational boundary components (P1,X1) of(P,X ), and we write (P1,X1)/W1 as (P1/W1,X1/W1). Note that each X1/W1 is a finitedisjoint union of hermitian symmetric domains, and that X embeds canonically into X ∗since it is the disjoint union of all X1 for all improper boundary components. Then onedefines a certain topology on X ∗, called the Satake-topology. We will not give its originaldefinition (cf. [AMRT] ch.III §6.1 p.257), but an equivalent one. For every connectedcomponent X 0 of X , and for every (P1,X1) such that X 0 maps to X1, let ψ1 be the projectionX1 → X1/W1, and fix a convex core D ⊂ C(X 0, P1). Note that the inverse image underψ1 of any subset of C(X 0, P1) lies in the subset X 0 ⊂ X1. For any subset Y ⊂ X1/W1 andevery positive number Λ ∈ R>0 consider the subset∐

ψ2(im−1(Λ ·D) ∩ ψ−11 (Y)) ⊂ X ∗,

where the disjoint union is extended over all rational boundary components (P2,X2) between(P,X ) and (P1,X1). Let x1 ∈ X1/W1, then if Y runs through all neighborhoods of x1 inX1/W1 and Λ through R>0, these sets form a fundamental system of neighborhoods of x1

in the Satake-topology on X ∗.Indeed, the equivalence of this definition of the Satake-topology with the usual one is

only a slight strengthening of [AMRT] ch.III p.259 thm 3, and its proof is almost the same.Let us give a few indications as to how the proof of [loc. cit.] can be adapted: Take anarithmetic subgroup Γ of P (Q) and let Γx := StabΓ(x). Then for any Siegel set S (see[loc. cit]) the set of all γ ∈ Γ such that γ · x ∈ S is a finite union of Γx-cosets. Withoutloss of generality we may assume that D is a standard core, and Y is stable under Γx,then our set above is Γx-invariant. It now easily follows from the usual definition of the

94

Satake-topoloy that there exists finitely many Siegel sets Sj with x ∈ Sj , such that our setsform a fundamental system of neighborhoods of x if and only if they do so on every Sj . Butthe proof of the lemma on p.263 in [AMRT] gives an explicit description of the topologyon the closure of a Siegel set. Using the maps ψi above for a certain rational boundarycomponents (Pi,Xi) between (P,X ) and (P1,X1), a fundamental system of neighborhoodsof x in Sj can be written as the union of ψi(Sn,t) for the Sn,t as in the cited lemma. Therest follows as in [loc. cit.] p.265.

The Baily-Borel compactification can now be defined as the double quotient

MKf (P,X )∗(C) := P (Q)\X ∗ × (P (Af )/Kf ),

where as in 3.1 P (Q) operates naturally on both factors from the left hand side. It is en-dowed with the quotient topology, and by a fundamental result of Baily and Borel it is thena compact Hausdorff space (see [AMRT] ch.II p.258 thm. 2). It contains MKf (P,X )(C) asan open dense subset. The obvious stratification of X ∗ by hermitian symmetric domainsinduces a stratification of MKf (P,X )∗(C) by finitely many locally closed subsets of theform ΓQ\(X1/W1)0, where (X1/W1)0 is a connected component of X1/W1, Q the admissi-ble Q-parabolic subgruop of P associated to P1 and ΓQ an arithmetic group of Q(Q). By[BB] thm. 10.4 there exists a unique structure of a projective normal complex space onMKf (P,X )∗(C), whose restriction to each stratum is isomorphic to that induced by thecomplex structure on the hermitian symmetric domain. The morphisms 3.4 (a) and (b)extend to holomorphic maps of the Baily-Borel compactification, if the respective groupsare reductive.

6.3. Stratification of the Baily-Borel compactification: We want to describea stratification of MKf (P,X )∗(C) by pure Shimura varieties associated to the rationalboundary components of (P,X ). For this fix a representative (P1,X1) for every P (Q)-conjugacy class of rational boundary components of (P,X ). Let Q be the admissble Q-parabolic subgroup of P associated to P1, then the conjugacy class of (P1,X1) contributesto MKf (P,X )∗(C) the subset

StabQ(Q)(X1)\(X1/W1)× (P (Af )/Kf ).

For fixed (P1,X1) choose a set of representatives pf of the double quotient

StabQ(Q)(X1) · P1(Af )\P (Af )/Kf .

Then we easily see that StabQ(Q)(X1)\(X1/W1) × (P (Af )/Kf ) is the disjoint union overthese pf of

(StabQ(Q)(X1) ∩ (P1(Af ) · pf ·Kf · p−1f ))\(X1/W1)× (P1(Af ) · pf ·Kf/Kf ).

Here observe that StabQ(Q)(X1)∩(P1(Af ) ·pf ·Kf ·p−1f ) is a subgroup of Q(Q), that contains

P1(Q) as a normal subgroup, so we can define

Λ1 := StabQ(Q)(X1) ∩ (P1(Af ) · pf ·Kf · p−1f ))/P1(Q).

95

Next with K1f := P1(Af ) ∩ pf ·Kf · p−1

f the map

P1(Af )/K1f → P1(Af ) · pf ·Kf/Kf , [p1,f ] 7→ [p1,f · pf ]

is bijective. This gives an identification of the above set with

Λ1\(P1(Q)\(X1/W1)× (P1(Af )/K1f ))

∼= Λ1\P1(Q)\(X1/W1)× (P1(Af )/W1(Af ) ·K1f ))

∼= Λ1\Mπ1(K1f )((P1,X1)/W1)(C).

By definition Λ1 is an arithmetic subgroup of (Q/P1)(Q). Since P1/W1 is an almost directfactor of Q/W1, a subgroup of finite index of Λ1 can be lifted to the centralizer of P1/W1 in

Q, hence acts trivially on Mπ1(K1f )((P1,X1)/W1)(C).Thus Λ1 acts through a finite quotient.

All in all we now have a stratification of MKf (P,X )∗(C) by finitely many quotients ofpure Shimura varieties by finite groups. Of course the connected components of these formprecisely the stratification mentioned in 6.2. Thus they are locally closed analytic subsets,and the embedding is a holomorphic map.

6.4. Admissible cone decompositions: Fix mixed Shimura data (P,X ) and con-sider the set C(P,X )× P (Af ). Like in the definition 3.1 of mixed Shimura varieties we areinterested in two commuting operations on this set: P (Q) operates from the left on bothfactors, namely by 4.23 resp. by left multiplication on the second factor, and P (Af ) actsfrom the right by multiplication in the second factor. We shall also need the left action ofP (Af ) by left multiplication on the second factor. We write “p·” for the former action, and“pf ·” for the latter: the presence or absence of an index “f” will distinguish the two leftactions. Each of these actions induces one subsets of C(P,X ) × P (Af ). For any rationalboundary component (P1,X1) of (P,X ), any connected component X 0 of X+, and anypf ∈ P (Af ) consider the subset C∗(X 0, P1)×pf ⊂ C(P,X )×P (Af ). These subsets forma covering, that is invariant under the above group actions. In the obvious way (“forgetpf”) we can interpret C∗(X 0, P1)× pf as a cone in the vector space U1(Q)(−1)⊗ R.

Now consider a collection Σ of subsets of C(P,X ) × P (Af ), such that any σ ∈ Σ iscontained in some C∗(X 0, P1) × pf. Define Σ(X 0, P1, pf ) := σ ∈ Σ | σ ⊂ C∗(X 0, P1) ×pf. Let Kf ⊂ P (Af ) be an open compact subgroup. We call Σ a Kf -admissible partialcone decomposition for (P,X ) if the following conditions hold:

(i) For all X 0, (P1,X1) and pf , Σ(X 0, P1, pf ) is a rational partial polyhedral decompositionof C∗(X 0, P1)× pf.(ii) Σ is invariant under right multiplication of Kf on the second factor of C(P,X )×P (Af ).

(iii) Σ is invariant under the left action of P (Q) on both factors of C(P,X )× P (Af ).

(iv) For all (P1,X1) the set⋃pf∈P (Af ) Σ(X 0, P1, pf ) is invariant under left multiplication of

P1(Af ) on P (Af ).

We call Σ finite if in addition the following condition holds:

(v) The double quotient P (Q)\Σ/Kf is finite, i.e. there exist only finitely many convexrational polyhedral cones in Σ modulo the two actions (ii) and (iii).

96

We call a Kf -admissible complete cone decomposition for (P,X ) if it is finite and thefollowing strengthening of (i) holds:

(i’) For all X 0, (P1,X1) and pf , Σ(X 0, P1, pf ) is a complete rational polyhedral decompo-sition of C∗(X 0, P1)× pf.

Let Σ a Kf -admissible partial cone decomposition for (P,X ). For every X 0, (P1,X1)and pf let ΓU ⊂ U1(Q) be the image of

(z ∈ Z(P )(Q) | z|X = id × U1(Q)) ∩ pf ·Kf · p−1f

under the projection Z(P ) × U1 → U1. We call Σ smooth with respect to Kf if and onlyif for all X 0, (P1,X1) and pf the rational partial polyhedral decomposition Σ(X 0, P1, pf ) issmooth with respect to the lattice ΓU (−1) ⊂ U1(Q)(−1).

Remarks. (a) Conditions (i’), (ii), (iii) and (v) together are equivalent to admissibledecompositions in the sense of [AMRT] ch.III p.252 for every connected component ofMKf (P,X )(C). Condition (iv) is the “arithmeticity” condition, and is not necessary todefine the toroidal compactification over C. But it is precisely the condition needed inorder to be able to describe the natural stratification of the toroidal compactification interms of other mixed Shimura varieties. Thus for our main purpose, the construction ofthe canonical model for the toroidal compactification, it is the essential new ingredent.(However, in general it is not strictly necessary condition to be able to define the toroidalcompactification over the reflex field.)

(b) As in 6.3 one can find finitely many rational boundary components (P1,X1) andfinitely many pf ∈ P (Af ) such that Σ is the union of all left P (Q)- and right Kf -conjugatesof the Σ(X 0, P1, pf ) for these (P1,X1) and pf . Thus the finiteness of Σ is equivalent tothe finiteness of every quotient ΓQ\Σ(X 0, P1, pf ), where Q is the admissible Q-parabolicsubgroup of P associated to P1, and ΓQ is an arithmetic subgroup of Q(Q).

(c) The study of the existence of admissible cone decompositions will be postponed tochapter 9, where we shall in fact construct a special type of these.

6.5. Operations on admissble cone decompositions: Let (P,X ) be mixedShimura data, Kf an open compact subgroup of P (Af ), and Σ a Kf -admissible partialcone decompositiion for (P,X ).

(a) For pf ∈ P (Af ) let [ ·pf ]∗Σ be the set of all “cones” (u, p′f · p−1f ) | (u, p′f ) ∈ σ for

all σ ∈ Σ. This is a pf · Kf · p−1f -admissible partial cone decomposition for (P,X ). It is

finite, respectively complete if and only if Σ is finite, resp. complete.

(b) Let φ : (P,X ) → (P,X ) be an automorphism. It induces an automorphism ofC(P,X ) × P (Af ) that we again denote by φ. Let φ∗Σ := φ−1(σ) | σ ∈ Σ. This is aφ−1(Kf )-admissible partial cone decomposition for (P,X ), and finite, resp. complete if andonly if Σ has the same property.

(c) Let (P1,X1) be a rational boundary component of (P,X ). We define a (P1(Af )∩Kf )-admissible partial cone decomposition for (P1,X1), denoted by Σ|(P1,X1), according to therule

(Σ|(P1,X1))(X 01 , P2, p1,f ) = Σ(X 0, P2, p1,f )

97

for all p1,f ∈ P1(Af ), every rational boundary component (P2,X2) of (P1,X1), and every pairof corresponding connected components X 0 ⊂ X and X 0

1 ⊂ X1 such that X 0 → X 01 → X2.

By 4.22 (b) this indeed defines a (P1(Af ) ∩Kf )-admissible partial cone decomposition for(P1,X1). Note that Σ|(P1,X1) inherits in general neither finiteness nor completeness.

(d) Define Σ0 as the set of all σ ∈ Σ such that σ ⊂ C(X 0, P1) × P (Af ) for someimproper rational boundary component (P1,X1). We say that Σ is concentrated in theunipotent fibre if Σ = Σ0.

6.6. Torus embedding along the unipotent fibre: Let (P,X ) be mixed Shimuradata and Kf an open compact subgroup of P (Af ). Let (P ′,X ′) := (P,X )/U and K ′f theimage of Kf in P ′(Af ). Let Σ be a Kf -admissible partial cone decomposition for (P,X )that is concentrated in the unipotent fibre. We want to define a canonical torus embeddingassociated to Σ along the fibres of [π′] : MKf (P,X )(C)→MK′f (P ′,X ′)(C).

Let us first assume that Kf is neat. Let X 0 be a connected component of X , let x ∈ X 0

and pf ∈ P (Af ). By 3.14 the fibre over [π′]([(x, pf )]) is isomorphic to ΓU\U(C), where ΓUis the image of

(z ∈ Z(P )(Q) | z|X = id × U(Q)) ∩ pf ·Kf · p−1f

under the projection Z(P )× U → U . This is an algebraic torus, whose cocharacter groupHomC(C×,ΓU\U(C)) is canonically isomorphic to Hom(Z(1),ΓU ) = ΓU (−1). Now the ra-tional partial polyhedral decomposition Σ(X 0, P, pf ) of U(R)(−1) = ΓU (−1)R defines atorus embedding of this fibre. By the functoriality of torus embeddings, conditions 6.4 (ii)and (iii) ensure that this embedding is independant of the choice of x and pf , up to isomor-phism. Moreover these torus embeddings form a holomorphic family of torus embeddingsover MK′f (P ′,X ′)(C). Denote the total space of this embedding by MKf (P,X ,Σ)(C). It isa normal complex space.

SinceKf is neat, MKf (P,X )(C) is smooth. Thus by definition 6.4 and by 5.3MKf (P,X ,Σ)(C)is smooth if and only if Σ is smooth with respect ot Kf . If we no longer require Kf to beneat, we can construct an embedding as follows. Choose a open normal subgroup K∗f ofKf which is neat. Clearly the action of K∗f/Kf through the morphism 3.4 (a) extends to

MK∗f (P,X ,Σ)(C) (this is a special case of the functoriality 6.7 (a) below), so we may defineMKf (P,X ,Σ)(C) as a quotient of MK∗f (P,X ,Σ)(C) by this action. By 5.4 this definitiondoes not depend on the choice of K∗f .

6.7. Functoriality: The morphism of 3.4 extend to torus embeddings along theunipotent fibre in the following cases. This follows easily from the definition 6.6 and from5.3–5 and 5.8.

(a) Let (P,X ) be mixed Shimura data, pf ∈ P (Af ), and Kf , K ′f ⊂ P (Af ) open compact

subgroups, such that K ′f ⊂ pf ·Kf ·p−1f . Let Σ be a Kf -admissible and Σ′ be a K ′f -admissible

partial cone decomposition for (P,X ), both concentrated in the unipotent fibre. Supposethat for every σ′ ∈ Σ′ there exists σ ∈ Σ such that σ′ ⊂ (u, p′f · p

−1f ) | (u, p′f ) ∈ σ, or

equivalently τ ∈ [ ·pf ]∗Σ such that σ′ ⊂ τ . Then the map of 3.4 (a) extends uniquely to aholomorphic map

[ ·pf ] = [ ·pf ]K′f ,Kf : MK′f (P,X ,Σ′)(C)→MKf (P,X ,Σ)(C).

98

It is an isomorphism if K ′f = pf ·Kf · p−1f and Σ′ = [ ·pf ]∗Σ. If K ′f = Kf and Σ′ ⊂ Σ, then

it is an open embedding. In that case the complement of the image is a closed analyticsubset. If Σ′ = Σ, K ′f = Kf and pf ∈ Kf , then [ ·pf ] is the identity map. If Σ′ = Σ and

K ′f is a normal subgroup of Kf , then the map [ ·1]K′f ,Kf identifies MKf (P,X ,Σ)(C) with

the quotient of MK′f (P,X ,Σ)(C) by the finite group K ′f/Kf , acting from the right hand

side through the morphisms [ ·kf ] on MK′f (P,X ,Σ)(C).

(b) Consider a morphism φ : (P1,X1) → (P2,X2) and open compact subgroups K1f ⊂

P1(Af ) and K2f ⊂ P2(Af ) such that φ(K1

f ) ⊂ K2f . Let Σ1 be a K1

f -admissible partial cone

decomposition for (P1,X1) and Σ2 a K2f -admissible partial cone decomposition for (P2,X2),

each concentrated in the unipotent fibre. Suppose that for every σ1 ∈ Σ1 there existsσ2 ∈ Σ2 such that φ(σ1) ⊂ σ2. Then the map of 3.4 (b) extends to a holomorphic map

[φ] = [φ]K1f ,K

2f

: MK1f (P1,X1,Σ1)(C)→MK2

f (P2,X2,Σ2)(C).

If φ : (P,X ) → (P,X ) is an automorphism, φ(K1f ) = K2

f and Σ1 = φ∗Σ2, then [φ] is anisomorphism.

6.8. Reinterpretation of 6.6: To illustrate the “arithmeticity condition” 6.4 (iv)we want to interpret the torus embedding along the unipotent fibre in a different way. Weshall show that this torus embedding correponds to an embedding of a certain standardtorus. Let (P,X ), (P ′,X ′), Kf neat, K ′f and Σ be as in 6.6. Suppose this time that (P,X )is irreducible. Then by 2.14 (a) P acts through the scalar character φ : P → Gm,Q on U .Let (Gm,Q,H0) be as in 2.8, and let (P∗,X∗)→ (Gm,Q,H0) be the unipotent extension wihP∗ = UoGm,Q, with Gm,Q acting on U through the character φ. In a canonical way, (P,X )→(P ′,X ′) is a ((P∗, h(X∗)) → (Gm,Q, h(H0)))-torsor. It can be made (noncanonically) intoa ((P∗,X∗) → (Gm,Q,H0))-torsor if and only if the morphism ((P ′,X ′) → (Gm,Q, h(H0))factors through (Gm,Q,H0). Let KU

f be the image of (z ∈ Z(P )(Q) | z|X = id×U(Af ))∩Kf under the projection Z(P )× U → U , and K∗f := KU

f o φ(Kf ).

By condition 6.4 (iv), the rational partial polyhedral decomposition Σ(X 0, P, pf ) ofU(R)(−1) does not depend on pf at all. Note that since P (Q) operates by scalars onU(R(−1), this decomposition is stable under all those p ∈ P (Q) that act through positivescalars on U(R)(−1). If there exists a p ∈ P (Q) that stabilizes a connected component andacts through a negative scalar, then Σ(X 0, P, pf ) is also independant of X 0, and invariantunder multiplication by −1. In this case Σ is just the pullback of a K∗f -admissible partialcone decomposition Σ∗ for (P∗, h(X∗)). Otherwise the morphism (P ′,X ′)→ (Gm,Q, h(H0))factors through (Gm,Q,H0), and Σ is the pullback of a K∗f -adimissible partial cone decom-position Σ∗ for (P∗,X∗).

Now by 3.12 (b), MKf (P,X )(C)→MK′f (P ′,X ′)(C) is a torsor underMK∗f (P∗,X∗)(C)→Mφ(Kf )(Gm,Q,H0)(C), resp. the same with X∗ and H0 replaced by h(X∗), h(H0). So by 5.5the torus embedding

MK∗f (P∗,X∗)(C) →MK∗f (P∗,X∗,Σ∗)(C),

resp. the same with h(X∗) and h(H0), also defines a torus embedding of MKf (P,X )(C)

along the fibres over MK′f (P ′,X ′)(C). One easily verifies (fibrewise) that this is isomorphic

99

to that defined in 6.6. Next we describe the torus embedding in the fundamental case(P∗,X∗).

Remark. Usually the morphism (P ′,X ′)→ (Gm,Q, h(H0)) does factor through (Gm,Q,H0).In fact, 4.15 (d) implies this whenever (P,X ) is a proper rational boundary component ofsome other mixed Shimura data.

6.9. The case of a standard torus: We first show that giving a K∗f -admissiblecone decomposition Σ∗ for (P∗,X∗) is equivalent to giving a rational partial polyhedraldecomposition ΣU of U(R). Consider a connected component X 0

∗ of X∗, and denote itsimage in H0 = Isom(Z,Z(1)) by λ. After a twist by −1 it corresponds to an isomorphismλ : U(R)(−1) → U(R). For any p∗,f ∈ P∗(Af ), Σ∗(X 0

∗ , P∗, p∗,f ) is a rational partialpolyhedral decomposition of U(R)(−1) that, by 6.4 (iv), depends only on X 0

∗ . ClearlyΣU := λ(Σ∗(X 0

∗ , P∗, p∗,f )) is a rational partial polyhedral decomposition of U(R) that isindependant of X 0

∗ . The converse is obvious, since (P∗,X∗) possesses no rationl boundarycomponents except itself.

Now let ΓU := U(Q)∩KUf , and consider the unipotent extension (P0,X0)→ (Gm,Q,H0)

defined in 2.24. Our aim is to prove:

Claim. The isomorphism of 3.16 induces canonical isomorphisms in the commutativediagram

MK∗f (P∗,X∗)(C) ∼−−→ C× ⊗ ΓU × Mφ(Kf )(Gm,Q,H0)(C)∩ ∩ ‖

MK∗f (P∗,X∗,Σ∗)(C) ∼−−→ (C× ⊗ ΓU )ΣU × Mφ(Kf )(Gm,Q,H0)(C)

Proof. Any Z-basis of ΓU determines an isomorphism between (P∗,X∗) and the dim(U)-fold fibre product of (P0,X0) with itself over (Gm,Q,H0). Using the canonical identificationU ∼= U0 ⊗ ΓU , we thus get the isomorphism in the first line. Take X 0

∗ and λ as above, andtf ∈ Z×, u ∈ U0(C) = C, and γ ∈ ΓU . If e : (Gm,Q,H0) → (P∗,X∗) denotes the givensplitting, then this isomorphism is explicitly given as

[((u⊗ γ) · e(λ), e(tf ))] 7→ (exp(λ(u))⊗ γ, [(λ, tf )]).

By 6.6, the cocharacter group of the torus on the left is ΓU (−1) = λ−1(ΓU ), and the torusembedding is that with respect ot the decomposition Σ∗(X 0

∗ , P∗, p∗,f ) = λ−1(ΣU ). By theexplicit formula for the isomorphism, the isomorphy in the second line follows. q.e.d.

As an example of this take (P∗,X∗) = (P0,X0) and ΣUR≥0, 0. Then the torusembedding Σ0 := Σ∗ corresponds to C× → C, so we get

MK0f (P0,X0,Σ0)(C) ∼= C×Mφ(Kf )(Gm,Q,H0)(C).

6.10. A covering of MKf (P,X )(C): For the rest of the chapter we fix mixed Shimuradata (P,X ), an open compact subgroup Kf ⊂ P (Af ), and a Kf -admissible partial conedecomposition Σ for (P,X ). We shall eventually construct the toroidal compactification onMKf (P,X )(C) associated to Σ. We begin by describing a certain possibly ramified coveringof MKf (P,X )(C) by mixed Shimura varieties associated to all its boundary components

100

For every rational boundary component (P1,X1) of (P,X ) and every pf ∈ P (Af ) letK1f := P1(Af )∩pf ·Kf ·p−1

f . Let X+ ⊂ X be associated to (P1,X1) as in 4.11, and consider

the following open subset of MK1f (P1,X1)(C):

U(P1,X1, pf ) := P1(Q)\X+ × (P1(Af )/K1f ) ⊂ P1(Q)\X1 × (P1(Af )/K1

f ).

Consider the map

β(P1,X1, pf ) : U(P1,X1, pf )→MKf (P,X )(C), [(x, p1,f )] 7→ [(x, p1,f · pf )].

Let U be the disjoint union of the U(P1,X1, pf ) for all (P1,X1) and pf , and let β : U →MKf (P,X )(C) be the map induced by the β(P1,X1, pf ). This is a surjective ramifiedcovering. For all u, u′ ∈ U write u ∼ u′ if and only if u and u′ are mapped to thesame point in MKf (P,X )(C). Then MKf (P,X )(C) is isomorphic to U/ ∼ as a set. Aconnected component of U is of the form Γ1\X 0 for X 0 a connected component of X andΓ1 an arithmetic subgroup of P1(Q). Its image under β is Γ1\X 0, where Γ is an arithmeticsubgroup of P (Q). This shows that MKf (P,X )(C) is isomorphic to U/ ∼ as a topologicalspace, where U/ ∼ carries the quotient topology. Moreover the equivalence relation ∼ isinduced by analytic automorphisms of X , and locally on U the map β is a finite possiblyramified covering, so the structure of MKf (P,X )(C) as a complex space is induced fromthat of U . Consider the graph (u, u′) ∈ U × U | u ∼ u′ of ∼. Since MKf (P,X )(C) isHausdorff, this is a closed subset of U × U .

6.11. Explicit description of the equivalence relation: The description ofMKf (P,X )(C) as U/ ∼ would not be useful if one could not describe the equivalencerelation ∼ explicitly. To give such a description we first consider a number of specialcases. Let (P1,X1) and (P ′1,X ′1) be rational boundary components of (P,X ), and let pf ,p′f ∈ P (Af ). We are interested in certain maps U(P1,X1, pf )→ U(P ′1,X ′1, p′f ).

(a) Suppose that (P ′1,X ′1) = (P1,X1) and p′f = pf · kf for kf ∈ Kf . Then K1f =

P1(Af ) ∩ pf · Kf · p−1f = P1(Af ) ∩ p′f · Kf · p′−1

f , hence the identity on MK1f (P1,X1)(C)

induces an isomorphism

[id] : U(P1,X1, pf )→ U(P1,X1, pf · kf ).

(b) Suppose that (P ′1,X ′1) = (P1,X1) and p′f = p−11,f · pf for some p1,f ∈ P1(Af ). Then

the map [ ·p1,f ] : MK1f (P1,X1)(C)→MK1′

f (P1,X1)(C) induces an isomorphism

[ ·p1,f ] : U(P1,X1, pf )→ U(P1,X1, p−11,f · pf ).

(c) Suppose that (P ′1,X ′1) = int(p)(P1,X1) = (p · P1 · p−1, p · X1) and p′f = p · pf for

some p ∈ P (Q). Then the map [int(p)] : MK1f (P1,X1)(C) → MK1′

f (P ′1,X ′1)(C) induces anisomorphism

[int(p)] : U(P1,X1, pf )→ U(p · P1 · p−1, p · X1, p · pf ).

(d) Suppose that (P1,X1) is a rational boundary component of (P ′1,X ′1), and thatp′f = pf . Then we have a canonical map

β(P1,X1, P′1,X ′1, pf ) : U(P1,X1, pf )→ U(P ′1,X ′1, pf ), [(x, p1,f )] 7→ [(x, p1,f )].

101

These maps commutes with each other according to the rules:

[id] [ ·p2,f ] = [ ·p2,f ] [id],

[id] [int(p)] = [int(p)] [id],

[id] β(P1,X1, P′1,X ′1, pf ) = β(P1,X1, P

′1,X ′1, pf · kf ) [id],

[int(p)] [ ·p1,f ] = [ ·p · p1,f · p−1] [int(p)],

[int(p)] β(P1,X1, P′1,X ′1, pf ) = β(p · P1 · p−1, p · X1, p · P ′1 · p−1, p · X ′1, p · pf ) [int(p)],

[ ·p1,f ] β(P1,X1, P′1,X ′1, pf ) = β(P1,X1, P

′1,X ′1, p1,f · pf ) [ ·p1,f ],

where in the last case it is necessary to have p1,f ∈ P1(Af ). Also in each of the cases (a)through (d) it is easy to see that the diagram

U(P1,X1, pf ) → U(P ′1,X ′1, p′f )

β(P1,X1, pf ) ↓ β(P ′1,X ′1, p′f ) cr MKf (P,X )(C)

is commutative. Thus its graph is a subset of the graph of ∼. The following lemma showshow the equivalence relation ∼ can be described in terms of these maps.

6.12. Lemma: (a) Let u ∈ U(P1,X1, pf ) and u′ ∈ U(P ′1,X ′1, p′f ). Then u ∼ u′

if and only if there exists a rational boundary component (P2,X2) of (P,X ), p ∈ P (Q),p2,f ∈ P2(Af ) and kf ∈ Kf such that

(i) (P1,X1) is a rational boundary component of (P2,X2),

(ii) (P ′1,X ′1) is a rational boundary component of int(p)(P2,X2),

(iii) p′f = p · p2,f · pf · kf , and

(iv) u and u′ have the same image under the following maps:

U(P1,X1, pf ) 3 u u′ ∈ U(P ′1,X ′1, p′f )

β(P1,X1, P2,X2, pf ) ↓ ↓ β(P ′1,X ′1, pP2p−1, pX2, p

′f )

U(P2,X2, pf ) U(p · P2 · p−1, p · X2, p′f )

[ ·p2,f ]xo xo[int(p)]

U(P2,X2, p2,f · pf )[id]−−−→∼ U(P2,X2, p2,f · pf · kf ).

(b) The equivalence relation ∼ is generated by the graphs of the maps 6.11 (a) to (d).

Proof. (b) follows directly from (a). The “if”-part of (a) is clear from 6.11. So

suppose that u = [(x, p1,f )] ∈MK1f (P1,X1)(C) and u′ = [(x′, p′1,f )] ∈MK1

f ′(P ′1,X ′1)(C) have

the same image in MKf (P,X )(C). Then there exist p ∈ P (Q) and kf ∈ Kf such thatp · x = x′ and p · p1,f · pf · kf = p′1,f · p′f . Let (P2,X2) be the (unique) improper rationalboundary component of (P,X ) such that (P1,X1) is a boundary component of (P2,X2).From x′ = p · x ∈ p · X2 it follows that (P ′1,X ′1) is a boundary component of int(p)(P2,X2).Now both p1,f and p−1 · p′1,f · p lie in P2(Af ), hence

p2,f : = p−1 · p′f · k−1f · p

−1f

= (p−1 · p′−11,f · p) · (p

−1 · p′1,f · p′f · k−1f · p

−1f )

= (p−1 · p′−11,f · p) · p1,f

102

lies in P2(Af ). One easily verifies that with theses choices for (P2,X2), p, p2,f , and kf thepoints u and u′ have the same image in the above diagram. q.e.d.

6.13. Compactification of the covering: We want to construct the toroidalcompactification of MKf (P,X )(C) as the quotient of a partial compactification of U by asuitable extended equivalence relation. First we construct the partial compactification Uof U .

For every rational boundary component (P1,X1) of (P,X ), and every pf ∈ P (Af ) letΣ0

1 := (([ ·pf ]∗Σ)|(P1,X1))0. By 6.5 this is a K1

f -admissible partial cone decomposition for

(P1,X1), which is concentrated in the unipotent fibre. Now MK1f (P1,X1)(C) is an open

subset of MK1f (P1,X1,Σ

01)(C), and we let U(P1,X1, pf ) be the interior of the closure of

U(P1,X1, pf ) in MK1f (P1,X1,Σ

01)(C). It contains U(P1,X1, pf ) as an open dense subset. Let

us explicitly determine which parts of which strata of MK1f (P1,X1,Σ

01)(C) are contained in

U(P1,X1, pf )

Let Γ1\X 01 be a connected component of MK1

f (P1,X1)(C). Let im : X1 → U1(R)(−1) bethe map “imaginary part” of 4.14, it is equivariant under Γ1. Assume first that K1

f is neat,

then Γ1 commutes with U1, hence im |X 01

factors through a map Γ1\X 01 → U1(R)(−1). Since

by 6.6 the cocharacter group of the relevant torus is an arithmetic subgroup of U1(Q)(−1),one easily sees that this is a map “ord” as in 5.8. Now by 4.15 (b) the corresponding con-nected component of X is X 0 = im−1(C(X 0, P1)), hence the corresponding connected com-ponent of U(P1,X1, pf ) is Γ1\X 0 = ord−1(C(X 0, P1)). Let σ ∈ Σ1, then 5.9 (a) shows that

the closure of U(P1,X1, pf ) meets the σ-stratum precisely in the set πσ(ord−1(C(X 0, P1))).Hence the intersection of U(P1,X1, pf ) with the σ-stratum is just πσ(ord−1(C(X 0, P1))) =πσ(Γ1\X 0). If σ0 ⊂ C(X 0, P1), then C(X 0, P1) + R · σ0 = U1(R)(−1), hence for such σ

U(P1,X1, pf ) is a neighborhood of the whole σ-stratum of MK1f (P1,X1,Σ

01)(C). If K1

f isnot neat, the same assertion follows by taking quotients.

For later use let us denote by ∂U(P1,X1, pf ) the union of all σ-strata in U(P1,X1, pf )such that σ0 ⊂ C(X 0, P1). By 5.3 and the above description of the strata it is a closed

analytic subset of U(P1,X1, pf ), or even of MK1f (P1,X1,Σ

01)(C). Note that if (P1,X1) is an

improper boundary component, then ∂U(P1,X1, pf ) = U(P1,X1, pf ).

6.14. Extension of the equivalence relation: Let U be the disjoint union of theU(P1,X1, pf ) for all (P1,X1) and pf , this contains U as an open dense subset. We wish toextend the equivalence relation ∼ to U such that the quotient U/ ∼ is the desired partialtoroidal compactification of MKf (P,X )(C). This extended equivalence relation must satisfythe following conditions:

(a) Its graph in U × U is closed (so that the quotient topology on U/ ∼ is Hausdorff).

(b) Locally on U it is given by analytic automorphisms, and locally on U the projectionU → U/ ∼ is a finite, possibly ramified, covering (so that U/ ∼ inherits the structure ofthe complex space).

Moreover we want to be able to explicitly describe the topology on U/ ∼. Observe that by(a) the graph of the extension must contain the closure of the graph of ∼. We shall definean explicit extension, whose graph will turn out to be equal to this closure.

103

6.15. Extension of the maps of 6.11: To approximate the desired extension of ∼we first consider the maps of 6.11. In 6.11 (a), (b) or (c) the isomorphism U(P1,X1, pf )→U(P ′1,X ′1, p′f ) is induced from an isomorphism MK1

f (P1,X1)(C) → MK1f ′(P ′1,X ′1)(C) as in

3.4. It is easily checked that in each of these cases the hypotheses of 6.7 (a) or (b) hold,

so by 6.7 the isomorphism MK1f (P1,X1)(C) → MK1

f ′(P ′1,X ′1)(C) extends uniquely to anisomorphism

MK1f (P1,X1,Σ

01)(C)→MK1

f ′(P ′1,X ′1,Σ′01 )(C).

By the definition 6.13 of U(P1,X1, pf ) we thus get a unique isomorphism

U(P1,X1, pf )→ U(P ′1,X ′1, p′f ).

Now consider the map 6.11 (d). Let Σ′′01 := Σ

′01 |(P1,X1), this consists of all σ ∈ Σ0

1

which are already contained in C(P ′1,X ′1) × P1(Af ) ⊂ C(P1,X1) × P1(Af ). Thus by 6.7 (a)

MK1f (P1,X1,Σ

′′01 )(C) is an open subset of MK1

f (P1,X1,Σ01)(C). Now MK1

f (P1,X1)(C) →MK1

f (P1,X1,Σ′′01 )(C) and MK1

f ′(P ′1,X ′1)(C) → MK1f ′(P ′1,X ′1,Σ

′01 )(C) are torus embeddings

with respect to the same torus. Hence the map β(P1,X1, P′1,X ′1, pf ) : U(P1,X1, pf ) →

U(P ′1,X ′1, pf ) of 6.11 (d) extends to a unique holomorphic map

β(P1,X1, P′1,X ′1, pf ) : U(P1,X1, pf ) ∩MK1

f (P1,X1,Σ′′01 )(C)→ U(P ′1,X ′1, pf ).

Note that the description of U(P1,X1, pf ) in 6.13 implies that the image of this map is aunion of connected components of U(P ′1,X ′1, pf ). Let us also observe the following fact. Forevery point u ∈ U(P1,X1, pf ) consider all rational boundary components (P ′1,X ′1) between(P,X ) and (P1,X1) such that this map is defined at u. There exists a unique maximal such(P ′1,X ′1), for which u is mapped into ∂U(P ′1,X ′1, pf ).

We have thus defined an extension for each of the maps of 6.11. Since U(P1,X1, pf ) is

open and dense in U(P1,X1, pf ), resp. in U(P1,X1, pf ) ∩MK1f (P1,X1,Σ

′′01 )(C) in the case

6.11 (d), the graphs of these extended maps are contained in the closure of the graph of∼. We shall see below that this closure is equal to the graph of the equivalence relationgenerated by these extended maps.

6.16. Definition of the extended relation: Let (P1,X1) and (P ′1,X ′1) be rationalboundary components of (P,X ) and pf , p′f ∈ P (Af ). Let u ∈ U(P1,X1, pf ) and u′ ∈U(P ′1,X ′1, p′f ). In analogy to 6.12 define u ∼ u′ if and only if there exists a rational boundarycomponent (P2,X2) of (P,X ), p ∈ P (Q), p2,f ∈ P2(Af ) and kf ∈ Kf such that

(i) (P1,X1) is a rational boundary component of (P2,X2),

(ii) (P ′1,X ′1) is a rational boundary component of int(p)(P2,X2),

(iii) p′f = p · p2,f · pf · kf , and

(iv) u and u′ have the same image under the following maps:

MK1f (P1,X1,Σ

′′01 )(C) ∩ U(P1,X1, pf ) 3 u u′ ∈ U(P ′1,X ′1, p′f ) ∩MK1

f ′(P ′1,X ′1,Σ′′01 )(C)

β(P1,X1, P2,X2, pf ) ↓ ↓ β(P ′1,X ′1, pP2p−1, pX2, p

′f )

U(P2,X2, pf ) U(p · P2 · p−1, p · X2, p′f )

104

[ ·p2,f ]xo xo[int(p)]

U(P2,X2, p2,f · pf )[id]∼−−−−→ U(P2,X2, p2,f · pf · kf ).

Remark. Clearly the graph of this relation is contained in the closure of the graph ofthe old relation on U .

6.17. Lemma: The relation ∼ defined in 6.16 is the equivalence relation on Ugenerated by the graphs of the maps 6.15.

Proof. It suffices to prove that ∼ is an equivalence relation. The reflexivity is obvious,and the symmetry follows from the commutativity properties of the maps 6.11 (a) to (c),which also hold for their extensions 6.15. For the transitivity consider u ∈ U(P1,X1, pf )and u′ ∈ U(P ′1,X ′1, p′f ), and u′′ ∈ U(P ′′1 ,X ′′1 , p′′f ), such that u ∼ u′ ∼ u′′. Let (P2,X2), p,p2,f and kf be as above for u ∼ u′ and (P ′2,X ′2), p′, p′2,f and k′f analogously for u′ ∼ u′′.

Let σ ∈ Σ′1 be such that u′ is in the σ-stratum of U(P ′1,X ′1, p′f ), then by assumption σ is

contained in C∗(X 0, p·P2 ·p−1) as well as C∗(X 0, P ′2). Let (P3,X3) be the rational boundarycomponent of (P,X ) such that σ0 ⊂ C(X 0, P3), then it follows that both (p ·P2 ·p−1, p ·X2)and (P ′2,X ′2) are boundary components of (P3,X3). Note the relation

β(P ′1,X ′1, P3,X3, p′f ) = β(P ′2,X ′2, P3,X3, p

′f ) β(P ′1,X ′1, P ′2,X ′2, p′f )

= β(p · P2 · p−1, p · X2, P3,X3, p′f ) β(P ′1,X ′1, p · P2 · p−1, p · X2, p

′f )

on U(P ′1,X ′1, p′f ) ∩MK1f ′(P ′1,X ′1,Σ

′′01 )(C) where Σ

′′01 := Σ0

3|(P ′1,X ′1) contains σ. In other com-mutativity relations 6.11, which extend automatically to the maps 6.15, now readily showsthat u ∼ u′′ with (p−1 · P3 · p, p−1 · X3) in place of (P2,X2), p−1 · p′2,f · p · p2,f in place ofp2,f , kf · k′f in place of kf , and p′ · p in place of p. q.e.d.

6.18. The action of the normalizer: We first study the equivalence relationinduced by some of the maps of 6.15 on a fixed U(P1,X1, pf ). Let Q be the admissibleQ-parabolic subgroup of P associated to P1. The group

StabQ(Q)(X1) ∩ (P1(Af ) · pf ·Kf · p−1f )

acts from the left hand side on X1× (P1(Af ) · pf ·Kf/Kf ) by multiplication in each factor.As in 6.3 define

∆1 := (StabQ(Q)(X1) ∩ (P1(Af ) · pf ·Kf · p−1f ))/P1(Q),

this is an arithmetic subgroup of (Q/P1)(Q). Since the map P1(Af )/K1f → (P1(Af ) · pf ·

Kf/Kf ), [p1,f ] 7→ [p1,f · pf ] is bijective, we get an action of ∆1 on

P1(Q)\X1 × (P1(Af ) · pf ·Kf/Kf ) ∼= P1(Q)\X1 × (P1(Af )/K1f )

= MK1f (P1,X1)(C).

The induced action on Mπ1(K1f )((P1,X1)/W1)(C) is the same as in 6.3, in particular ∆1 acts

on Mπ1(K1f )((P1,X1)/W1)(C) through a finite quotient.

105

Next we want to express the ∆1-action on U(P1,X1, pf ) in terms of the maps 6.11 (a)to (c). For this let q ∈ Q(Q), p1,f ∈ P1(Af ), and kf ∈ Kf be such that [q] ∈ ∆1 and

q = p1,f · pf · kf · p−1f . Denote by L[q] the corresponding automorphism of MK1

f (P1,X1)(C).One easily verifies that the maps

[int(q)]K1f q·K

1f ·p−11,f ,K

1f

: MK1f (P1,X1)(C)→M q·K1

f ·q−1

(P1,X1)(C)

and[ ·p1,f ]p1,f ·K1

f ·P−11,f,K1

f : Mp1,f ·K1f ·p−11,f (P1,X1)(C)→MK1

f (P1,X1)(C)

are well-defined and isomorphisms. Likewise the relations q ·K1f · q−1 = p1,f ·K1

f · p−11,f and

L[q] = [ ·p1,f ] [int(q)]

are straightforward. Together with the map [id] of 6.11 (a) this shows that the restrictionof L[q] to U(P1,X1, pf ) can indeed be expressed in terms of the maps 6.11 (a) to (c). By

6.15 this implies that the ∆1-action extends canonically to an action on U(P1,X1, pf ).

We just showed that if two points u, u′ ∈ U(P1,X1, pf ) are equivalent under ∆1, thenthey are equivalent under the extensions 6.15 of the maps 6.11 (a) to (c). The converse alsoholds: Indeed, the latter condition is equivalent to the existence of a diagram as in 6.16with (P2,X2) = (P1,X1) = (P ′1,X ′1) and p′f = pf . Then with q := p−1 and p1,f := p2,f wehave u = L[q](u

′) for [q] ∈ ∆1, as desired.

The map from U(P1,X1, pf ) to the desired toroidal compactification of MKf (P,X )(C)now factors through ∆1\U(P1,X1, pf ). The next step is therefore to show that this quotientis a “good” object. For this we need the following important reduction theoretic fact, whichis a slight generalization of [AMRT] ch.II cor. p.116.

6.19. Theorem: Let (P,X ) be mixed Shimura data, (P1,X1) a rational boundarycomponent, Q the corresponding admissible Q-parabolic subgroup of P , ρ : Q → GL(U1)its natural representation on U1, and X 0 a connected component of X+. Let Γ be anarithmetic subgroup of ρ(StabQ(Q)(X 0)) ⊂ ρ(Q)(Q).

(a) If σ and τ are convex rational polyhedral cones in C∗(X 0, P1), then the set γ ∈ Γ |γσ ∩ τ ∩ C(X 0, P1) 6= ∅ is finite.

(b) There exists a convex rational polyhedral cone σ in C∗(X 0, P1) such that Γ · σ =C∗(X 0, P1).

Proof. If P is reductive, then the decomposition [AMRT] ch.III §4.1 of Q implies thatρ(Q) is the group of linear automorphisms of the cone C(X 0, P1). In this case (a) theassertion (i) of the corollary [AMRT] ch.II p.116, and by (ii) of [loc. cit.] there exists aconvex rational polyhedral cone σ in C∗(X 0, P1) such that Γ · σ ∩ C(X 0, P1) = C(X 0, P1).To get the full assertion of (b) note that there are only finitely many Γ-conjugacy classesof rational boundary components (Pi,Xi) between (P,X ) and (P1,X1). Applying [loc. cit.](ii) to each of these with StabΓ(C(X 0, P1)) in place of Γ we find finitely many convexrational polyhedral cones σi such that C∗(X 0, P1) =

⋃i Γ · σi. Letting σ :=

∑i σi we get

assertion (b).

106

In the general case we may without loss of generality assume that (P,X ) is irreducible.Next note that if dim(U) > 0, then by 4.15 (c) we have C∗(X 0, P1) = C∗(X 0, P1) +U(R)(−1). Hence for both assertions we may replace σ by σ + U(R)(−1) and τ by τ +U(R)(−1). Consider the situation after dividing by U , i.e (P1,X1)/U as rational boundarycomponent of (P,X )/U . Since U ⊂ U1 ⊂ P1, the group Γ maps isomorpically to thecorresponding group for (P,X )/U , so the assertions are equivalent to those for the imageof σ and τ in C∗(X 0/U(C), P1/U). Thus it is enough to treat the case dim(U) = 0. In thiscase we could mutatis mutandum repeat the proof of [AMRT] ch.II §4.3. But with a trickwe can reduce the assertions to the reductive case.

Let us first prove (b). Let (G,H) := (P,X )/V . Let U1 and P 1 be the image of U1,resp. P1 in G, and π the projection U1 → U1. Choose a convex rational polyhedral coneτ ⊂ C∗(X 0, P1) such that

π(Γ · τ) = C∗(H0, P 1).

Let Γρ(V ) := ρ(V )(Q) ∩ Γ, this is a lattice in ρ(V )(Q). Thus there exists a compact subsetK ⊂ ρ(V )(R) such that Γρ(V ) ·K = ρ(V )(Q). The first assertion of 4.15 (c) implies that

ρ(V )(Q) acts transitively on the fibres of the projection π : C∗(X 0, P1) → C∗(H0, P 1), soit follows

Γ ·K · τ = C∗(X 0, P1).

Thus it suffices to show that K · τ is contained in some convex rational polyhedral cone inC∗(X 0, P1). Note that ρ(V ) acts trivially on V ∩U1 and U1 = U1/(V ∩U1). Thus the actionof ρ(V ) on U1 is given by v · u = u + B(v, π(u)) for some bilinear map B : ρ(V ) × U1 →V ∩ U1. Choose a finite set of elements vi ∈ ρ(V )(Q) whose convex closure contains K.Let uj ∈ U1(Q)(−1) be finitely many elements that generate τ as a convex cone. Thenevery element of K · τ is of the form (

∑i αi · vi) · (

∑j βj · uj) with αi, βj ∈ R≥0 such that∑

i αi = 1. We calculate

(∑i

αi · vi) · (∑j

βj · uj) =∑j

βj · (∑i

αi · vi) · uj

=∑j

βj · (uj +B(∑i

αi · vi, π(uj)))

=∑j

βj ·∑i

αi · (uj +B(vi, π(uj)))

=∑i

∑j

αi · βj · vi · uj .

This shows that K · τ is contained in the convex rational polyhedral cone generated by theelements vi · uj ∈ U1(Q)(−1) ∩ C∗(X 0, P1), as desired.

For the proof of (a) we first reduce to a special case. Note that for an embedding(P,X ) → (P ′,X ′), Γ embeds into the corresponding group for (P ′,X ′), so it suffices toprove the assertion for (P ′,X ′). Choose an embedding

(P,X ) → (G,H)× (V2,g o CSp2g,Q,H′2g)

as in the proof of 2.26 (b), where (V2,g o CSp2g,Q,H′2g) is the mixed Shimura data definedin 2.25. Since we already know the assertion for (G,H), it remains to treat the case

107

(P,X ) = (V2,goCSp2g,Q,H′2g). But this isomorphic to (P2g,X2g)/U2g, so by the equivalenceabove we are reduced to the case (P,X ) = (P2g,X2g). By 4.25 this is a rational boundarycomponent of (P ′,X ′) := (CSp2(g+1),H2(g+1)), it is this fact that we shall exploit. By 4.20(P1,X1) is also a rational boundary component of (P ′,X ′).

Since C(X ′0, P ) is a half-line in U(R)(−1), it is of the form R>0 · u0 for some u ∈C(X ′0, P ). By 4.21 (a) we have C(X 0, P1) = C(X ′0, P1) + U(R)(−1), since u0 lies in theclosure of C(X ′0, P1), it follows that C(X 0, P1) = C(X ′0, P1) + R≤0 · u0. Likewise by 4.22(b) we have C∗(X 0, P1) = C∗(X ′0, P1) + R≤0 · u0. Thus any convex rational polyhedralcone σ ⊂ C∗(X 0, P1) is contained in the convex cone generated by ±u0 and finitely manyuα ∈ C∗(X ′0, P1) ∩ U(R)(−1). Let σ′ be the convex rational polyhedral cone generated byu0 and these uα, then we have σ′ ⊂ C∗(X ′0, P1) and σ = σ′+R≤0 ·u0. Thus to prove (a) itsuffices to consider σ = σ′+R≤0 ·u0 and τ = τ ′+R≤0 ·u0 for two convex rational polyhedralcones σ′, τ ′ ⊂ C∗(X ′0, P1).

Now let γ ∈ Γ such that γσ ∩ τ ∩ C(X 0, P1) 6= ∅. Then there exists λ, µ and ν ∈ R≤0

such that

γ(σ′ + (−λ) · u0) ∩ (τ ′ + (−µ) · u0) ∩ C(X ′0, P1) + (−ν) · u0) 6= ∅.

Replacing λ, µ and ν by λ′ := maxλ, µ, ν implies that

(γσ′ ∩ τ ′ ∩ C(X ′0, P1)) + (−λ′) · u0) 6= ∅,

hence γσ′ ∩ τ ′ ∩C(X ′0, P1) 6= ∅. Let Q′ be the normalizer of P1 in P ′, this is the parabolicsubgroup of P ′ associated to (P1,X1) as boundary component of (P ′,X ′), and it containsQ. Thus Γ is contained in the corresponding group for (P1,X1) relative to (P ′,X ′). Thusthe assertion follows from the reductive case. q.e.d.

6.20. Proposition: ∆1 acts properly discontinuously on U(P1,X1, pf ).

Remark. In fact we shall prove that modulo an arithmetic subgroup of Z(P )(Q) theaction is properly discontinuous in the usual sense.

Proof. (Compare [AMRT] ch.III p.272 prop. 2 and p.157 thm. (i). Note that Σneed not be finite.) Let p1,f ∈ P1(Af ), X 0

1 a connected component of X1 and X 0 thecorresponding connected component of X . The stabilizer of X 0

1 × p1,f · pf ·Kf in Q(Q) is

ΓQ := StabQ(Q)(X 0) ∩ (p1,f · pf ) ·Kf · (p1,f · pf )−1.

Without loss of generality we may replace Kf by a subgroup of finite index, so we mayassume that ΓQ is neat. Let Γ1 := P1(Q) ∩ ΓQ and ΓU the image of (z ∈ Z(P )(Q) | z|X =id × U1(Q)) ∩ ΓQ in U1(Q). By neatness Γ1 acts trivially on the torus U1(R)(−1)/ΓU ,

and the corresponding connected component of MK1f (P1,X1,Σ

01)(C) is the torus embed-

ding Γ1\X 01 → (Γ1\X 0

1 )Σ01(X 0

1 ,P1,p1,f ) with respect to the torus U1(C)/ΓU . The corre-

sponding connected component of U(P1,X1, pf ) is the interior of the closure of Γ1\X 0 =ord−1(C(X 0, P1)), where as in 6.13 the map ord : Γ1\X 0

1 → U1(R)(−1) is induced from themap “imaginary part”. Since ΓZ := Z(P )(Q) ∩ ΓQ acts trivially on X 0, the action of ΓQon (Γ1\X 0

1 )Σ01(X 0

1 ,P1,p1,f ) factors through ΓQ/ΓZ · Γ1. We shall show that this is a properlydiscontinuous action in the usual sense.

108

Let σ ∈ Σ01(X 0

1 , P1, p1,f ) and u in the σ-stratum of U(P1,X1, pf ). Let λ ∈ C(X 0, P1)such that u ∈ πσ(ord−1(λ)). There exists a convex rational polyhedral cone τ ⊂ C∗(X 0, P1)such that τ0 is a neighborhood of λ. After replacing τ by σ+ τ we have τ0 = τ0 +σ, henceby 5.9 (a)

U :=∐

ρ face of σ

πρ(ord−1(τ0))

is an open neighborhood of u in U(P1,X1, pf ). Likewise consider σ′ ∈ Σ01(X 0

1 , P1, p1,f ),u′ in the σ′-stratum of U(P1,X1, pf ), and let τ ′ be a convex rational polyhedral cone inC∗(X 0, P1) such that

U ′ :=∐

ρ′ face of σ′

πρ′(ord−1(τ ′0))

is an open neighborhood of u′ in U(P1,X1, pf ). To prove the desired assertion it suffices toshow that there exist at most finitely many [γ] ∈ ΓQ/ΓZ · Γ1. such that γU ∩ U ′ 6= ∅.

If γU ∩ U ′ 6= ∅, there exist faces ρ of σ and ρ′ of σ′ such that

γ(πρ(ord−1(τ0))) ∩ πρ′(ord−1(τ ′0)) 6= ∅.

But this implies that γ · ρ = ρ′, hence by the ΓQ-equivariance of ord

πρ′(ord−1(γ · τ0)) ∩ πρ′(ord−1(τ ′0)) 6= ∅.

This is equivalent to ord−1(γ ·τ0)∩Tρ′ ·ord−1(τ ′0) 6= ∅, where Tρ′ is the subtorus of U1(C)/ΓUwith cocharacter group ΓU ∩ R · ρ′. But Tρ′ · ord−1(τ ′0) = ord−1(τ ′0 + R · ρ′), hence

(γ · τ0) ∩ (τ ′0 + R · ρ′) 6= ∅.

Since ρ′ is a convex cone, every element of R · ρ′ can be written as a difference of twoelements of ρ′. This implies

((γ · τ0) + ρ′) ∩ (τ ′0 + ρ′) 6= ∅.

But ρ′ is a face of σ′, and σ′ is contained in τ ′, so τ ′0 + ρ′ = τ ′0. Likewise it follows that(γ · τ0) + ρ′ = γ · (τ0 + ρ) = γ · τ0, hence

γ · τ0 ∩ τ ′0 6= ∅.

Since τ0 is contained in C(X 0, P1), this implies

γ · τ ∩ τ ′ ∩ C(X 0, P1) 6= ∅.

Thus by 6.19 (a) there are finitely many possibilities for ρ(γ). But the kernel of ρ|Γ iscommensurable with ΓZ ·Γ1, so there are only finitely many possibilities for [γ] ∈ ΓQ/ΓZ ·Γ1,as desired. q.e.d.

6.21. Proposition: Let π be the projection (P,X )→ (P,X )/W . Then the compositemap U →MKf (P,X )(C)→Mπ(Kf )((P,X )/W )(C) extends uniquely to a holomorphic map[π]∗ : U →Mπ(Kf )((P,X )/W )∗(C). This map factors through U/ ∼.

109

Proof. Since U is dense in U , the extension is unique if it exists. Moreover since Uis a normal complex space, if a continuous extension exists, then by Riemann’s extensiontheorem (see [GR] ch.7 §4.2 p.144) this extension is already holomorphic. We first definean explicit extension.

For all (P1,X1) and pf we define it on ∂U(P1,X1, pf ) ⊂ U(P1,X1, pf ) as the composite

∂U(P1,X1, pf ) →MK1f (P1,X1)(C) ı

ı ∆1\Mπ1(K1f )((P,X )/W )(C) →Mπ(Kf )((P,X )/W )∗(C).

Every point in U(P1,X1, pf ) r ∂U(P1,X1, pf ) is mapped to ∂U(P ′1,X ′1, pf ) for a uniquerational boundary component (P ′1,X ′1) between (P,X ) and (P1,X1) under the extension6.15 of the map 6.11 (d). Thus there exists a unique way to extend this map to the wholeof U such that it is compatible with these maps. To show that this extended map factorsthrough U/ ∼ it thus remains to show that it is invariant under the extensions 6.15 of themaps 6.11 (a) to (c). This is an easy calculation, as in 6.11. Also it is easy to check thatthis map extends the given map U → MKf (P,X )(C) → Mπ(Kf )((P,X )/W )(C). To provethe assertion it remains to show that it is continious.

Fix (P1,X1), pf a point u ∈ U(P1,X1, pf ), and a neighborhood V ⊂Mπ(Kf )((P,X )/W )∗(C)of the image of u, then we have to show that the inverse image of V in U(P1,X1, pf ) is aneighborhood of u. If u 6∈ ∂U(P1,X1, pf ), then the map factors through U(P ′1,X ′1, pf ) for aboundary component (P ′1,X ′1) between (P,X ) and (P1,X1), and different from (P1,X1), sothe assertion follows by induction over (P1,X1). Thus we may suppose u ∈ ∂U(P1,X1, pf ).

Let (x1, p1,f ) ∈ (X1/W1)×P1(Af )be a representative for π1(u) ∈Mπ1(K1f )((P1,X1)/W1)(C),

then by the definition 6.2 of the topology on (X/W )∗ we may take V to be the image of

(∐

ψ2(im−1(D) ∩ ψ−11 (Y)))× p1,f · pf

in Mπ(Kf )((P,X )/W )∗(C), where D is an open core in C(X 0, P1), Y a neighborhood ofx1 in X1/W1, and the disjoint union is extended over all rational boundary components(P2,X2) between (P1,X1) and (P,X ).

The connected component of U(P1,X1, pf ) that contains u is the torus embedding of(Γ1\X 0)×p1,f with respect to the cone decomposition Σ(X 0, P1, p1,f ·pf ). LetW denotethe image of (im−1(D)∩ψ−1

1 (Y))×p1,f ·pf in U(P1,X1, pf ), and πσ the canonical projectionto the σ-stratum in the torus embedding, for each σ ∈ Σ(X 0, P1, p1,f ·pf ). Then the inverseimage of V certainly contains the union of all πσ(W). By assumption u lies in the τ -stratumfor some τ ∈ Σ(X 0, P1, p1,f ·pf ) such that τ0 ⊂ C(X 0, P1). We also have ψ−1

1 (Y) = π−1τ (Y ′)

for a neighborhood Y ′ of u in the τ -stratum. Moreover in terms of the torus embeddingthe map im : X 0

1 → U1(R)(−1) is a map ord as in 5.8, and by 6.1 we may without loss ofgenerality assume that D is open and equal to D+C(X 0, P1), hence in paticular to D+ τ .Thus the inverse image of V contains the set

∐πσ(ord−1(D + τ) ∩ π−1

τ (Y ′)), which by 5.9(a) is a neighborhood of u. q.e.d.

6.22. Proposition Fix (P1,X1), pf , and a relatively compact subset Z1 ⊂ ∆1\Mπ1(K1f )((P1,X1)/W1)(C).

Consider the morphism [π]∗ : U →Mπ(Kf )((P,X )/W )∗(C) of 6.21 and the projection [π1] :

110

U(P1,X1, pf ) → ∆1\Mπ1(K1f )((P1,X1)/W1)(C). There exists a neighborhood V1 of Z1 in

Mπ(Kf )((P,X )/W )∗(C), and a ∆1-invariant neighborhoodW1 of ∂U(P1,X1, pf )∩[π1]−1(Z1)in [π]∗−1(V1) ∩ U(P1,X1, pf ) such that

(a) Every element of [π]∗−1(V1) ⊂ U is under ∼ equivalent to an element of W1.

(b) Any two elements of W1 are equivalent under ∼ if and only they are equivalent underthe action of ∆1.

Proof. There exist a connected component (X1/W1)0 of X1/W1, a relatively compactopen subset Y1 ⊂ (X1/W1)0, and a finite subset p1,i,f ⊂ P1(Af ) such that Z1 lies in

the image of Y1 × p1,i,f in ∆1\Mπ1(K1f )((P1,X1)/W1)(C). Consider the subset E1 :=

im−1(D) ∩ ψ−11 (Y1) of X for some arbitrarily small open core D ⊂ C(X 0, P1). Let V1 be

the image of (∐ψ2(E1)) × p1,i,f · pf in Mπ(Kf )((P,X )/W )∗(C), by the definition of the

Satake-topology this is a neighborhood of Z1. Also let F1 be the image of E1 × p1,i,f ⊂X 0×P1(Af ) in U(P1,X1, pf ) andW1 :=

∐πσ(F1), where the union is extended over all cones

σ ∈ Σ(X 0, P1, p1,i,f · pf ) for some i. By 5.9 (a) this is a neighborhood of ∂U(P1,X1, pf ) ∩[π1]−1(Z1) in [π]∗−1(V1) ∩ U(P1,X1, pf ), and it is clearly ∆1-invariant.

Consider an element of U in the inverse image of V1. By the explicit descriptionof the map [π]∗ in the proof of 6.21 it is equivalent to an element of ∂U(P ′1,X ′1, p′f ) for

some (P ′1,X ′1) and p′f . Let [(x′1, p′1,f )] be its image in Mπ(Kf )((P,X )/W )∗(C), for some

x′1 ∈ X ′1/W ′1 and p′1,f ∈ P ′1(Af ). Then by the definition of the Baily-Borel compactification

there exist p ∈ P (Q) and kf ∈ Kf such that (P2,X2) := (p ·P ′1 ·p−1, p ·X ′1) is between (P,X )and (P1,X1), that p · x′1 ∈ ψ2(E1), and that p · p′1,f · p′f · kf = p1,i,f · pf for some p1,i,f in theabove finite set. After applying in order the extensions 6.15 of the maps 6.11 (a) for kf (c)for p, and (b) for p · (p′1,f )−1 ·p−1 ·p1,i,f we are left with an element of ∂U(P2,X2, pf ), whose

image in ∆2\Mπ2(K2f )((P2,X2)/W2)∗(C) is represented by (x2, p1,i,f ) for some x2 ∈ ψ2(E1).

It lies in the σ-stratum for some σ ∈ Σ(X 0, P2, p1,i,f · pf ) ⊂ Σ(X 0, P1, p1,i,f · pf ), so underthe extension 6.15 of the map 6.11 (d) it is equivalent to an element of πσ(F1), hence ofW1. This proves (a).

For (b) consider two elements u, u′ ∈ W1 such that u ∼ u′. Write u = πσ([(x1, p1,i,f )])for some x ∈ E1 and for some i, and likewise u′ = πσ′([(x

′1, p1,i′,f )]). Then we have the

diagram of 6.16 with the additional relation p · p1,i,f · p−12,f · p

−1 = p1,i′,f . If p stabilizes(P1,X1), then this equation implies p2,f ∈ P1(Af ), so the three lower maps in the diagramof 6.16 all lift to corresponding maps of the U(P1,X1, . . .). Thus by 6.18 u and u′ areequivalent under ∆1, as desired. It remains to show that p stabilizes (P1,X1), if D waschosen sufficiently small.

For this observe that the equation p ·p1,i,f ·p−12,f ·p

−1 = p1,i′,f together with that in 6.16(iii) implies

p ∈ ((p1,i′,f · pf ) ·Kf · (p1,i′,f · pf )−1) · (p1,i′,f · (p1,i,f )−1).

Here the first factor is an open compact subgroup of P (Af ), while the second factor runsthrough a finite set. Since p is an element of P (Q), it lies in a finite union of cosets under anarithmetic subgroup Γ of P (Q). Thus for any fixed p0 ∈ P (Q) we have to know that, if Dis sufficiently small, then any p ∈ Γ · p0 with (

∐ψ2(E1)) ∩ p · (

∐ψ2(E1)) 6= ∅ must stabilize

(P1,X1). But this property of the Satake-topology is a direct consequence of [AMRT] ch.III§6.1 thm. 1(iv). q.e.d.

111

6.23. Corollary: The graph of the equivalence relation ∼ on U is closed. In fact itis the closure of the graph of ∼ on U .

Proof. The second assertion follows from the first and the remark in 6.16. Consider twopoints u, u′ ∈ U such that for all neighborhoods W of u and W ′ of u′ there exist elementsw ∈ W and w′ ∈ W ′ with w ∼ w′. Then we have to show that u ∼ u′. Now by assumptionand by 6.21 all neighborhoods of [π]∗(u) and of [π]∗(u′) in Mπ(Kf )((P,X )/W )∗(C) have acommon point. Since Mπ(Kf )((P,X )/W )∗(C) is Hausdorff, this implies [π]∗(u) = [π]∗(u′).Choose V1 and W1 as in 6.22 for Z1 = [π]∗(u). Then by 6.22 (a) both u and u′ areequivalent to a point in W1. If for each of these points we fix the parameters in thediagram of 6.16, then we see that every sufficiently small neighborhood of u, resp. of u′,corresponds to a neighborhood of its (fixed) conjugate in W1. Thus we are reduced to thecase u, u′ ∈ W1. Then 6.20 implies that u and u′ are already conjugate under ∆1, henceunder ∼, as desired. q.e.d.

6.24. Definition: Let (P,X ) be mixed Shimura data, Kf an open compact subgroupof P (Af ), and Σ a Kf -admissible partial cone decomposition for (P,X ). Let U be as in 6.14,and ∼ the equivalence relation on U defined in 6.16. We define the toroidal compactificationof MKf (P,X )(C) with respect to Σ as

MKf (P,X ,Σ)(C) := U/ ∼ .

Endowed with the quotient topology, it is a Hausdorff topological space, since by 6.23 thegraph of ∼ is closed. By 6.22 it is locally isomorphic to certain open subsets ∆1\W1 of∆1\U(P1,X1, pf ) for various (P1,X1) and pf , which by 6.20 is itself locally isomorphic to thequotient of a normal complex space by a finite group that acts analytically. Thus it is a nor-mal complex space. It contains MKf (P,X )(C) as an open dense subset. If Σ is concentratedin the unipotent fibre, then it is canonically isomorphic to the relative torus embedding de-fined in 6.6. By 6.21 the projection [π] : MKf (P,X )(C)→Mπ(Kf )((P,X )/W )(C) extendsuniquely to a holomorphic map [π]∗ : MKf (P,X ,Σ)(C)→Mπ(Kf )((P,X )/W )∗(C), that isdescribed explicitly in the proof of 6.21.

Actually the term “compactification” is somewhat misused, since in generalMKf (P,X ,Σ)(C)will not be compact. Nevertheless we keep this term, since the compactness ofMKf (P,X ,Σ)(C).

6.25. Proposition: The morphisms of 3.4 extend ta arbitrary torus embeddings inthe following cases (compare 6.7). More precisely:

(a) Let (P,X ) be mixed Shimura data, pf ∈ P (Af ), and Kf ,K′f ⊂ P (Af ) open compact

subgroups, such that K ′f ⊂ pf ·Kf · p−1f . Let Σ be a Kf -admissible and Σ′ a K ′f -admissible

partial cone decomposition for (P,X ). Suppose that for every σ′ ∈ Σ′ there exists τ ∈[ ·pf ]∗Σ such that σ′ ⊂ τ . Then the map of 3.4 (a) extends uniquely to a holomorphic map

[ ·pf ] = [ ·pf ]K′f ,Kf : MK′f (P,X ,Σ′)(C)→MKf (P,X ,Σ)(C).

It is an isomorphism if K ′f = pf ·Kf · p−1f and Σ′ = [ ·pf ]∗Σ. If K ′f = Kf and Σ′ ⊂ Σ, then

it is an open embedding. In that case the complement of the image is a closed analyticsubset. If Σ′ = Σ, K ′f = Kf and pf ∈ Kf , then [ ·pf ] is the identity map. If Σ′ = Σ and K ′f

112

is a normal subgroup of Kf , then the map [ ·1]K′f ,Kf identifies MKf (P,X ,Σ)(C) with the

quotient of MK′f (P,X ,Σ)(C) by the finite group K ′f/Kf acting from the right hand side

through the morphisms [ ·kf ] on MK′f (P,X ,Σ)(C).

(b) Consider a morphism φ : (P1,X1) → (P2,X2) and open compact subgroups K1f ⊂

P1(Af ) and K2f ⊂ P2(Af ) such that φ(K1

f ) ⊂ K2f . Let Σ1 be a K1

f -admissible partial cone

decomposition for (P1,X1) and Σ2 a K2f -admissible partial cone decomposition for (P2,X2).

Suppose that for every σ1 ∈ Σ1 there exists σ2 ∈ Σ2 such that φ(σ1) ⊂ σ2. Then the mapof 3.4 (b) extends to a holomorphic map

[φ] = [φ]K1f ,K

2f

: MK1f (P1,X1,Σ1)(C)→MK2

f (P2,X2,Σ2)(C).

If φ : (P,X ) → (P,X ) is an automorphism, φ(K1f ) = K2

f and Σ1 = φ∗Σ2, then [φ] is anisomorphism.

Outline of proof: To extend the respective maps to the toroidal compactification,one constructs an obvious extension of the corresponding map on the level of U . In case(b), one uses the functoriality of boundary components (4.16). Then one shows that thisextension is compatible with the equivalence relation of 6.16. This is just a matter of goingthrough the definitions, so the details are left to the reader. Most of the properties of thesemaps carry over directly to their extensions. The rest is again easily checked on the levelof U .

Let us indicate one of the more delicate points. It concerns the last statement of (a).

There we have Σ′ = Σ and K ′f is a normal subgroup of Kf . Let U ′ be the analogue of Ufor K ′f in place of Kf , and consider two points u1, u2 ∈ U

′with, say, u1 ∈ U

′(P1,X1, pf ).

Then the crucial step is to verify that the image of u1 and u2 in U are equivalent under ∼if and only if there exists kf ∈ Kf such that u2 ∼ [id](u1), where [id] is the isomorphism

U ′(P1,X1, pf ) → U ′(P1,X1, pf · kf ) of 6.11 (a), resp. 6.15. But that is clear from thedefinition of ∼. q.e.d.

6.26 Proposition: Suppose that Kf is neat and Σ is smooth with respect to Kf .Then MKf (P,X ,Σ)(C) is smooth.

Proof. SinceMKf (P,X ,Σ)(C) is locally isomorphic to open subsets of ∆1\U(P1,X1, pf )for various (P1,X1) and pf , it suffices to show that every such quotient is smooth. By theproof of 6.20 every connected component of this quotient is an open subset of ΓQ\(Γ1\X 0

1 )Σ01(X 0

1 ,P1,p1,f ),

where X 01 is a connected component of X1, p1,f ∈ P1(Af ), ΓQ = StabQ(Q)(X 0

1 ) ∩ p1,f · pf ·Kf · p−1

f , Γ1 = P1(Q) ∩ ΓQ, and Σ01 = (([ ·pf ]∗Σ)|(P1,X1))

0. Since ΓQ is neat, its imagein (Q/(Z(P ) · P1))(Q) is torsion free. Define ΓZP1 := (Z(P ) · P1)(Q) ∩ ΓQ, then by 6.20ΓQ/ΓZP1 acts freely on our connected component, so it suffices to show that the quotientΓZP1\(Γ1\X 0

1 )Σ01(X 0

1 ,P1,p1,f ) is smooth.

Now (Γ1\X 01 )Σ0

1(X 01 ,P1,p1,f ) is a relative torus embedding over Γ1\X 0

1 /U1(C), so we havethe map

ΓZP1\(Γ1\X 01 )Σ0

1(X 01 ,P1,p1,f ) → ΓZP1\X 0

1 /U1(C).

113

Let Γ0 be the subgroup of all elements γ ∈ ΓZP1 which operate trivially on X 01 /U1(C).

Clearly it maps to the center Z(P1/U1)(Q), but since ΓZP1 is neat and Z(P1/U1) is generatedby the image of Z(P ) and a torus that is an extension of a Q-split torus with a torus ofcompact type, it must be contained in (Z(P ) · U1)(Q). Now 2.11 implies that any elementz ∈ Z(P )(R), which fixes one connected component of X , must operate trivially on X . Thisshows the equalities

Γ0 = (Z(P ) · U1)(Q) ∩ ΓZP1 = (z ∈ Z(P )(Q) | z|X = id × U1(Q)) ∩ ΓZP1 .

In particular ΓZP1/Γ0 is isomorphic to the image of ΓZP1 in ((Z(P ) · P1)/(Z(P ) · U1))(Q),which is again neat, hence torsion free. On the other hand ΓZP1 acts properly discontiuouslyon X 0

1 in the usual sense, so ΓZP1/Γ0 acts freely on Γ0\X 01 /U1(C). Thus it suffices to prove

that Γ0\((Γ0 ∩ Γ1)\X 01 )Σ0

1(X 01 ,P1,p1,f ) is smooth.

If we let ΓU be the image of Γ0 under the projection Z(P ) × U1 → U1, we mayuse 5.4 to write this as (ΓU\X 0

1 )Σ01(X 0

1 ,P1,p1,f ). This is a relative torus embedding over

a smooth base with respect to the torus U1(C)/ΓU , so by 5.3 it suffices to show thatΣ0

1(X 01 , P1, p1,f ) is smooth with respect to the lattice ΓU (−1) in U1(Q)(−1). Applying

6.5 we find that Σ01(X 0

1 , P1, p1,f ) is the same rational partial polyhedral decomposition ofC∗(X 0, P1) ⊂ C∗(X 0

1 , P1) as Σ(X 0, P1, p1,f · pf ). Also we get that ΓU is the image of

(z ∈ Z(P )(Q) | z|X = id × U1(Q)) ∩ (p1,f · pf ) ·Kf · (p1,f · pf )−1

under the projection Z(P )×U1 → U1. Thus the desired property is precisely that requiredin the definition 6.4 of the smoothness of Σ. q.e.d.

6.27. Proposition: Suppose that Σ is a complete Kf -admissible cone decompositionfor (P,X ). Then MKf (P,X ,Σ)(C) is compact.

Proof. Since Mπ(Kf )((P,X )/W )∗(C) is compact, it suffices to show that for all suffi-ciently small V1 andW1 as in 6.22 the image ofW1 is relatively compact inMKf (P,X ,Σ)(C).But this image is isomorphic to ∆1\W1, so it suffices to show that it is relatively compactin ∆1\U(P1,X1, pf ).

In 6.22 we may replaceW1 by the closure ofW1∩U(P1,X1, pf ), hence it suffices to showthat ∆1\(W1 ∩ U(P1,X1, pf )) is relatively compact. By the proof of 6.22 the intersectionof W1 with a connected component of U(P1,X1, pf ) is the image of im−1(D) ∩ ψ−1

1 (Y1)in Γ1\X 0. There is an arithmetic subgroup ΓQ of Q(Q) such that the image of Γ1\X 0 in∆1\U(P1,X1, pf ) is isomorphic to ΓQ\X 0. After replacing ΓQ by a smaller subgroup, whichdoes not change the desired assertion, we may assume that ΓQ = Γ1 · Γ′, where Γ′ ⊂ ΓQcentralizes P1/W1. Then Γ′ acts trivially on X 0

1 /W1, hence it maps the subset ψ−11 (Y1) ⊂ X 0

to itself. On the other hand by 6.19 (b) there exists a convex rational polyhedral coneτ ⊂ C∗(X 0, P1) such that C∗(X 0, P1) = Γ′ · τ . Without loss of generality we may assumethat D is a standard core, invariant under Γ′, then it follows that D = Γ′ · (τ ∩D). Thisimplies that W1 is the union of all Γ′-conjugates of the image of im−1(τ ∩D) ∩ ψ−1

1 (Y1) inΓ1\X 0. Thus it suffices to show that this image is relatively compact in U(P1,X1, pf ).

For this we may suppose that Γ1 is neat, then the projections

(Γ1\X 0)Σ(X 0,P1,pf )[π′]−−−→→ Γ1\X 0

1 /U1(C) ı Γ1\(X1/W1)0,

114

are topologically locally trivial bundles. The fibres of the second projection are compact, sosince Yi is relatively compact, the image of our set in Γ1\X 0

1 /U1(C) is relatively compact.Thus it suffices to show the ralative compactness in every fibre of the first projection. Sincethe map im : X 0

1 ı U1(R)(−1) induces a map “ord” as in 5.8, we are reduced to the torusembedding ΓU\U1(C)Σ0

1(X 01 ,P1,p1,f ), where we have to show that the subset ord−1(τ ∩D) is

relatively compact in the subset∐σ∈Σ(X 0,P1,pf )

πσ(ord−1(C(X 0, P1))).

Now we use the completeness of Σ. It says on the one hand that Σ(X 0, P1, pf ) is a completerational polyhedral decompostion of C∗(X 0, P1), hence τ ∩ C(X 0, P1) is covered by theτ ∩ σ ∩ C(X 0, P1) for all σ ∈ Σ(X 0, P1, pf ). On the other hand the finiteness implies thatthere exist only finitely many such σ for which this set is non-empty. Indeed, it suffices toconsider only those σ with σ0 ⊂ C(X 0, P1). By 6.4 (v) the double quotient P (Q)\Σ/Kf

is finite, which implies that there are only finitely many Γ′-orbits [σ] in Σ(X 0, P1, pf ) withσ0 ⊂ C(X 0, P1). Moreover for every representative σ, 6.19 (a) says that there exist onlyfinitely many Γ′-conjugates γσ such that τ ∩ γσ ∩ C(X 0, P1) is nonempty.

Thus altogether it follows that τ ∩C(X 0, P1) is contained in the union of finitely manyσ ∈ Σ(X 0, P1, pf ). Observe that it is enough to take only those σ with R · σ = U1(R)(−1),since by completeness every other cone in Σ(X 0, P1, pf ) is a face of such a cone. So inparticular τ ∩D is contained in the union of finitely many σ∩D for σ ∈ Σ(X 0, P1, pf ) withR · σ = U1(R)(−1). Thus it is enough to prove that for every convex rational polyhedralcone σ ⊂ C∗(X 0, P1) with R · σ = U1(R)(−1) the set ord−1(σ ∩D) is relatively compact inthe subset ∐

ρ face of σ

πρ(ord−1(C(X 0, P1)))

of the affine torus embedding (ΓU\U1(C))σ.

By the following lemma 6.28 there exists a compact subset V ⊂ C(X 0, P1) such thatσ ∩ D ⊂ V + σ. The assumption R · σ = U1(R)(−1) means that the σ-stratum is just apoint, hence by 5.9 (b) ∐

ρ face of σ

πρ(ord−1(V + σ))

is compact. This set contains im−1(σ ∩ D), and is contained in∐ρ face of σ πρ

(ord−1(C(X 0, P1))), as desired. q.e.d.

6.28. Lemma: Let (P,X ) be mixed Shimura data, (P1,X1) a rational boundarycomponent and X 0 a connected component of X which maps to X1. Let σ ⊂ C∗(X 0, P1)be a convex rational polyhedral cone, and D ⊂ C(X 0, P1) a core. Then σ ∩D ⊂ V + σ forsome compact subset V ⊂ C(X 0, P1).

Proof. Without loss of generality we may assume that D is the standard core D0 of 6.1and that σ0 ⊂ C(X 0, P1). If (P1,X1) is an improper boundary component of (P,X ), thenD0 = U(R)(−1), so the assumption holds with V = 0. Otherwise σ is not completelycontained in C(X 0, P1), since for instance 0 ∈ σ. Observe that if σ is the union of finitely

115

many other convex rational polyhedral cones, then it is enough to prove for each of those.Thus after splitting up σ in a suitable way we may assume that there exists a rationalboundary component (P2,X2) between (P1,X1) and (P,X ), different from (P1,X1), suchthat σ ⊂ C∗(X 0, P1)

∐C(X 0, P1). Let ρ := σ ∩ C∗(X 0, P2), and choose finitely many

ui ∈ σ ∩ C(X 0, P1) ∩ U1(Q)(−1) which together with ρ generate σ as a convex cone. Wemay suppose that we have at least on ui, since otherwise σ = ρ, so σ∩D = σ∩C(X 0, P1) = ∅anyway.

Choose a linear form ` : U1(Z)(−1)→ Z which vanishes on C∗(X 0, P2) and is strictlypositive on C(X 0, P1). Then the definition of D0 implies `(D0) ⊂ R≥1. After replacing theui by positive rational multiples we may assume that `(ui) = 1. Let V be the convex closureof these ui, this is a compact subset of u ∈ C(X 0, P1) | `(u) = 1. Now σ = ρ + R≥0 · V ,so if an element u+λ · v ∈ σ for u ∈ ρ, λ ∈ R≥0, v ∈ V lies in D0, then λ = `(u+λ · v) ≥ 1.Thus σ ∩D0 is contained in ρ+ R≥1 · V = ρ+ V + R≥0 · V = V + σ, as desired. q.e.d.

116

Chapter 7

Stratification of toroidalcompactification

In this chapter we study in detail the natural stratification of the toroidal compactification.In particular we describe the closure of a stratum as the toroidal compactification of adifferent mixed Shimura variety. If the stratum lies in the torus embedding along theunipotent fibre, then this can be done without extra restrictions. In the general case,however, the closure of a stratum may have self-intersections. To prohibit these we assumea certain condition 7.12 for Kf and the cone decomposition. Also, for simplicity and sincewe do not need the general case, we restrict attention to the case where Σ is complete,when dealing with an arbitrary stratum. Under these assumptions we shall even identifyan open neighborhood of the closure of the stratum with an open neighborhood in a simplertoroidal compactification.

In 7.1–4 we describe the natural stratification of MKf (P,X ,Σ)(C), induced by its defini-tion as a toroidal compactification, in terms of other mixed Shimura varieties. Proposition7.5 allows to translate certain local questions at the boundary between the two types ofcompactification, which will be needed in 8.1. The rest of the chapter deals with the closureof a stratum. First we consider the Baily-Borel compactification (7.6). Then we explain(7.7) how a cone decomposition induces one for a boundary stratum, and in 7.8 we provethat the properties “finite” and “complete” are preserved under this operation. In 7.9 wedescribe the closure of a stratum that already lies in the torus embedding along the unipo-tent fibre. In 7.11 we explain the main construction in the general case. In 7.12 we statethe condition on (Kf ,Σ), and in 7.13 we show it is usually not a big restriction. In 7.15–17we prove the remaining assertions about the closure of an arbitrary stratum and an openneighborhood of this closure.

7.1. Stratification of a torus embedding along the unipotent fibre: Let (P,X )be irreducible mixed Shimura data, Kf an open compact subgroup of P (Af ), and Σ aKf -admissible partial cone decomposition for (P,X ) that is concentrated in the unipotentfibre. Then MKf (P,X ,Σ)(C) possesses a natural stratification as a torus embedding. Wewant to describe this stratification in terms of other mixed Shimura varieties.

Since (P,X ) is irreducible, it is its own boundary component, so by 6.4 (iv) Σ(X 0, P, pf )

117

is independant of pf ∈ P (Af ). Thus the data conveyed by Σ consists of a rational par-tial polyhedral decomposition of U(R)(−1) for every connected component X 0 of X , suchthat the whole is invariant under the left action of P (Q). Let [σ] be a double coset inP (Q)\Σ/P (Af ). Since (P,X ) is irreducible, by 2.14 (a) P (Q) acts through homotheties onU(R)(−1). Thus if we view σ as a cone in U(R)(−1), the subspace R · σ ⊂ U(R)(−1) doesnot depend on the choice of the representative σ ∈ [σ]. Moreover, since σ is a convex ra-tional polyhedral cone, this subspace comes from a unique rational subspace of U(Q)(−1).Thus there exists a unique subgroup 〈[σ]〉 ⊂ U such that R · σ = 〈[σ]〉(R)(−1) for everyrepresentative σ ∈ [σ]. Let us also note that P (Q) ∩ P (R)0 maps X 0 and every cone toitself, so the double coset [σ] consists of finitely many P (Af )-orbits. In other words the set[σ]/P (Af ) is finite.

We now define P[σ] := P/〈[σ]〉 and U[σ] := U/〈[σ]〉, and let X[σ] be the P[σ] · (R) ·U[σ](C)-orbit generated by (X 0/〈[σ]〉(C)) × σ · P (Af ) in (X 0/〈[σ]〉(C)) × ([σ]/P (Af )), such thatσ ∈ [σ] ∩ Σ(X 0, P, 1). Clearly this does not depend on the choice of the representativeσ. Since [σ]/P (Af ) is a finite set with left action of P (Q)/(P (Q) ∩ P (R)0) ∼= π0(P[σ](R) ·U[σ](C)), the pair (P[σ],X[σ]) is mixed Shimura data together with the obvious map X[σ] →Hom(SC, P[σ],C). The canonical morphism (P[σ],X[σ]) → (P,X )/〈[σ]〉 is an isomorphism ifand only if every element of StabP (Q)(X 0) acts by a positive scalar on U . For example, thisis so whenever (P,X ) is a proper boundary component of some other mixed Shimura data.We denote the canonical projection P → P[σ] by π[σ].

7.2. Proposition: Consider (P,X ), Kf , and Σ as in 7.1. There is a canonicalstratification

MKf (P,X ,Σ)(C) ∼=∐

Mπ[σ](Kf )(P[σ],X[σ])(C),

where the disjoint union is extended over all double cosets [σ] in P (Q)\Σ/P (Af ). Each

stratum Mπ[σ](Kf )(P[σ],X[σ])(C) is a locally closed analytic subset. Each such stratum is a

finite union of strata in the natural stratification of MKf (P,X ,Σ)(C) as a torus embedding.

Construction-Proof: Without loss of generality we may assume that Kf is neat.Fix X 0, x ∈ X 0 and pf ∈ P (Af ). The definition of X[σ] implies that the fibre of X[σ] →X/〈[σ]〉(C) over [([x]), pf )] is canonically isomorphic to [σ] ∩ Σ(X 0, P, pf ). With the nota-

tions of 6.6 there is a canonical morphism Mπ[σ](Kf )(P[σ],X[σ])(C) → MK′f (P ′,X ′)(C) andx and pf determine an isomorphism of the fibre over [π′]([(x, pf )]) under this map with

(ΓU\U(C)/〈[σ]〉(C))× ([σ] ∩ Σ(X 0, P, pf )).

By 5.3 this is canonically isomorphic to the union of all strata associated to [σ]∩Σ(X 0, P, pf )in the torus embedding of Γ(pf )\U(C) with respect to Σ(X 0, P, pf ). This gives the desired

map Mπ[σ](Kf )(P[σ],X[σ])(C) → MKf (P,X ,Σ)(C). The remaining assertions follows from5.2 and 5.3. q.e.d.

Remark. In general the assertion would be false for (P,X )/〈[σ]〉 in place of (P[σ],X[σ]).

For example take (P,X ) = (P0, h(X0)) as in 2.24 and Σ(X 0, P, pf ) = √−1 ·R≤0, 0,

√−1 ·

R≥0 for every pf ∈ P (Af ). Then there are exactly two double cosets in P (Q)\Σ/P (Af ),and by the proposition the closed orbit is canonically isomorphic to Mπ(Kf )(Gm,Q,H0)(C).

118

This Shimura variety cannot be written as the disjoint union of other Shimura varieties,and replacing H0 by h(H0) in general yields a proper quotient. This situation was one ofthe reasons to allow for finite coverings of h(X ) in the definition 2.1.

7.3. Stratification of an arbitrary toroidal compactification: Now let (P,X ) bearbitrary and mixed Shimura data, Kf an open compact subgroup, and Σ a Kf -admissiblepartial cone decomposition for (P,X ). Then MKf (P,X ,Σ)(C) is stratified by the in-

verse images of the strata ∆1\Mπ1(K1f )((P1,X1)/W1)(C) of Mπ(Kf )((P,X )/W )∗(C) as in

6.3. This inverse image is isomorphic to ∆1\∂U(P1,X1, pf ). In fact, by the proof of 6.21∂U(P1,X1, pf ) ⊂ U maps to this stratum, and by 6.18 the resulting map factors through∆1\∂U(P1,X1, pf ). By 6.22 this map is a closed embedding over some neighborhood of ev-

ery relatively compact subgroup of ∆1\Mπ1(K1f )((P1,X1)/W )(C). Thus it is locally closed

embedding.

Going one step further, we stratify ∆1\∂U(P1,X1, pf ) as in 7.2. Let Σ1 := ([ ·pf ]∗Σ|(P1,X1)

and consider a double coset [σ] in P1(Q)\Σ01/P1(Af ) such that σ0 ⊂ C(X 0, P1). Let

Mπ[σ](K1f )(P[σ],X[σ])(C) be the corresponding stratum of MK1

f (P1,X1,Σ01)(C) according to

7.2, by 6.13 it is contained in ∂U(P1,X1, pf ). Since by definition P1(Q) acts trivially onP1(Q)\Σ0

1/P1(Af ), the action of StabQ(Q)(X1)∩(P1(Af ) ·pf ·Kf ·p−1f ) on Σ0

1 factors through

∆1. Thus the image of Mπ[σ](K1f )(P[σ],X[σ])(C) in ∆1\∂U(P1,X1, pf ) is isomorphic to

Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X1,[σ])(C).

Here note that Stab∆1([σ]) acts through a finite quotient on MK1f (P1,X1,Σ

01)(C), hence

on Mπ[σ](K1f )(P1,[σ],X1,[σ])(C), since by 6.19(a) for every arithmetic subgroup ΓQ ⊂ Q(Q)

a subgroup of finite index of StabΓQ(σ) lies in Z(P ) · P1. As [σ] runs through a set ofrepresentatives for the ∆1-action on the double quotient P1(Q)\Σ0

1/P1(Af ), these subsetsform a stratification of ∆1\∂U(P1,X1, pf ) by locally closed analytic subsets.

If Σ is finite, then there are only finitely many such strata. In fact, by 6.4 (v) Σ is finiteif and only if the double quotient P1(Q)\Σ/Kf is finite. This implies the finiteness of

(StabQ(Q)(X1) ∩ (P1(Af ) · pf ·Kf · p−1f ))\Σ0

1/Kf ,

hence that of ∆1\(P1(Q)\Σ01/P1(Af )), as desired. In that case we get a finite stratification

of MKf (P,X ,Σ)(C).

Assume now that Kf is neat. Then Q(Af ) ∩ pf · Kf · p−1f is again neat, so by 0.6 its

image in PGL(U1)(Af ) is again neat. Since this image contains that of ∆1, which is anarithmetic subgroup, it follows that the image of ∆1 in PGL(U1)(Af ) is is torsion free. Let[σ] be a double coset in P1(Q)\Σ0

1/P1(Af ) such that σ0 ⊂ C(X 0, P1). From 6.19 (a) itfollows that the image of Stab∆1([σ]) in PGL(U1)(Q) is finite, so by the neatness this imageis trivial. This means that Stab∆1([σ]) acts trivially on the torus with respect to which

Mπ[σ](K1f )(P1,[σ],X1,[σ])(C) is a torus embedding.

7.4. Stratification and functoriality: Consider the situation of 6.25 (a) or (b), andthe stratification of 7.3 on both sides. It is easy to see that every stratum is mapped to a

119

stratum, and this image-stratum can be described explicitly. In fact, consider for examplethe case 6.25 (b). Let

Stab∆11([σ1])\Mπ[σ1](K11f )(P11,[σ1],X11,[σ1])(C)

be a stratum ofMK1f (P1,X1,Σ1)(C), associated to a rational boundary component (P11,X11)

of (P1,X1), p1,f ∈ P1(Af ), and σ1 ∈ Σ1(X 01 , P11, p1,f ). Let (P21,X21) be the rational bound-

ary component of (P2,X2) associated to (P11,X11) by functoriality (4.16), p2,f := φ(p1,f ),and σ2 ∈ Σ2(X 0

2 , P21, p1,f ) the unique cone so that φ(σ1) ⊂ σ2. Then the image of the

above stratum lies in the stratum of MK2f (P2,X2,Σ2)(C) associated to (P21,X21), p2,f , and

σ2. For the maps 6.25 (a) the result is analogous.

For later use we prove the following fact.

7.5. Proposition: Recall that with Z1, V1 and W1 as in 6.22, ∆1\W1 is isomorphicto the inverse image of V1 in MKf (P,X ,Σ)(C). Given Z1, the other two sets can be chosen

such that there exists a unique holomorphic map V1 → ∆1\Mπ1(K1f )((P1,X1)/W1)(C) with

which the following diagram commutes:

MKf (P,X ,Σ)(C) ⊃ ∆1\W1 → ∆1\MK1f (P1,X1,Σ

01)(C)

↓ ↓ ↓Mπ(Kf )((P,X )/W )∗(C) ⊃ V1 −→ ∆1\Mπ1(K1

f )((P1,X1)/W1)(C)

Proof. The fibre in ∆1\W1 over a point inMπ(Kf )((P,X )/W )(C)∩V1 is by the extension6.15 of the map 6.11 (d) contained in a fibre of the map on the right hand side. Thuswe already have the desired map on an open dense subset of V1. If this map extendsholomorphically to V1, then the diagram also commutative for this extension. By Riemann’sextension theorem (see [GR] ch.7 §4.2 p.144) it suffices to construct a continuous extensionof this map.

As in the proof of 6.22 let D ⊂ C(X 0, P1) be a sufficiently small core, Y1 a relativelycompact subset of (X1/W1)0, p1,i,f a finite subset of P1(Af ), E1 := im−1(D)∩ψ−1

1 (Y1) ⊂X 0, and V1 the image of (

∐ψ2(E1)) × p1,i,f · pf in Mπ(Kf )((P,X )/W )∗(C). Then the

proof of 6.22 shows that V1 is isomorphic to the image of (∐ψ2(E1))× p1,i,f · pf in

(StabQ(Q)(X 0) ∩ P1(Af ) · pf ·Kf · p−1f )\(((X/W )∗)0 × P1(Af ) · pf ·Kf/Kf ).

For every boundary component (P2,X2) between (P1,X1) and (P,X ) let

ψ12 : (X2/W2)0 −→ (X1/W1)0

be the unique equivariant map with ψ1 = ψ12 ψ2. These maps patch together to a mapψ∗1 :

∐(X2/W2)0 → X1. By the definition of the Satake-topology

∐(X2/W2)0 is an open

subset of (X/W )∗, and ψ∗1 is a continuous map. Thus the induced map

(StabQ(Q)(X 0) ∩ P1(Af ) · pf ·Kf · p−1f )\(

∐(X2/W2)0 × P1(Af ) · pf ·Kf/Kf ) −→

−→ (StabQ(Q)(X 0) ∩ P1(Af ) · pf ·Kf · p−1f )\((X1/W1)0 × P1(Af ) · pf ·Kf/Kf )

= ∆1\Mπ1(K1f )((P1,X1)/W1)(C)

120

is continuous. Its restriction to V1 is the desired extension. q.e.d.

7.6. Closure of a stratum in the Baily-Borel compactification: Consider

the stratum ∆1\Mπ1(K1f )((P1,X1)/W1)(C) of MKf (P,X )∗(C) as in 6.3. The locally closed

embedding of this stratum into MKf (P,X )∗(C) extends uniquely to a holomorphic map

∆1\Mπ1(K1f )((P1,X1)/W1)∗(C) −→MKf (P,X )∗(C).

In fact, note that (X1/W1)∗ is a closed subset of X ∗. As in 6.3 the map

(X1/W1)∗ × P1(Af )→ P (Q)\X ∗ × P (Af )/Kf , (x, p1,f ) 7→ (x, p1,f · pf )

factors through ∆1\(P1(Q)\(X1/W1)∗ × (P1(Af )/K1f )), as desired. By construction this

induced map is continuous, and extends the embedding of the stratum. Thus by Riemann’s

extension theorem (see [GR] ch.7 §4.2 p.144) it is holomorphic. Since ∆1\Mπ1(K1f )((P1,X1)/W1)∗(C)

is compact, its image is the closure of the stratum. By checking what happens to the strata

of ∆1\Mπ1(K1f )((P1,X1)/W1)∗(C) one easily sees that the map is finite. In general, however,

we cannot hope that it is injective.

We now want to do the same for the toroidal compactification.

7.7. Induced cone decompositions: Consider irreducible mixed Shimura data(P,X ), an open compact subgroup Kf ⊂ P (Af ), and Kf -admissible partial cone decom-position Σ. Consider a double coset [σ] ⊂ P (Q)\Σ/P (Af ) and the associated mixedShimura data (P[σ],X[σ]) defined in 7.1. Let (P1,[σ],X1,[σ]) be a rational boundary com-ponent of (P[σ],X[σ]). By definition every connected component of X[σ] is of the form(X 0

[σ] = X 0/〈[σ]〉(C)) × σ · P (Af ) where X 0 is a connected component of X and σ a

representative of [σ] such that σ ∈ Σ(X 0, P, pf ) for some pf . If X 0[σ] embeds into X1,[σ], let

(P1,X1) be the unique rational boundary component of (P,X ) such that P1 := π−1[σ] (P1,[σ])

and X 0 embeds into X1. Clearly it depends only on (P1,[σ],X1,[σ]). Let pf,[σ] be the imageof pf in P[σ](Af ).

We can now define a π[σ](Kf )-admissible partial cone decomposition Σ[σ] for (P[σ],X[σ])as follows. For all X 0, pf , σ, and (P1,[σ],X1,[σ]) as above we define

Σ[σ](X 0[σ], P1,[σ], p1,[σ]) := τ mod〈[σ]〉 | τ ∈ Σ(X 0, P1, pf ) such that σ is a face of τ.

One easily verfies that this is indeed well-defined and satisfies the conditions 6.4 (i)–(iv).

Denote the double coset in P1(Q)\(Σ|(P1,X1))0/P1(Af ) generated by σ again by [σ]. Then

applying the construction of 7.1 to (P1,X1) and [σ], we get mixed Shimura data which isclearly canonically isomorphic to (P1,[σ],X1,[σ]). Moreover with the above construction wehave the relation (Σ|(P1,X1))[σ] = (Σ[σ])|(P[σ],X[σ]).

7.8. Proposition: Let (P,X ) be mixed Shimura data, Kf ∩ P (Af ) an open compactsubgroup, and Σ a Kf -admissible partial cone decomposition for (P,X ). Let pf ∈ P (Af ),(P1,X1) a rational boundary component of (P,X ), and Σ1 := ([ ·pf ]∗Σ)|(P1,X1). Let [σ] be

121

a double coset in P1(Q)\Σ01/P1(Af ) such that σ0 ⊂ C(X 0, P1, p

′f ) for some X 0 and p′f . If Σ

is finite, resp. complete, then Σ1,[σ] possesses the same property.

Proof. Both properties are invariant under [ ·pf ]∗, so we may assume pf = 1. Sup-pose that Σ is finite. In the notation of 7.7 replace (P,X ) by (P1,X1), (P1,X1) by(P2,X2) and so on, and let Q12 be the admissible Q-parabolic subgroup of P1 associ-ated to (P2,X2). To prove the finiteness of Σ1,[σ] we then have to show that the quotientΓQ12\Σ1,[σ](X 0

1,[σ], P2,[σ], p1,f,[σ]) is finite, where ΓQ12 is an arithmetic subgroup of Q1(Q).

Let Q, resp. Q2 be the admissible Q-parabolic subgroups of P associated to (P1,X1), resp.(P2,X2), then we have Q12 = P1 ∩ Q2. By definition Σ1,[σ](X 0

1,[σ], P2,[σ], p1,f,[σ]) consists

of all τ mod〈[σ]〉 for τ ∈ Σ1(X 01 , P2, p1,f ) such that σ is a face of τ . By definition of Σ1

we have Σ1(X 01 , P2, p1,f ) = Σ(X 0, P2, p1,f ), so by the finiteness of Σ these τ lie in finitely

many ΓQ2-orbits for every arithmetic subgroup ΓQ2 of Q2(Q). We may suppose that ΓQ12

is contained in ΓQ2 , then it suffices to show that all such τ which lie in a fixed ΓQ2-orbit,lie in finitely many ΓQ12-orbits.

Let us fix such a τ , and consider all γ ∈ ΓQ2 such that σ is a face of γτ . Then γ−1σis a face of τ , and since τ possesses only finitely many different faces, we are reduced toconsidering only those γ ∈ ΓQ2 with γσ = σ. Every such γ stabilizes (P1,X1), hence liesin an arithmetic subgroup of Q(Q). Since by assumption σ0 ⊂ C(X 0, P1), 6.19 (a) impliesthat all such γ lie in finitely many coset under Z(P ) · P1. Thus we are reduced to γ in anarithmetic subgroup of (Z(P ) ·P1)(Q). Since every sufficiently small arithmetic subgroup ofZ(P )(Q) acts trivially, we are even reduced to γ ∈ P1(Q). Thus the τ under considerationlie in finitely many orbits under an arithmetic subgroup of P1(Q) ∩ Q2(Q) = Q12(Q), asdesired. This proves the finiteness of Σ1,[σ].

If Σ is complete, then C∗(X 0, P2) is the union of all cones in Σ(X 0, P2, p1,f ), and we haveto show that C∗(X 0

1 , P2) is the union of all τ + 〈[σ]〉(R)(−1) where τ ∈ Σ1(X 01 , P2, p1,f ) =

Σ(X 0, P2, p1,f ) such that σ is a face of τ . By 4.22 (b) we have C∗(X 01 , P2) = C∗(X 0, P2) +

U1(R)(−1). Also since σ0 ⊂ C(X 0, P1), we have U1(R)(−1) = C(X 0, P1) + 〈[σ]〉(R)(−1) =C∗(X 0, P1) + 〈[σ]〉(R)(−1). Together this shows C∗(X 0

1 , P2) = C∗(X 0, P1) + 〈[σ]〉(R)(−1),so it suffices to show that every element u ∈ C∗(X 0, P1) lies inside τ + 〈[σ]〉(R)(−1) forsome τ ∈ Σ(X 0, P2, p1,f ) such that σ is a face of τ

To prove this fix u1 ∈ σ0, and consider the convex rational polyhedral cone ρ inC∗(X 0, P2) generated by u and u1. If ρ0 does not lie in C(X 0, P2), then we may re-place (P2,X2) by a rational boundary component between (P1,X1) and (P2,X2), so wemay assume ρ0 ⊂ C(X 0, P2). Fix an arithemetic subgroup ΓQ2 of Q2(Q), then by 6.19(a) ρ0 has non-empty intersection at most with finitely many cones in every ΓQ2-orbit inΣ(X 0, P2, p1,f ). Since Σ is finite, there are only finitely many such orbits, so ρ0 ∩ τ 6= ∅ foronly finitely many different τ ∈ Σ(X 0, P2, p1,f ). But the completeness of Σ ρ0 is covered bythe ρ0 ∩ τ0 for these τ . Thus there exists a τ ∈ Σ(X 0, P2, p1,f ) which contains the elementλ · u + (1 − λ) · u1 for all sufficiently small λ > 0. Then clearly u ∈ τ + 〈[σ]〉(R)(−1), asdesired. Moreover since τ is closed, it contains u1. But u1 lies in σ0 as well as in the interiorof a unique face of τ . Thus σ is a face of τ , which finishes the proof. q.e.d.

7.9. Proposition: Let (P,X ) be irreducible mixed Shimura data, Kf an open com-pact subgroup of P (Af ), and Σ a Kf -admissible partial cone decomposition. Let [σ] be a

122

double coset in P (Q)\Σ0/P (Af ), and Mπ[σ](Kf )((P[σ],X[σ])(C) the corresponding stratum of

MKf (P,X ,Σ0)(C) according to 7.2. Then the closure of this stratum in MKf (P,X ,Σ)(C)is canonically isomorphic to Mπ[σ](Kf )(P[σ],X[σ],Σ[σ])(C).

Proof. If Σ is concentrated in the unipotent fibre, then on every connected componentin the assertion follows from the constrution 6.6 of MKf (P,X ,Σ)(C) and the correspondingassertion 5.3 for abstract torus embeddings. For Σ arbitrary consider U , U(P1,X1, pf ) etc.as in chapter 6, and denote the analogous object for (P[σ],X[σ]) in place of (P,X ) by U [σ],

U [σ](P1,[σ],X1,[σ], pf,[σ]) etc. To relate U and U [σ] we shall construct a certain subset of U .

For every (P1,[σ],X1,[σ]) and pf as in 7.7, we have by 6.13 Σ01 = (([ ·pf ]∗Σ)|(P1,X1))

0,and (Σ[σ])

01 = (([ ·pf,[σ]]

∗(Σ[σ]))|(P1,[σ],X1,[σ]))0 analoguous. The definitions imply (Σ[σ])

01 =

(Σ01)[σ]. Since MK1

f (P1,X1,Σ01)(C) is a torus embedding along the unipotent fibre, we get

a closed embedding

Mπ[σ](K1f )(P1,[σ],X1,[σ](Σ[σ])

01)(C) −→MK1

f (P1,X1,Σ01)(C).

The explicit description of U(P1,X1, pf ) in 6.13 shows that this induces a closed embedding

U [σ](P1,[σ],X1,[σ], pf,[σ]) −→ U(P1,X1, pf ).

Denote by U[σ] the union of the images af all these maps. This is a closed subset of U ,and stable under each of the maps 6.15, hence under the equivalence relation ∼. ThusU[σ]/ ∼ is a closed subset of MKf (P,X ,Σ)(C), and it is this that we shall identify with

Mπ[σ](K1f )(P[σ],X[σ]Σ[σ])(C).

By definition U [σ] is the disjoint union∐U [σ](P1,[σ],X1,[σ], pf,[σ]),

indexed by all rational boundary components (P1,[σ],X1,[σ]) and all pf,[σ] ∈ P[σ](Af ). Now

U[σ] is the same disjoint union, but indexed by all (P1,[σ],X1,[σ]) and all pf ∈ P (Af ), so there

is a natural surjective map U[σ] → U [σ]. Let ∼[σ] be the natural equivalence relation on

U [σ], analogous to ∼ on U . It remains to show that the given map induces an isomorphism

U[σ]/ ∼ ∼= U[σ]/ ∼[σ].

For this we have to show that any two elements of U[σ] are equivalent under ∼ if andonly if their images are equivalent under ∼[σ]. The “only if” part is clear, since it is so foreach of the maps 6.15. For the other direction suppose first that their images are equal.Then we have the same (P1,[σ],X1,[σ]) and pf,[σ], but two different lifts of pf,[σ] to P (Af ).If these are pf and u · pf with uf ∈ 〈[σ]〉(Af ), then the extension 6.15 of the map 6.11 (b)with p1,f = uf is an isomorphism [ ·uf ] : U(P1,X1, uf · pf )→ U(P1,X1, pf ). Thus the twopoints are equivalent under ∼ if the following diagram is commutative:

U(P1,X1, uf · pf )[ ·uf ]−−−−−→ U(P1,X1, pf )

↑∪

↑∪

U [σ](P1,[σ],X1,[σ], pf,[σ])

123

But this is a straightforward functorial property of the stratification of 7.2. Using this, itnow remains to prove that whenever two points of U [σ] are equivalent under ∼[σ], thenthere exist points in the fibres that are equivalent under ∼. But this is clear, since it isso for each of the maps of 6.15. Thus we have proved that Mπ[σ](Kf )(P[σ],X[σ]Σ[σ])(C) =

U [σ]/ ∼[σ]∼= U[σ]/ ∼ is a closed subset of MKf (P,X ,Σ)(C), as desired. q.e.d.

7.10. Proposition: Consider an embedding φ : (P1,X1) → (P,X ) as in 2.13, i.e.where P1 is a normal subgroup of P , for instance (P1,X1) is an improper rational boundarycomponent of (P,X ). Let Kf be an open compact subgroup of P (Af ). There exist finitelymany pνf ∈ P (Af ), such that with Kν

f := P1(Af )∩pνf ·Kf ·(pνf )−1 and Σν := ([ ·pνf ]∗Σ)|(P1,X1),the group

∆ν := (StabP (Q)(X1) ∩ (P1(Af ) · pνf ·Kf · (pνf )−1))/P1(Q),

defined as in 6.18, acts through a finite quotient on MKνf (P1,X1,Σν)(C), and the maps 6.25

induce an isomorphism∐ν

∆ν\MKνf (P1,X1,Σν)(C)

∐[ ·pνf ][φ]

−−−−−−−−→MKf (P,X ,Σ)(C).

Moreover if Kf is neat, and P/P1 is an extension of a Q-split torus with a torus of compacttype, then all ∆ν = 1.

Proof. Choose the pνf as a system of representatives of

(StabP (Q)(X1) · P1(Af ))\P (Af )/Kf .

Denoting the projection P → P/P1 by ψ, the group ∆ν is contained in (P/P1)(Q)∩ψ(Kf ),which is an arithmetic subgroup of a torus. By the condition 2.1 (viii) the quotient P/(P1 ·Z(P )) is the extenison of a Q-split torus with a torus of compact type, so every sufficientlysmall arithmetic subgroup of (P/P1)(Q) is contained in the image of Z(P )(Q)∩Z(P )(R)0.But this group acts trivially on X1, hence trivially on MKν

f (P1,X1,Σν)(C). Thus ∆ν alwaysacts through a finite quotient. If Kf is neat, then so is ψ(Kf ) by 0.6, and if P/P1 is anextension of a Q-split torus with a torus of compact type, then ∆ν must be trivial.

Next if Σ is the trivial cone decomposition, then we are dealing MKf (P,X )(C) andMKν

f (P1,X1)(C), and the assertion is a special case of 7.3. For arbitrary Σ we still have ageneric isomorphism, and since MKf (P,X ,Σ)(C) is normal, the images of any two distinctindividual maps MKν

f (P1,X1,Σν)(C)→ MKf (P,X ,Σ)(C) are disjoint. Thus it remains toshow that every such map induces an open and closed embedding of ∆ν\MKν

f (P1,X1,Σν)(C).The proof of this is almost the same as that of 7.9, so we leave it to the reader. q.e.d.

7.11. The closure of an arbitrary stratum: While in the preciding propositionwe studied the closure of a stratum of MKf (P,X ,Σ0)(C), we now turn to the closure of anarbitrary stratum of MKf (P,X ,Σ)(C). It is not difficult to extend the inclusion

Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X[1,σ])(C) −→MKf (P,X ,Σ)(C)

124

of 7.3 to a holomorphic map

Stab∆1([σ])\Mπ[σ](K1f )((P1,[σ],X[1,σ],Σ1,[σ])(C) −→MKf (P,X ,Σ)(C).

But in general this map will not be injective. Also we are, for later purposes, interesed notonly in the closure of the stratum, but also in a canonical description of a small neighbor-hood of it. For simplicity we shall assume throughout that Σ is complete, since this is theonly case that we shell need in the sequel. We begin by constructing a certain map, whichwe shall later prove to be an open embedding, if a certain condition is met.

So fix (P,X ), Kf , Σ, (P1,X1), pf , K1f , and Σ1 as in 7.3. Consider U =

∐U(P2,X2, p

′f )

etc. as in chapter 6, and denote the analogous objects for (P1,X1), K1f and Σ1 in place of

(P,X ), Kf and Σ by U1 =∐U1(P2,X2, p1,f ) etc. Observe that in U the union is extended

over all rational boundary components (P2,X2) of (P,X ), and all p′f ∈ P (Af ), while in U1

only over those of (P1,X1) and p1,f ∈ P1(Af ). For the latter ones U1(P2,X2, p1,f ) is an

open subset of MK2f (P2,X2,Σ

02)(C), where

K2f = P2(Af ) ∩ p1,f ·K1

f · p−11,f = P2(Af ) ∩ p1,f · pf ·Kf · (p1,f · pf )−1

andΣ0

2 = (([ ·p1,f ]∗Σ1)|(P2,X2))0 = (([ ·p1,f · pf ]∗Σ)|(P2,X2))

0.

Thus U(P2,X2, p1,f · pf ) is also an open subset of MK2f (P2,X2,Σ

02)(C), and by the explicit

description given in 6.13 it is contained in U1(P2,X2, p1,f ).

So letting U (1) be the disjoint union of U(P2,X2, p1,f · pf ) for all rational boundarycomponents (P2,X2) of (P1,X1) and all p1,f ∈ P (Af ), we have canonical embeddings ofU (1) into U and into U1.

Let ∼ be the equivalence relation on U defined in 6.16, and ∼1 the analogous one onU1. By comparing the definitions of ∼ and ∼1 we easily find that U (1) → U1 is stable

under ∼1, that U (1)/∼1→ U1/∼1

= MK1f (P1,X1,Σ1)(C) is stable under the action of ∆1

on MK1f (P1,X1,Σ1)(C), and that the map U (1) → U factors through a map ∆1\U (1)/∼1

→U/ ∼= MKf (P,X ,Σ)(C). In other words we have extended the maps MKf (P,X )(C) ←∆1\U(P1,X1, pf ) → ∆1\MK1

f (P1,X1)(C) of 6.10 canonically to holomorphic maps

∆1\U (1)/∼1open−−−−→ ∆1\MK1

f (P1,X1,Σ1)(C)↓

MKf (P,X ,Σ)(C)

Since U (1)contains ∂U(P2,X2, p1,f · pf ) for all(P2,X2) and p1,f , it follows from 7.9 that

the closure of ∂U(P1,X1, pf ) in MK1f (P1,X1,Σ1)(C) is contained in the image of U (1)/ ∼1.

Fix a double coset [σ] ∈ P1(Q)\Σ01/P1(Af ) with σ0 ⊂ C(X 0, P1) × 1 for some X 0, then

in particular U (1)/ ∼1 is an open neighborhood of Mπ1,[σ](K1f )((P1,[σ],X1,[σ],Σ1,[σ])(C) ⊂

MK1f (P1,X1,Σ1)(C). Consider an arbitrarily small open neighborhood V ⊂ U (1)/ ∼1 of

this closure. Since Stab∆1([σ]) acts on MK1f (P1,X1,Σ1)(C) through a finite quotient, we

125

may without loss of generality assume that V is Stab∆1([σ])-invariant. Consider the maps

Stab∆1([σ])\V open−−−−→ Stab∆1([σ]) \MK1

f (P1,X1,Σ1)(C)↓

MKf (P,X ,Σ)(C)

Since the assumption Σ is complete, 6.27 implies that Mπ1,[σ](K1f )(P1,[σ],X1,[σ]Σ1,[σ])(C) is

compact. Hence so is its image in MKf (P,X ,Σ)(C), so it is equal to the closure of the

corresponding stratum Stab∆1([σ])\Mπ1,[σ](K1f )(P1,[σ],X1,[σ])(C) of MKf (P,X ,Σ)(C). So

far this is the same situation as in 7.6. But in the toroidal compactification things arebetter. In 7.15 we shall even prove that under certain conditions the vertical map is anopen embedding. Then we get in particular a canonical description of the closure of astratum of MKf (P,X ,Σ)(C) as a toroidal embedding of another mixed Shimura variety.Loosely speaking the closure of the stratum has no self-intersections. Moreover we get acanonical isomorphism of every sufficiently small open neighborhood of this closure with aneighborhood of the closure of a stratum in a different mixed Shimura variety, where thesituation is much simpler, namely where the stratum is that of a torus embedding alongthe unipotent fibre.

7.12. A certain condition on cone decompositions: Consider mixed Shimuradata (P,X ), an open compact subgroup Kf ⊂ P (Af ), and a Kf -admissible partial conedecomposition Σ. Consider pf ∈ P (Af ), a connected component X 0 of X , a rationalboundary component of (P1,X1) of (P,X ), and a rational boundary component (P2,X2)of (P1,X1). Let τ ∈ Σ(X 0, P2, pf ) and σ ∈ Σ(X 0, P1, pf ) ⊂ Σ(X 0, P2, pf ) such that σ is aface of τ . Let p ∈ P (Q), p1,f ∈ P (Af ), and kf ∈ Kf , then with the operations defined in6.4 p · p1,f · σ · kf is a again an element of Σ. Consider the following condition on the pair(Kf ,Σ):

(*) For all pf , (P1,X1), (P2,X2), τ , σ, p, p1,f , and kf as above, if both σ and p · p1,f · σ · kfare faces of τ , then p · p1,f · σ · kf = σ.

This condition will turn out to be sufficient for the purpose decribed in 7.11. If it holdsfor given (Kf ,Σ), then it clearly holds for any smaller open compact subgroup. It shouldbe noted that this condition is not related to neatness of Kf . Nevertheless the followingproposition shows that it is only a mild restriction.

7.13. Proposition: Let (P,X ), Kf and Σ be as in 7.12. If Σ is finite, then thereexists an open subgroup K ′f of Kf such that the condition of 7.12 holds for (K ′f ,Σ).

Proof. We first show that it is enough to prove that for all fixed τ and σ there existsK ′f ⊂ Kf such that the condition in 7.12 holds. In fact, by finiteness of Σ, and since everyconvex rational polyhedral cone has only finitely many different faces, there are, up to theleft action of P (Q) and the right action of Kf , only finitely many pairs (σi, τi) of conesin Σ such that σi is a face of τi. Suppose that the assertion holds for each of these withK ′f,i ⊂ Kf . Choose an open normal subgroup K ′f of Kf that is contained in every K ′f,i.Then for every p ∈ P (Q) and kf ∈ Kf the conditions also holds for K ′f and (p·σi·kf , p·τi·kf ).Thus it holds for K ′f and every (σ, τ), as desired.

126

Next fix pf , (P2,X2) and τ as in 7.12. Let (R≥0 · ui) × pf for i = 1, . . . , r be thedifferent one dimensional faces of τ . Clearly it suffices to prove the assertion for each oftheses, say for σ = (R≥0 · ui)× pf. Consider p ∈ P (Q), p1,f ∈ P1(Af ), and kf ∈ Kf . Bydefinition of the various operations, p·p1,f ·σ·kf is a face of τ if and only if p·p1,f ·pf ·Kf = pf ,p · X 0 = X 0, and R≥0 · p · ui = R≥0 · ui for some (unique) i ∈ 1, . . . , r. We have to provethat for Kf sufficiently small these conditions imply p · p1,f · σ · kf = σ. Since this equalityholds if and only if i = 1, we have to show that R≥0 · p · ui 6= R≥0 · ui for all i 6= 1 and allp ∈ StabP (Q)(X 0) ∩ pf ·Kf · p−1

f · P1(Af ), provided that Kf is sufficiently small. Note that

since R≥0 · u1 and R≥0 · ui are distinct faces of the same convex rational polyhedral cone,we have R · u1 6= R · ui. Thus we may apply the following lemma 7.14, which proves whatwe want. q.e.d.

7.14. Lemma: Let (P,X ) be mixed Shimura data, and (P1,X1), (P2,X2) two rationalboundary components of (P,X ). Let 0 6= ui ∈ Ui(Q)(−1) for i = 1, 2 such that R·u1 6= R·u2.Then there exists an open compact subgroup Kf ⊂ P (Af ) such that R · p · u1 6= R · u2 forall p ∈ P (Q) ∩Kf · P1(Af ).

Proof. Fix any isomorphism λ : Q → Q(1), and identify Ui with Lie(Ui) ⊂ Lie(P ).Then by assumption we have two different lines Q · λ(ui) in Lie(P )(Q). We can also viewthem as two distinct points [λ(ui)] in (PLie(P ))(Af ) := (Lie(P )(Af )−0)/Gm(Af ), and itsuffices to find Kf such that [λ(u2)] 6∈ Kf ·P1(Af ) · [λ(ui)]. By the irreducibility of (P1,X1),P1 acts through scalars on U1. Thus it fixes the point [λ(u1)], and Kf · P1(Af ) · [λ(u1)] =Kf · [λ(u2)]. Choose a neighborhood of [λ(u1)] in (PLie(P ))(Af ) such that a certain [λ(u2)].By the continuity of the P (Af )-action we can choose Kf such that Kf · [λ(u2)] is containedin this neighborhood, so we are done. q.e.d.

7.15. Proposition: Consider the situation of 7.11. If Σ is complete and the conditionof 7.12 holds, then for every sufficiently small Stab∆1([σ])-invariant open neighborhood of

V in MK1f (P1,X1,Σ1)(C) of Mπ1,[σ](K

1f )(P1,[σ],X1,[σ],Σ1,[σ])(C) the map

Stab∆1([σ])\V →MKf (P,X ,Σ)(C)

of 7.11 is an open embedding.

Proof. Consider the map

[π1]∗ : MK1f (P1,X1,Σ1)(C)→Mπ(K1

f )((P1,X1)/W1)∗(C).

We shall prove that for any two points x2, x3 ∈Mπ(K1f )((P1,X1)/W1)∗(C) there exist open

neighborhoods Vi of xi, and open neighborhoods V ′i of Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ1,[σ])(C) ∩

[π1]∗−1(Vi), contained in U (1)/ ∼1 such that either the image of V ′i in MKf (P,X ,Σ)(C) are

disjoint, or the restriction of the map in question to the image of V ′2∪V ′3 in Stab∆1([σ])\MK1f (P1,X1,Σ1)(C)

is an open embedding. By the compactness of Mπ(K1f )((P1,X1)/W1)∗(C) we can then choose

finitely many such V ′i, whose union of V is a Stab∆1([σ])-invariant open neighborhood of

Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ1,[σ])(C) in U (1)/ ∼1, such that the above property holds for every

127

pair of such V ′i. This implies that the restriction to Stab∆1([σ])\V of the map in questionis injective and locally an open embedding. Thus it is an open embedding, as desired.

To prove the existence of such V ′i, fix rational boundary components (Pi,Xi) of (P1,X1),and p1,i,f ∈ P1(Af ) for i = 2, 3. We may suppose that a fixed connected component X 0

1

of X1 maps to Xi. Applying 6.22 to this situation we get a ∆1i-invariant open subsetW1i ⊂ ∂U1(Pi,Xi, p1,i,f ) such that ∆1i\W1i is equal to [π1]∗−1(Vi) as above. Here if Qi isthe admissible Q-parabolic subgroup of P associated to (Pi,Xi), then ∆1i is in analogy to6.18 defined as

(Stab(P1∩Qi)(Q)(Xi) ∩ Pi(Af ) · p1,i,f ·K1f · p−1

1,i,f )/Pi(Q).

By the proof of 6.22 W1i can be given as follows. There are relatively compact open subsetYi ⊂ Xi/Wi, a finite subset pi,α,f ⊂ Pi(Af ), and an open core D1i ⊂ C(X 0

1 , Pi), such thatFi is the image of

(im−1(D1i) ∩ ψ−1i (Yi))× pi,α,f ⊂ X 0

1 × Pi(Af )

in U1(Pi,Xi, p1,i,f ), and W1i :=∐πτ (Fi), where the union is extended over all cones τ ∈

Σ1(X 01 , Pi, pi,α,f ·p1,i,f ) = Σ(X 0, Pi, pi,α,f ·p1,i,f ·pf ) for some α. The intersection of ∆1i\W1i

in MK1f (P1,X1,Σ1)(C) with Mπ1,[σ](K

1f )(P1,[σ],X1,[σ],Σ1,[σ])(C) is the image of the union of

all πτ (Fi) such that some representative of [σ] is a face of τ . We can therefore chooseV ′i := ∆1i\W ′i for every small ∆1i-invariant open subset W ′i ⊂ W1i ∩ U(Pi,Xi, p1,i,f · pf )that containes all these πτ (Fi).

Let Di be an open core in C(X 0, Pi). If the both Di and D1i are given as the interiorof the standard core with respect to the same lattice, then D1i = Di + U1(R)(−1). Weshall prove below that W ′i can be chosen to satisfy the following additional property: itsintersection with U(Pi,Xi, p1,i,f ·pf ) is equal to the image of (im−1(Di∩

∐τ0)∩ψ−1

i (Yi))×pi,α,f in U(Pi,Xi, p1,i,f · pf ), where the disjoint union

∐τ0 is extended over all τ ∈

Σ(X 0, Pi, pi,α,f · p1,i,f · pf ) for all α such that some representative of [σ] is a face of τ .Note that such W ′i is contained in some Wi as in 6.22, applied to (Pi,Xi) as a boundarycomponent of (P,X ).

If the images of V ′2 and V ′3 in MKf (P,X ,Σ)(C) are not disjoint, then the same holdsfor W ′2 and W ′3, hence also for W2 and W3 above. Thus by 6.22, if the Di are chosensufficiently small, the two ∂U(Pi,Xi, p1,i,f · pf ) must map to the same boundary stratumof Mπ(Kf )((P,X )/W )∗(C). In other words there exist p ∈ P (Q) and kf ∈ Kf such thatint(p)((P2,X2)) = (P3,X3) and p · p1,2,f · pf · kf = p1,3,f · pf . Thus by 4.20 (iii) there existsp ∈ P (Q) such that int(p1)((P2,X2)) = (P3,X3), so after applying the extenion 6.15 of themap [int(p1)] of 6.11 (c) we may assume that (P3,X3)) = (P2,X2). Then every such p liesin Q2(Q).

Now for all such p we also have

p · (im−1(D2 ∩∐

τ0) ∩ ψ−12 (Y2)) ∩ (im−1(D3 ∩

∐τ ′0) ∩ ψ−1

3 (Y3)) 6= ∅,

hence (p ·∐τ0 ∩

∐τ ′0 6= ∅. Thus there exist τ ∈ Σ(X 0, P2, p2,α,f · p1,2,f · pf ) and τ ′ ∈

Σ(X 0, P3, p3,β,f · p1,3,f · pf ), such that both τ and τ ′ have a face in [σ], and p · τ0 ∩ τ ′0 6= ∅.The latter relation implies p·τ = τ ′, which should really be written in the form p·τ ·kf = τ if

128

we view these cones as subsets of C∗(X 0, P2)×P (Af ). Let σ, resp. σ′ be the representativesof [σ] such that σ is a face of τ , and σ′ is a face of τ ′, then there exist p1 ∈ P1(Q) andp1,f ∈ P1(Af ) such that σ′ = p1 · p1,f · σ. It follows that both σ and (p−1 · p1) · p1,f · σ · kfare faces of τ . Thus the conditon in 7.12 implies (p−1 · p1) · p1,f · σ · kf = σ. As a firstconsequence, p−1 · p1 maps C(X 0, P1) to itself, so p−1 · p1, hence also p, stabilizes (P1,X1).Thus p lies in StabQ(Q)(X1). Using this, the relation p ·p1,2,f ·pf ·kf = p1,3,f ·pf now implies

that p also lies in P1(Af ) · pf · Kf · p−1f . Thus it defines a class [p] in ∆1, and this class

stabilizes [σ]. Hence after applying [int(p−1)] to U(P2,X2, p1,2,f · pf ) we are reduced to thecase where p1,3,f = p1,2,f .

In this case we may enlarge theW ′i so thatW ′3 =W ′2. It remains to show that the mapW ′2 →W2 →MKf (P,X ,Σ)(C) factors through an open embedding of the image of W ′2 in

Stab∆1([σ])\MK1f (P1,X1,Σ1)(C) . Now by 6.22 the map W2 → MKf (P,X ,Σ)(C) factors

through an open embedding of ∆2\W2, where

∆2 := (StabQ2(Q)(X2) ∩ (P2(Af ) · p2,f · pf ·Kf · (p2,f · pf )−1))/P2(Q).

The argument above shows that for any [q] ∈ ∆2 with [q] · W ′2 ∩W ′2 6= ∅ we already haveq ∈ StabQ1(X1) ∩ (P1(Af ) · pf · Kf · p−1

f ), and the class [q] ∈ ∆1 stabilizes [σ]. Thus themap W ′2 → W2 → ∆2\W2 factors through an open embedding of the image of W ′2 in

Stab∆1([σ])\MK1f (P1,X1,Σ1)(C), as desired.

It remains to prove the assertion about the special choise of W ′i. This is done in thefollowing lemma. q.e.d.

7.16. Lemma: In the notation of the proof of 7.15 there exists a ∆1i-invariant opensubset W ′i ⊂ W1i ∩ U(Pi,Xi, p1,i,f · pf ) with the properties:

(i) It contains the τ -stratum of W1i for all α and every τ ∈ Σ(X 0, Pi, pi,α,f · p1,i,f · pf ), suchthat some representative of [σ] is a face of τ .

(ii) Its intersection with U(Pi,Xi, p1,i,f · pf ) is equal to the image of (im−1(Di ∩∐τ0) ∩

ψ−1i (Yi)) × pi,α,f in U(Pi,Xi, p1,i,f · pf ), where the disjoint union

∐τ0 is extended over

all τ ∈ Σ(X 0, Pi, pi,α,f · p1,i,f · pf ) for all α such that some representative of [σ] is a face ofτ .

Proof. For every α and every ρ ∈ Σ(X 0, Pi, pi,α,f · p1,i,f · pf ), such that some represen-tative of [σ] is a face of ρ, let Fi,ρ be the image of

(im−1(Di ∩∐

τ0) ∩ ψ−1i (Yi))× pi,α,f ⊂ X 0

1 × Pi(Af )

in U(Pi,Xi, p1,i,f ), where the disjoint union is extended over all τ ∈ Σ(X 0, Pi, pi,α,f ·p1,i,f ·pf )such that ρ is a face of τ . Let W ′i,ρ be the union of πρ′(Fi,ρ) for all faces ρ′ of ρ, and W ′ithe union of all theseW ′i,ρ. ClearlyW ′i is ∆1i-invariant and satisfies property (ii). To provethe rest it suffices to show that every W ′i,ρ is open and contains the ρ-stratum of W1i.

So fix ρ, and consider the subset V := Di ∩∐ρ face of τ τ

0 above. By definition itsatisfies V = V + ρ, so to prove the openness of W ′i,ρ it suffices by 5.9 (a) to show thatV is open in Ui(R)(−1). Likewise consider the ρ-stratum πρ(Fi) of W1i and the ρ-stratumπρ(Fi,ρ) ofW ′i,ρ. Since Fi is the image of (im−1(D1i∩ψ−1

i (Yi))×pi,α,f in U1(Pi,Xi, p1,i,f ),

129

and Fi,ρ that of (im−1(Di ∩∐τ0) ∩ ψ−1

i (Yi))× pi,α,f, they two ρ-strata are equal if andonly if R · ρ+D1i = R · ρ+ V .

To prove that V is open in Ui(R)(−1) fix u ∈ Di ∩ τ0 such that ρ is a face of τ . Sinceu ∈ Di ⊂ C(X 0, Pi), there exists a convex rational polyhedral cone π ⊂ C∗(X 0, P1) suchthat Di ∩ π is a neighborhood of u in Ui(R)(−1). As in the proof of 6.27 the completenessof Σ implies that π ∩ C(X 0, P1) is contained in the union of finitely many cones τ ′ ∈Σ(X 0, Pi, pi,α,f · p1,i,f · pf ). Thus a union of finitely many such τ ′ is a neighborhood of u.Since each τ ′ is closed, we may leave out those which do not contain u. On the other handif u ∈ τ ′, then τ0 ∩ τ ′ 6= ∅, so τ must be a face of τ ′. Hence ρ is a face of τ ′, so a union ofsuch τ ′ is a neighborhood of u. Since Di is open, this shows that V is a neighborhood of u,as desired.

To prove the equality R · ρ + D1i = R · ρ + V note that the inclusion “⊃” holds bydefinition. Thus it remains to show D1i ⊂ R · ρ+ V . Since

D1i = Di + U1(R)(−1)

= Di + C∗(X 0, Pi) + U1(R)(−1)

= Di + C∗(X 0, Pi) + R · σ= Di + R · σ⊂ Di + R · ρ,

this is equivalent to Di ⊂ R · ρ + V . To prove this fix u ∈ Di and u0 ∈ ρ0, and let π bethe convex rational polyhedral cone R≥0 · u + R≥0 · u0. By the same argument as aboveπ ∩C(X 0, Pi) is covered by finitely many τ ∈ Σ(X 0, Pi, pi,α,f · p1,i,f · pf ). Thus there existsone such τ such that λ ·u+(1−λ) ·u0 ∈ τ0 for all sufficiently samll λ > 0. This implies thatu0 is contained in the closure of τ0, so ρ must be a face of τ . It also shows that u+µ·u0 ∈ τ0

for every sufficiently big real number µ. Since Di = Di + C∗(X 0, Pi), and u ∈ Di, we alsohave u+ µ · u0 ∈ Di for every µ > 0. Thus u = −µ · u0 + (u+ µ · u0) ∈ R · ρ+Di ∩ τ0, asdesired. q.e.d.

7.17. Corollary: Consider (P,X ), Kf , Σ, (P1,X1), pf , K1f , Σ1, and [σ] as in 7.3.

Assume that Σ is complete, and that (Kf ,Σ) satisfies the condition in 7.12. Then:

(a) The closure of the stratum

Stab∆1([σ])\Mπ1,[σ](K1f )(P1,[σ],X1,[σ])(C)

in MKf (P,X ,Σ)(C) is canonically isomorphic to

Stab∆1([σ])\Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ1,[σ])(C),

where Σ1,[σ] is as defined in 7.7.

(b) Every sufficiently small open neighborhood of this closure is canonically isomorphic to

Stab∆1([σ])\V for some Stab∆1([σ])-invariant open neighborhood V ofMπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ1,[σ])(C),

in MK1f (P1,X1,Σ1)(C). In particular MKf (P,X )(C) →MKf (P,X ,Σ)(C) is a toroidal em-

bedding without self-intersection in the sence of [KKMS] ch.II §1 p.57.

Remark. The assertion also holds without the assumption that Σ is complete.

Proof. Both assertions are direct consequences of 7.11 and 7.15. q.e.d.

130

Chapter 8

Construction of ample line bundles

In this chapter we construct ample invertible sheaves on MKf (P,X ,Σ)(C) for certain conedecompositions Σ. This is the first step towards the algebraization of MKf (P,X ,Σ)(C);the second step is carried out in the next chapter, where we shall construct the necessarycone decompositions. The main result of this chapter is 8.14, which is a generalization ofTai’s theorem in [AMRT] ch.IV §2.1 p.312.

First, we express well-known facts about the canonical sheaves on Shimura varieties inour language. We show (8.1) that the canonical sheaf on MKf (P,X ,Σ)(C) with at mostlogarithmic poles along the boundary is isomorphic to the pullback of a canonically definedinvertible sheaf on the Baily-Borel compactification Mπ(Kf )((P,X )/W )∗(C). This sheaf isample (8.4). Thus, to construct an ample invertible sheaf on MKf (P,X ,Σ)(C) it remainsto construct one which is ample on every fibre of the projection [π]∗ : MKf (P,X ,Σ)(C)→Mπ(Kf )((P,X )/W )∗(C).

As explained in 8.15, we need a more sophisticated approach for this than in [AMRT]

ch.IV §2. The main idea is to interpret some morphismsMK′f (P ′,X ′,Σ′)(C)→MKf (P,X ,Σ)(C)of 6.25 as line bundles, and to give sufficient conditions for such line bundles to the rela-tively ample (or anti-ample) over the Baily-Borel compactification. In 8.5–6 we describethe conditions under which the above morphism represents a line bundle. The amplenesscondition has a geometric content like in 5.19, and is better expressed for anti-ample bun-dles. Thus in 8.13 we give a sufficient condition for the inverse of this line bundle to berelatively ample over Mπ(Kf )((P,X )/W )∗(C).

All in all, this shows that in certain cases an ample line bundle can be described interms of other mixed Shimura varieties and their internal geometry. This will play a rolein chapter 12.

8.1. Proposition: Let (P,X ) be mixed Shimura data and Kf a neat open subgroup ofP (Af ). There exists an invertible sheaf Lp on Mπ(Kf )((P,X )/W )∗(C), up to isomorphismdepending only on (P,X )/U , such that for every Kf -admissible partial cone decompositionΣ for (P,X ) there is an isomorphism

([π]∗)∗Lp ∼= ω[dlog],

131

where ω[dlog] is the canonical sheaf onMKf (P,X ,Σ)(C) with logarithic poles alongMKf (P,X ,Σ)(C)rMKf (P,X )(C), locally defined as in 5.26.

Proof. First suppose that (P,X ) is irreducible, and consider the restriction of ω[dlog] toMKf (P,X ,Σ0)(C). Then in the notations of 6.8 [π′] : MKf (P,X ,Σ0)(C)→Mπ′(Kf )((P,X )/U)∗(C)

is a relative torus embedding with respect to the family of tori MKUf oφ(Kf )(P∗,X∗)(C) →

Mφ(Kf )(Gm,Q,H0)(C). Fix an invariant differential form on every connected component ofMKfoφ(Kf )(P∗,X∗)(C). Then 5.27 induces an isomorphism between ω[dlog]|MKf (P,X ,Σ0)(C)

and the pullback of the canonical sheaf of Mπ′(Kf )((P,X )/U)(C). This isomorphism iscanonical up to a scalar factor on each connected component of MKf (P,X ,Σ0)(C).

Next by 3.14 every fibre of Mπ′(Kf )((P,X )/U)(C) → Mπ(Kf )((P,X )/W )(C) is a com-pact complex torus. Since this is a smooth family, the relative canonical sheaf is the pullbackof a unique invertible sheaf on Mπ(Kf )((P,X )/W )(C), the sheaf of invariant relative topdifferentials. For the (absolute) canonical sheaf the analogous fact follows. Thus there is anatural invertible sheaf on Mπ(Kf )((P,X )/W )(C), depending only on (P,X )/U , whose pull-back to MKf (P,X ,Σ0)(C) is isomorphic to ω[dlog]. It is important that this isomorphismis canonical up to a scalar factor on each connected component.

Let us now go back to the general case. The preceding remarks, applied to W1 ⊂MK1

f (P1,X1,Σ01)(C) as in 6.22, show that the restriction of ω[dlog] to ∆1\W1 ⊂MKf (P,X ,Σ)(C)

is isomorphic to the pullback of a natural invertible sheaf on ∆1\Mπ1(K1f )((P1,X1)/W1)(C).

Hence by 7.5 it is also isomorphic to the pullback of a natural invertible sheaf on V1 ⊂Mπ(Kf )((P,X )/W )∗(C). Since these isomorphisms, for a covering ofMπ(Kf )((P,X )/W )∗(C),are canonical up to a scalar on each connected component, they define a global isomorphismbetween ω[dlog] and the pullback of an invertible sheaf Lp on Mπ(Kf )((P,X )/W )∗(C).Clearly it depends only on (P,X )/U , and not on Σ or even the existence of Σ with certainproperties, e.g. completeness. q.e.d.

8.2. Projectivity of the Baily-Borel compactification: Let (P,X ) be pureShimura data and Kf a neat open subgroup of P (Af ). Denote by MKf (P,X )+(C) theunion of MKf (P,X )(C) with all boundary strata of codimension 1. This is smooth, its com-pletement in MKf (P,X )∗(C) has codimenion ≥ 2, and the complement of MKf (P,X )(C)in MKf (P,X )+(C) is a smooth divisor. Hence we may consider the sheaf ω[dlog] of all topdifferential forms on MKf (P,X )+(C) with at most simple poles along MKf (P,X )+(C) rMKf (P,X )(C).

For any integer n the space of sections

Γ(MKf (P,X )+(C), ω[dlog]⊗n)

is isomorphic to the space of holomorphic automorphic forms of a certain weight. IndeedΓ(MKf (P,X )(C), ω[dlog]⊗n) is isomorphic to a space of holomorphic automorphic forms“without condition at ∞” (compare [AMRT] ch.IV §1). By the Koecher principle thecondition at∞ is necessary only for each simple factor of P ad that is isomorphic to PGL2,Q.In that case it is well-known and easy to check that the condition at ∞ is equivalent to theextendability to a section of ω[dlog]⊗n.

132

Fix an integer n ≥ 1 and consider the projective space

PN := (Γ(MKf (P,X )+(C), ω[dlog]⊗n) r 0)/C×.

If ω[dlog]⊗n on MKf (P,X )+(C) is generated by global sections, then there is a canonicalholomorphic map MKf (P,X )+(C) → PN . Now [BB] thm. 10.11 asserts that for somen ≥ 1, ω[dlog]⊗n is generated by global sections over MKf (P,X )+(C), and the canonicalmap MKf (P,X )+(C)→ PN extends to a closed embedding MKf (P,X )∗(C)→ PN . Observethat this then follows for every positive multiple of n.

Denote by O(1) the pullback of MKf (P,X )∗(C) of the standard ample sheaf on PN . Bythe above we have an isomorphism O(1)|MKf (P,X )+(C)

∼= ω[dlog]⊗n. On the other hand we

have by construction LP |MKf (P,X )+(C)

∼= ω[dlog], since P is reductive. The following lemma

now implies O(1) ∼= L⊗nP , so LP is ample.

8.3. Lemma: Let X be a normal complex space and Z a closed subspace of codimen-sion at least 2. Let L1 and L2 be two invertible sheaves on X, whose restrictions to X rZare isomorphic. Then they are already isomorphic over X.

Proof. We shall show that any given isomorphism L1|XrZ∼= L2|XrZ extends to an

isomorphism L1∼= L2. Since this is a local question, we may assume that L1 = L2 = OX .

Then the isomorphism corresponds to a section f ∈ Γ(X rZ,O∗X). Now by Riemann’s ex-tension theorem (see [GR] ch.7 §4.2 p.144) both f and f−1 extend to holomorphic functionson X. Thus f extends to a section in Γ(X,O∗X), as desired. q.e.d.

More generally, we have the following proposition (which will not be needed in thesequel).

8.4. Proposition: For arbitrary mixed Shimura data, the inverible sheaf LP definedin 8.1 is ample.

Sketch of proof: Since ampleness is invariant under finite coverings, using 7.10 wemay reduce to the case where (P,X ) is irreducible. Next, by 8.1 LP depends only on(P,X )/U , so we may assume that U = 1. Put LrelP := LP⊗(LP/W )⊗−1. Since by 8.2 LP/Wis ample, it suffices to show that some positive tensor power of LrelP is generated by globalsections. If V = 1, then we are done. Otherwise let 2 ·g = dim(V ) > 0, then as in the proofof 2.26 we can find a morphism φ : (P,X )/V → (CSP2g,Q,H2g) and an isomorphism

(P,X ) ∼= (V2g o CSP2g,Q,H′2g)×(CSP2g,Q,H2g) (P,X )/V.

Abbreviating (P ′,X ′) ∼= (V2g o CSP2g,Q,H′2g), it is easy to check from the definition that

LrelP ∼= [φ]∗LrelP ′ , where LrelP ′ = LP ′ ⊗ (LP ′/W ′)⊗−1. Thus it suffices to prove the same

assertion for LrelP ′ .By 8.2 this clearly follows from the well-known formula LP ′/W ′ ∼= (LrelP ′ )⊗(g+1). To prove

it, note that the restriction of LrelP ′ to MKf (CSP2g,Q,H2g)(C) is isomorphic to the pullbackof the sheaf of relative top differentials under the zero section, and LP ′/W ′ isomorphic to

the canonical sheaf. Over each connected of MKf (CSP2g,Q,H2g)(C) the formula follows

133

from an easy calculation with the automorphy factors. If g ≥ 2, then by lemma 8.3 theformula follows over the whole Baily-Borel compactification, as desired. In the case g = 1one must have a closer look at the boundary to get the same result. Aliter: Let (P ′′,X ′′)be the fibre product of (P ′,X ′) with itself over (P ′,X ′)/V ′. As above, one easily showsthat (LrelP ′ )⊗2 ∼= LrelP ′′ , and the assertion follows from that for LrelP ′′ . q.e.d.

8.5.C×-torsors and cone decompositions: Consider the situation of 3.17, i.e. a((P0,X0) → (Gm,Q,X0))-torsor (P ′,X ′) → (P,X ). Let K ′f be a neat open compact sub-group of P ′(Af ), and Kf its image in P (Af ). By 3.17 the corresponding mixed Shimuravarieties form a C×-torsor. Consider the associated line bundle on MKf (P,X )(C), and someadmissible cone decompositions Σ′ resp. Σ. So we want to show that under certain condi-tions, MK′f (P ′,X ′,Σ′)(C) → MKf (P,X ,Σ)(C) has a canonical structure as a line bundle,extending the given line bundle on MKf (P,X )(C).

Consider a connected component X ′0 of X ′, a rational boundary component (P ′1,X ′1) of(P ′,X ′) such that X ′0 maps to X ′1, and p′f ∈ P ′(Af ). They determine corresponding objects

X 0, (P1,X1) , and pf for (P,X ). Then Σ′(X ′0, P ′1, p′f ) is a rational partial polyhedral decom-

position of C∗(X ′0, P ′1) ⊂ U ′1(R)(−1), and Σ(X 0, P1, pf ) one of C∗(X 0, P1) ⊂ U1(R)(−1).Let ΓU ′ ⊂ U ′1(Q) be the image of

(z ∈ Z(P ′)(Q) | z|X ′ = id × U ′1(Q)) ∩ p′f ·K ′f · p′−1f

under the projection Z(P ′) × U ′1 → U ′1, and define ΓU ⊂ U1(Q) likewise. It is easy to seethat the short exact sequence 0→ U0 → U ′1 → U1 → 0 induces a short exact sequence

0→ U0(Q) ∩ ΓU ′ → ΓU ′ → ΓU → 0.

Recall the definition of the cone decomposition Σ0 for (P0,X0) in 6.9: Every connectedcomponent X 0

O of XO corresponds to an isomorphism λ : UO(R)(−1) → UO(R) = R, andfor every pO<f ∈ PO(Af ) we have ΣO(X 0

O, PO, pO,f ) = 0, λ−1(R≥0) × pO,f. SinceX ′0 maps to a unique connected component of XO, we get a canonical convex rationalpolyhedral cone

σ0 := λ−1(R≥0) ⊂ UO(R)(−1) ⊂ C∗(X ′0, P ′1) ⊂ U ′1(R)(−1).

that depends only on X ′0.

Now 5.9 together with the proof of 6.26 tells us which condition to impose on theK ′f -admissible cone decomposition Σ′. Namely:

(*) For every X ′0, (P ′1,X ′1) and p′f , and every σ ∈ Σ(X 0, P1, pf ) there exists a splittingeσ : U1 → U ′1, which induces a splitting for the short exact sequence 0 → UO(Q) ∩ ΓU ′ →ΓU ′ → ΓU → 0, such that

Σ′(X ′0, P ′1, p′f ) =⋃

σ∈Σ(X 0,P1,pf )

eσ(σ), eσ(σ) + σ0.

8.6. Proposition: If the condition (*) in 8.5 holds, then

MK′f (P ′,X ′,Σ′)(C)→MKf (P,X ,Σ)(C)

134

has a canonical structure as a line bundle. Its restriction to MKf (P,X )(C) is a line bundle

associated to the C×-torsor MK′f (P ′,X ′)(C)→MKf (P,X )(C) of 3.17.

Proof. By the proof of 6.26 the given map is locally isomorphic to the map

(ΓU ′\X ′1)Σ′(X ′0,P ′1,p′f ) −→ (ΓU\X1)Σ(X 0,P1,pf )

with ΓU ′ and ΓU as in 8.5. It is easily checked from their definition that they lie in a shortexact sequence 1 → ΓU0 → ΓU ′ → ΓU → 1, where ΓU0 := U0(Q) ∩ ΓU ′ . Thus these torusembeddings corresponds to the short exact sequence of tori over C

0→ U0(C)/ΓU0 → U ′(C)/ΓU ′ → U(C)/ΓU → 0,

so the assertion follows diretly from 5.11. q.e.d.

8.7. Pullback of line bundles: We want to show that the line bundles in 8.6behave nicely under pullback by the morphisms 6.25. We only consider the case 6.25 (b),the other case being more trivial. So in addition to the data in 8.5, consider a morphismφ : (P ∗,X ∗) → (P,X ), an open compact subgroup K∗f ⊂ P ∗(Af ), and a K∗f -admissible

cone decomposition Σ∗ for (P ∗,X ∗) such that the morphism [φ] : MK∗f (P ∗,X ∗,Σ∗)(C)→MKf (P,X ,Σ)(C) of 6.25 (b) exists.

Define the fibre product of (P ′∗,X ′∗) := (P ′,X ′)×(P,X )(P∗,X ∗) and K ′∗f := K ′f×KfK∗f .

Consider a rational boundary component (P ∗1 ,X ∗1 ) of (P ∗,X ∗), and let (P1,X1) be theassociated rational boundary component of (P,X ), and (P ′1,X ′1) that of (P ′,X ′). Then(P ′∗1 ,X ′∗1 ) ∼= (P ′1,X ′1) ×(P1,X1) (P ∗1 ,X ∗1 ) is the associated rational boundary component of(P ′∗,X ′∗), and in particular U ′∗1

∼= U∗1 ×U1 U′1. This shows that if σ∗ ∈ Σ∗ and σ′ ∈ Σ′ both

map to σ ∈ Σ, then σ′∗ := σ′ ×σ σ∗ is a convex rational polyhedral cone in U ′∗1 (R)(−1), forsome (P ∗1 ,X ∗1 ). We define Σ′∗ as the set of all these σ′∗.

It is now straightforward to prove: MK′∗f (P ′∗,X ′∗,Σ′∗)(C)→MK∗f (P ∗,X ∗,Σ∗)(C) sat-isfies the condition (*) of 8.5, and the diagram

MK′∗f (P ′∗,X ′∗,Σ′∗)(C) −→ MK′f (P ′,X ′,Σ′)(C)↓ ↓

MK∗f (P ∗,X ∗,Σ∗)(C) −→ MKf (P,X ,Σ)(C)

is cartesian. In other words, the left hand side is the pullback of the line bundle on theright hand side.

Later in this chapter we shall need the following two lemmas, which allow us to localizeassertions about the line bundle in 8.6.

8.8. Lemma: Consider the situation of 8.6. If K ′f is replaced by K ′f · KUf for an

arbitrary open compact subgroup KUf of UO(Af ), then the condition in 8.6 still hold. The

line bundle associated to K ′f ·KUf is isomorphic to a strictly positive tensor power of the

line bundle associated to K ′f .

Proof. Consider the situation in the proof of 8.6. By definition UO(Q) ∩ ΓU ′ is theimage in UO(Q) of

(z ∈ Z(P ′)(Q) | z|X ′ = id × U0(Q)) ∩ (p′1,f · p′f ) ·K ′f · (p′1,f · p′f )−1.

135

This can be written as the intersection with U0(Q) of the image of

(z ∈ Z(P ′)(Q) | z|X ′ = id × U0(Af )) ∩ (p′1,f · p′f ) ·K ′f · (p′1,f · p′f )−1

in U0(Af ), since K ′f contains an open subgroup of U0(Af ). Let KUf be the image of (z ∈

Z(P ′)(Q) | z|X ′ = id × U0(Af )) ∩ K ′f in U0(Af ), then it follows that U0(Q) ∩ ΓU ′ =

U0(Q) ∩ int(p′1,f · p′f )(KUf ). If K ′f is replaced by K ′f ·KU

f , then clearly KUf is replaced by

KUf · KU

f , and the only effect on the objects considered in 8.5 is that ΓU ′ is replaced by

ΓU ′ + Γ′U0with Γ′U0

= U0(Q)∩ int(p′1,f · p′f )(KUf ). Thus clearly the condition (*) of 8.5 still

holds with K ′f ·KUf in place of K ′f .

More precisely we find that the index

[ΓU ′ + Γ′U0: ΓU ′ ] = [U0(Q) ∩ ΓU ′ + Γ′U0

: U0(Q) ∩ ΓU ′ ]

= [U0(Q) ∩ int(p′1,f · p′f )(KUf ·KU

f ) : U0(Q) ∩ int(p′1,f · p′f )(KUf )]

= [KUf ·KU

f : KUf ]

is independant of the choice of (P ′1,X ′1), p′f , p′1,f and X 0. If this index is n, then locally 5.10

gives a canonical isomorphism of the “new” line bundle with the nth tensor power of the“old” line bundle. These must patch together to a global isomorphism, as desired. q.e.d.

8.9. Lemma: Consider the situation of 8.6, and let (P ′1,X ′1), p′f be as in 8.5.

Define K ′1f := P ′1(Af ) ∩ p′f · K ′f · p′−1f , Σ′1 := ([ p′f ]∗Σ′)|(P ′1,X ′1), W ′1 ⊂ U

′(P ′1,X ′1, p′f ) ⊂

MK′1f (P ′1,X ′1,Σ′01 )(C), and ∆′1 in analogy toK1f , Σ1,W1 ⊂ U(P1,X1, pf ) ⊂MK1

f (P1,X1,Σ01)(C),

and ∆1 as in 6.10, 6.13, 6.18, and 6.22. By 8.6 the embeddings ∆1\W1 →MKf (P,X ,Σ)(C)

and ∆′1\W ′1 →MK′f (P ′,X ′,Σ′)(C) make ∆′1\W ′1 → ∆1\W1 a line bundle.

Then by replacing K ′f by K ′f ·KUf for a sufficiently big open compact subgroup KU

f ⊂U0(Af ) we can achieve that

(a) the assumptions of 8.6 are also satisfied for

MK′1f (P ′1,X ′1,Σ′01 )(C)→MK1f (P1,X1,Σ

01)(C),

so W ′1 →W1 carries a canonical structure as a line bundle, and

(b) this line bundle is isomorphic to the pullback toW1 of the line bundle ∆′1\W ′1 → ∆1\W1

above.

Proof. Let ΓU ′ and ΓU be as in 8.5, but with p′1,f · p′f in place of p′f , and define Γ1U ′

and Γ1U likewise, with (P ′,X ′) in place of (P ′1,X ′1) and p′1,f in place of p′f . Explicitely ΓU ′

is the image in U ′1(Q) of

(z ∈ Z(P ′)(Q) | z|X ′ = id × U ′1(Q)) ∩ (p′1,f · p′f ) ·K ′f · (p′1,f · p′f )−1,

while Γ1U ′ is the image of the possibly smaller subgroup

(z ∈ Z(P ′1)(Q) | z|X ′ = id × U ′1(Q)) ∩ p′1,f ·K ′1f · (p′1,f )−1 =

= (z ∈ Z(P ′1)(Q) | z|X ′ = id × U ′1(Q)) ∩ (p′1,f · p′f ) ·K ′f · (p′1,f · p′f )−1.

136

Thus we have the inclusion Γ1U ′ ⊂ ΓU ′ and Γ1

U ⊂ ΓU . If K ′f is replaced by K ′f ·KUf for a

sufficiently big open compact subgroup KUf ⊂ U0(Af ), then U0(Q)∩ΓU ′ is already equal to

U0(Q)∩ (p′1,f · p′f ) ·K ′f · (p′1,f · p′f )−1. Assume that this replacement has been made, then it

follows that ΓU0 := U0(Q) ∩ ΓU ′ = U0(Q) ∩ Γ1U ′ .

Now in (a) everthing is clear except that in the condition (*) of 8.5 for all σ ∈Σ1(X 0

1 , P1, p1,f ) = Σ(X 0, P1, p1,f · pf ) the splitting eσ induces a splitting of the correspond-ing lattices Γ1

U → Γ1U ′ . Denote the projection U ′1 → U1 by π0, then this is equivalent to

saying that id − eσ0 π0 : U ′1 → U0 ⊂ U ′1 induces a splitting Γ1U ′ → U0(Q) ∩ Γ1

U ′ . But byassumption we have this property for ΓU ′ and ΓU , that this id− eσ π0 induces a splittingΓ1U ′ → ΓU0 . Thus by the equality ΓU0 = U0(Q) ∩ Γ1

U ′ we are done.

For (b) we have to show that the obvious map

U ′(P ′1,X ′1, p′f ) −→ U(P1,X1, pf )×(∆1\U(P1,X1,pf )) (∆′1\U′(P ′1,X ′1, p′f ))

is an isomorphism. By the proof of 6.20 every connected component of ∆′1\U′(P ′1,X ′1, p′f )

is of the form Γ′Q\((Γ′1\X ′0)Σ′1(X ′01 ,P ′1,p′1,f )), where Γ′Q = StabQ′(Q)(X ′0) ∩ ((p′1,f · p′f ) · K ′f ·

(p′1,f · p′f )−1 and Γ′1 = P ′1(Q) ∩ Γ′Q, and the analogous formula holds for the image of

this connected component in ∆1\U(P1,X1, pf ). Abbreviate X ′ := (Γ′1\X ′0)Σ′1(X ′01 ,P ′1,p′1,f )

and X := (Γ1\X 0)Σ1(X 01 ,P1,p1,f ), then X ′ → X is a line bundle, and the image connected

component is isomorphic to Γ′Q\X. Since K ′f is neat, so is Γ′Q, hence it acts trivially on

U0(Q). This implies that Γ′Q-action commutes with the action of the torus C× on X ′.Moreover the assumption on K ′f above implies

γ ∈ Γ′Q | γ|X 0 = id = U0(Q) ∩ Γ′Q = U0(Q) ∩ Γ′1,

henceX ′ ∼= X ×(Γ′Q\X) (Γ′Q\X ′),

as desired. q.e.d.

8.10. Towards an ampleness criterion: In the situation of 8.5 let us now assumethat Σ is complete, so by 6.27 MKf (P,X ,Σ)(C) is a compact complex space. In particularevery fibre of the map [π]∗ : MKf (P,X ,Σ)(C)→ Mπ(Kf )((P,X )/W )∗(C) is compact. Ournext aim is given a sufficient condition for the inverse of the line bundle of 8.6 to be ampleon every such fibre. Theorem 5.19 suggests what that condition could look like. Note thatby 7.3 and 7.8 in every fibre we are dealing with the union of torus embeddings of unipotnetfibres of certain other mixed Shimura varieties. Thus on the one hand it is plausible torequire |Σ′(X ′0, P ′1, p′f )| to be strictly convex in the sence of 5.16. The other condition in5.19 means that certain line bundles on the abelian variety part of the unipotent fibre haveto be ample. By 3.21 this can be recognized in terms of polarizations of Hodge structures.To express this condition, like the first one, in terms of |Σ′(X ′0, P ′1, p′f )|, we make thefollowing definition.

8.11. Definition: Consider arbitrary mixed Shimura data (P,X ), and a connectedcomponent X 0 of X . Let Ψ : V × V → U be the pairing induced by the commutator

137

pairing on the unipotent radical W of P . Let√−1 ∈ C be a square root of −1, and define

B(X 0) := (2π√−1)−1 ·Ψ(v, hx(

√−1)v) | 0 6= v ∈ V (R), x ∈ X 0 ⊂ U(R)(−1).

As usual (cf. 1.11) this does not depend on the choice of√−1. More generally let (P1,X1)

be a rational boundary component of (P,X ) such that X 0 corresponds to a connectedcomponent X 0

1 of X1. Then we define B(X 0, P1) := B(X 01 ).

Observe that this definition is almost, but not quite functorial in (P,X ). In fact, it iseasy to see that it is functorial only with respect to morphisms which are injective on V1.By contrast B(X 0, P1) ∪ 0 is always functorial. The definition is made such that we cancheck on B(X 0, P1) whether a pairing on V , that factors through the commutator pairingin W , defines a polarization of Hodge structures. We shall need the following result aboutB(X 0, P1).

8.12. Proposition: B(X 0, P1) is contained in the closure of C(X 0, P1) in U1(R)(−1).If P is reductive, then 0 6∈ B(X 0, P1).

Proof. By 4.15 (c) we have C(X 0, P1) = C(X 0, P1) + (W ∩ U1)(R)(−1), so C(X 0, P1)is the inverse image of the corresponding cone for (P,X )/W . Thus the assertion followsfrom that for (P,X )/W , and we are reduced to the case that P is reductive. In that casethis is, at least up to sign, what is proved in [AMRT] ch.III §4.2 p.231 thm. 2. (There isa slight error in the formulation of that theorem: the proof only shows that every elementof B(X 0, P1) is nonzero, and is contained in the closure of C(X 0, P1). In fact B(X 0, P1) isnot always a subset of C(X 0, P1), even if P ad is Q-simple.)

We still have to make sure that the sign in [loc. cit.] is the same as above. So westart with x1 ∈ X 0

1 , the connected component of X1 associated to X 0. Since the assertion isinvariant under conjugation by U1(C), we may assume that x1 corresponds to some elementx ∈ X 0. Let ωx be as in 4.6 (b), fix a choice of

√−1, and let J := int(ωx(

√−1, 1))|V

as in [AMRT] p.230. Since by definition hx1 = ωx h∞ defines pure Hodge structure ofweight −1 on V , this operator J acts on V like int(ωx(h∞((

√−1))) = int(hx1(−

√−1)) =

−1 · int(hx1(√−1)). Thus for any 0 6= v ∈ V (R) it follows that

(2π√−1)−1 ·Ψ(v, hx(

√−1)v) = (2π

√−1)−1 · (−1) ·Ψ(v, Jv) =

√−1 · (2π)−1 ·Ψ(v, Jv).

To identify C(X 0, P1) with the correct cone in [AMRT] note that by 4.7 (a) we are still freeto choose h0 in 4.3. If we make the same choice as in [AMRT] p.174 by defining

h0(x+√−1y) := (x+

√−1y,

(x y−y x)),

then using the definition 4.3 of h∞ an easy calculation shows that

h∞(x+√−1y) = (x+

√−1y,

(x2+y2

√−1(1−x2−y2)

0 1

))

= int((1,(

1√−1

0 1

)))((x+

√−1y,

(x2+y2 00 1

))).

Thus the definition 4.14 of the imaginary part implies that im(x1) = ωx((1,(

1√−1

0 1

))) ∈

U1(R)(−1), and by 4.15 (c) C(X 0, P1) is the Q(R)0-orbit in U1(R)(−1) generated by this

138

element. On the other hand in [AMRT] p.227 the cone C ⊂ U1(R) is defined as the Q(R)0-orbit generated by ωx((1,

(1 10 0

))) so we have

√−1 · C = C(X 0, P1). Now as mentioned

above the proof of [AMRT] p.231 thm. 2 shows that for all 0 6= v ∈ V (R) we have0 6= Ψ(v, Jv) ∈ C. Putting everything together we thus get

0 6= (2π√−1)−1 ·Ψ(v, hx(

√−1)v) =

√−1 · (2π)−1 ·Ψ(v, Jv) ∈

√−1 · C = C(X 0, P1),

as desired. q.e.d.

8.13. Theorem: In the situation of 8.5 suppose that Σ is complete and condition(*) holds. Denote the homomorphism P ′ → P by π0, and for every connected componentX ′0 of X ′ let λ : U0(R)(−1)→ U0(R) = R be as in 8.5. Assume that for every X ′0, (P ′1,X ′1)and p′f we have:

(a) Σ′(X ′0, P ′1, p′f ) is strictly convex, and

(b) for every σ ∈ Σ(X 0, P1, pf ) with R · σ = U1(R)(−1) (i.e. of top dimension) the linearform λ (id− eσ π0) is strictly positive on B(X ′0, P ′1).

Then the inverse of the line bundle

MK′f (P ′,X ′,Σ′)(C)→MKf (P,X ,Σ)(C)

is ample on every fibre of the map

[π]∗ : MKf (P,X ,Σ)(C)→Mπ(Kf )((P,X )/W )∗(C).

Proof. We first assume that (P,X ) is irreducible, and prove the assertion overMπ(Kf )((P,X )/W )(C).

Consider the corresponding connected components Γ′\X ′0 ⊂MK′f (P ′,X ′)(C) and Γ\X 0 ⊂MKf (P,X )(C). By 6.6 the torus embeddings along the unipotnet fibre MK′f (P ′,X ′,Σ′0)(C)and MKf (P,X ,Σ0)(C) are on these connected components given as torus embeddings overΓ\X 0/U(C) with respect to tori T ′ = U ′(C)/ΓU ′ , resp. T = U(C)/ΓU , and the rationalpartial polyhedral decompositions Σ′(X ′0, P ′, p′f ), resp. Σ(X 0, P, pf ). Here ΓU ′ and ΓU arelattices in U ′(Q), resp. U(Q), which lie in a short exact sequence 1→ ΓU0 → ΓU ′ → ΓU → 1with ΓU0 := U0(Q) ∩ ΓU ′ . As in 3.16 U0(C)/ΓU0 is identified with C× by U0(C)/ΓU0 =C/d · Z 3 [u] 7→ exp(1

d · λ(u)) for d ∈ Z>0. Thus we are given a short exact sequence1 → C× → T ′ → T → 1, and the convex rational polyhedral cone σ0 defined in 8.5 corre-sponds to the cone σ0 = R≥0 ⊂ R = Y∗(Gm)R considered in 5.11 (hence the same notation).It suffices to verify the condition (b) of 5.19 on every fibre over Γ\X 0/(U(C) ·W (R)).

By assumption Σ(X 0, P, pf ) is a finite and complete rational polyhedral decompositionof U(R)(−1), and by condition (a) Σ′(X ′0, P ′1, p′f ) is strictly convex. Thus the first part of

5.19 (b) is satisfied. For the second part fix σ ∈ Σ(X 0, P1, pf ) such that R · σ = U(R)(−1),and let eσ be the associated splitting U → U ′ of 8.5. Then the invertible sheaf Meσ of5.19 is the invertible sheaf on Γ\X 0/(U(C) associated to the C×-torsor Γ′\X ′0/eσ(U)(C),with the C×-torsor structure given through the isomorphism U0

∼−−→ U ′/eσ(U). Since(P,X ) is irreducible, by 2.14 (a) eσ(U) is a normal subgroup of P ′, so we may considerthe mixed Shimura data (P ′,X ′)/eσ(U). To show that 5.19 (b) is verified in every fibre

139

of Γ\X 0/U(C) → Γ\X 0/(U(C) · W (R)), it thus remains to show that the invers of theC×-torsor

Mπ[σ](K′f )(P ′,X ′)/eσ(U)(C)→Mπ′(Kf )((P,X )/U)(C)

is ample on every fibre on Mπ′(Kf )((P,X )/U)(C)→Mπ(Kf )((P,X )/W )(C).

To prove this note that the canonical map U ′ → U ′/eσ(U) ∼= U0 ⊂ U ′ is equal toid− eσ π0. Thus if Ψ′ : V × V = V ′× V ′ → U ′ is the pairing induced by the commutatorin W ′, then for W ′/eσ(U) we get the pairing id − eσ π0 Ψ′ : V × V → U0 ⊂ U ′. Thusby 3.21 the line bundle in question is ample if the pairing

λ (id− eσ π0) Ψ′ : V × V → U0(1) = Q(1)

is a polarization of Hodge structures for every x ∈ X 0. By the definition of polarization in1.11 and the definition of B(X ′0) in 8.11 this is just the condition (b) above for (P ′1,X ′1) =(P ′,X ′). This finishes the special case.

Let us now do the general case. By 7.3 a fibre of [π]∗ is isomorphic to a fibre of the

map ∆1\∂U(P1,X1, pf ) → ∆1\Mπ1(K1f )((P1,X1)/W1)(C). Every irreducible component of

∂U(P1,X1, pf ) is by 7.3 and 7.9 isomorphic to Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ

01,[σ])(C) for some

double coset [σ] ∈ P1(Q)\Σ01/P1(Af ), and the induced map

Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ

01,[σ])(C) −→ ∆1\∂U(P1,X1, pf ).

is a finite holomorphic map. Thus it suffices to show the assertion for pullback of our line

bundle to Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ

01,[σ])(C). Now we can do the same steps for the mixed

Shimura variety MK′f (P ′,X ′,Σ′)(C) in place of MKf (P,X ,Σ). By lemma 8.8 we mayenlarge K ′f without changing the desired assertion. If we do this the conclusion in lemma8.9 holds, then it is easy to see that the condition of 8.5 hold also for (P ′1,[σ′],X

′1,[σ′]) and

Σ′01,[σ′] with σ′ = eσ(σ). It follows that the pullback to Mπ1,[σ](K1f )(P1,[σ],X1,[σ],Σ

01,[σ])(C) of

the line bundle in question is isomorphic to the line bundle

Mπ′

1,[σ](K′1f )

(P ′1,[σ′],X′1,[σ′],Σ

′01,[σ′])(C) −→Mπ1,[σ](K

1f )(P1,[σ],X1,[σ],Σ

01,[σ])(C).

But this is just the case we have covered in the first part of the proof, so it only remainsto show that the conditions (a) and (b) hold for (P ′1,[σ′],X

′1,[σ′]) and Σ′01,[σ′].

To prove this let Σ : |Σ′(X ′0, P ′1, p′1,f · p′f )| and Σ[σ] := |Σ′1,[σ′](X′01,[σ′], P

′1,[σ′], p

′1,f )|. We

first show that Σ[σ] = (Σ modR · σ′). By the definition 7.7 of Σ′1,[σ′], Σ[σ] is the union

of τ ′modR · σ′ for all τ ′ ∈ Σ′(X ′0, P ′1, p′1,f · p′f ) such that σ′ is a face of τ ′. Thus theinclusion “⊂” is clear. For the reverse inclusion note that by 7.8 Σ1,[σ] is complete, soΣ[σ] + R · σ0 = C∗(X ′01,[σ′], P

′1,[σ′]). This means that C∗(X ′0, P ′1) is contained in the union of

eτ (τ) + R · σ′ + R · σ0 = eτ (τ + R · σ0) + R · σ0 for all τ ∈ Σ(X 0, P1, p1,f · pf ) such that σ isa face of τ . It suffices to take only those τ that are of top dimension. For every such τ theconvexity condition (a) implies Σ ⊂ eτ (U1(R)(−1))+σ0. Thus it follows that Σ is containedin the union of eτ (τ + R · σ + σ0 = eτ (τ) + R · σ′ + σ0 for the same τ . But σ′ = eσ(σ) is aface of eτ (τ) + σ0, so the reverse inclusion (Σ modR · σ′) ⊂ Σ[σ] follows.

140

Now the equation Σ[σ] = (Σ modR · σ′) and the convexity of Σ implies that Σ[σ] isconvex. To show that Σ′1,[σ](X

′01,[σ], P

′1,[σ], p

′1,f ) is strictly convex, the definition in 5.16 tells

us that we have to consider eτ (τ) modR · σ′ for every τ ∈ Σ(X 0, P1, p1,f · pf ) such that σ isa face of τ . By the strict convexity of Σ′(X ′0, P ′1, p′1,f · p′f ) there exists a linear form ` onU ′1(Q) such that eτ (τ) = u ∈ Σ | `(u) = 0. Since σ′ = eσ(σ) is a face of eτ (τ), this linearform vanishes on R · σ′, and hence factors through a linear form `′ on U ′1,[σ′](Q). It follows

that eτ (τ) modR · σ′ = u ∈ Σ modR · σ′ | `′(u) = 0 = u ∈ Σ[σ] | `′(u) = 0, providingcondition (a).

For condition (b) note that the splitting e(τ modR·σ) : U1,[σ] → U ′1,[σ′] is just that inducedby eτ , that is more precisly we have the equation

id− eτ π0 = (id− e(τ modR·σ) π0) π[σ].

Since moreover

B(X ′01,[σ′], P′1,[σ′] = (B(X ′0, P ′1) modR · σ′) = π[σ](B(X ′0, P ′1)),

condition (b) follows from that for (P ′,X ′). q.e.d.

8.14. Corollary: Let (P,X ) be mixed Shimura data, Kf ⊂ P (Af ) a neat opencompact subgroup, and Σ a complete Kf -admissible cone decomposition for (P,X ). Sup-pose that there exist (P ′,X ′), K ′f , Σ′ such that the assumption of 8.13 are verified. Then

MKf (P,X ,Σ)(C) is projective.

More specifically let L be the invertible sheaf associated to the line bundleMK′f (P ′,X ′,Σ′)(C)→MKf (P,X ,Σ)(C) and ω[dlog] the canonical sheaf on MKf (P,X ,Σ)(C) with logarithmicpoles along MKf (P,X ,Σ)(C)rMKf (P,X )(C), locally defined as in 5.26. Then there existsa positive integer n, such that the invertible sheaf

L⊗−1 ⊗ (ω[dlog])⊗n

is ample.

Proof. Using 8.4 and 8.13 we have the situation of lemma 5.18, withX := MKf (P,X ,Σ)(C),Y := MKf (P,X )∗(C), L replaced by L⊗−1, and M := LP of 8.4. q.e.d.

8.15. Remarks: (a) If P is reductive, then 8.14 essentially implies, and is almostequivalent to, Tai’s theorem in [AMRT] ch.IV §2.1 p.312 (Compare also [Nam] thm. 7.23).In fact our construction of an explicit relatively ample sheaf can, as in the proof of [H2]ch.II thm. 7.17, be translated into the construction of an explicit sheaf of ideals, such thatMKf (P,X ,Σ)(C) is the normalization of the blowing up of MKf (P,X )∗(C) along this sheafof ideals. Thus the conclusion of 8.14 is equivalent to that in [loc. cit.], and it remains tocheck that the respective assumptions are more less the same.

We shall only indicate the main point. First note that since P is reductive, afterpossibly changing the center of P there exists, up to isomorphism, exactly one choicefor (P ′,X ′) as in 8.5. Let (P ′1,X ′1) be a rational boundary component of (P ′,X ′), and

141

(P1,X1) := (P ′1,X ′1)/U0. Any splitting P ′ ∼= U0 o P induces a splitting W ′1∼= U0 oW1, and

this is already a direct product since W1 acts trivially on U0. Since all splittings of P ′ areconjugate under U0, this splitting is canonical! Thus we have a canonical decompositionU ′1 = U0 × U1, and B(X ′0, P ′1) is contained in the second factor. In analogy to 5.20 we cannow describe Σ′(X ′0, P ′1, p′f ) in terms of a function φ : C∗(X 0, P1)→ R, where the splitting

associated to any σ ∈ Σ(X 0, P1, pf ) is, with respect to the canonical decomposition of U ′1,equal to (−λ−1 φ, id) on σ.

It is now easy to see that the condition (a) of 8.13 is equivalent to the strict convexityof φ in the sense of condition (2) of [AMRT] ch.IV §2.1 p.310. It can be shown that thecondition (1) of [loc, cit.] is equivalent to the assertion that for all σ of top dimension thelinear form λ (id− eσ π0) is strictly positive on C(X 0, P1) r 0. By 8.12 it thus impliesour condition (b). In general we do not have equivalence, but this is only a very minordifference.

(b) More generally one can describe the line bundle of 8.13 in terms of a function φ asin [AMRT] ch.IV §2 if and only if the extension 1→ U0 → P ′ → P → 1 splits. In fact, wehave described the line bundle locally as an equivariant torus embedding for a short exactsequence of tori 1 → Gm → T ′ → T → 1, and a global splitting of this extension amountsto a splitting P ′ ∼= U0 o P . But 3.21 shows that whenever the abelian variety part ofthe unipotent fibre is nontrivial, then one can get ample line bundles only with extensionsthat do not split. Thus we needed a more sophisticated description of relatively ample linebundles than in [AMRT]. A geometric illustration of this will be given in 9.39 (b).

142

Chapter 9

Algebraization of the toroidalcompactification

To finish the algebraization of MKf (P,X ,Σ)(C) we construct cone decompositions thatsatisfy the requirements of the preciding chapter. This construction is an analog of [AMRT]ch.II §§5.3–4. Under a mild condition on (P,X ), we begin (9.1–5) with the study of acertain set of splittings that satisfy the condition 8.13 (b). Our cone decompositions will bedescribed in terms of “locally polyhedral” subsets Σ ⊂ C∗(X ′0, P ′1), defined in 9.8. Theirproperties (9.9–11) lead to cone decompositions that satisfy the desired conditions (9.12–13), and it remains to construct compatible systems of such subsets Σ. These constructionsoccupy most of the rest of this chapter.

First study the behavior of Σ with respect to different boundary components (9.14–15).This leads to our first existence theorem 9.16: asserting that some such systems of Σ’s exists.Next we show how to obtain refinements with specific properties (9.18–20), in particularconcerning smoothness and the extendability of the maps 3.4. In 9.21–23 we extend thisto arbitrary (P,X ). These results show that MKf (P,X )(C), and certain MKf (P,X ,Σ)(C),possess canonical algebraic structures (9.24–26). In 9.28–33, we extend the existence ofan algebraic structure to all MKf (P,X ,Σ)(C) for which, loosely speaking, Σ is sufficientlyfine or Kf sufficiently small. This implies that there always exists a canonical structure ofalgebraic space over C; which, however, is not always a scheme (9.34–35).

Having constructed an algebraic structure on most MKf (P,X ,Σ)(C), it follows fromthe construction that the maps 6.25 are algebraic, and most properties carry over to thealgebraic side (9.25). in 9.36–38 we prove the same for the stratifications described inchapter 7, in particular that 7.17 (b) gives an isomorphism of the formal completions. Thechapter concludes with an example (9.39).

9.1. Proposition: Let (P,X ) be mixed Shimura data, and (P,X ) → (Gm,Q,H0)a morphism. As in 8.7 every connected component X ′0 determines an isomorphism λ :U0(R)(−1) → U0(R) = R and a convex rational polyhedral cone σ0 := λ−1(R≥0) ⊂U0(R)(−1).

(a) Suppose that (P,X ) is irreducible. Then there exists a ((P0,X0) → (Gm,Q,H0))-torsor(P ′,X ′)→ (P,X ), and a splitting µ : U ′ → U0, such that if Ψ : V ×V → U ′ is the pairing

143

induced by the commutator in W ′, then λµΨ defines a polarization of Hodge structureson V2g for every point in X ′0.

(b) Let (P ′,X ′) → (P,X ) be as in (a). Then there exists pure Shimura data (P∗, X∗), arational boundary component (P , X ) of (P∗, X∗), and an embedding (P ′,X ′) → (P , X ),subject to the following condition. For every connected component X ′0 of X ′ let X 0 andX 0∗ be the corresponding connected components of X resp. of X∗. Then the image of σ0 in

U(R)(−1) is contained in C(X 0∗ , P ).

Proof. (a) If V = 1, let (P ′,X ′) := (P0,X0) ×(Gm,Q,H0) (P,X ), then any splitting isokay. Otherwise it suffices to take (P ′,X ′) and µ be as in the proof of 2.26 (b).

(b) As in the proof of 2.26 (b) the set of all µ that satisfy (a) above is an open subset ofthe set of all splittings. This also holds if V = 1, since then all splittings are okay. Let 2g :=dim(V ), then every such splitting defines a morphism (P ′,X ′) → (P2g,X2g). As in 2.26any set of splittings in the above open set, that generates the Q-vector space Hom(U ′, U0),determines an embedding (P ′,X ′) → (P,X )/W ×(P2g,X2g)

r for some r. By 4.25 (P2g,X2g)can be interpreted as a rational boundary component of (CSP2(g+1),Q,H2(g+1)). So let

(P∗, X∗) := (P,X )/W × (CSP2(g+1),Q,H2(g+1))r, and let (P , X ) be the (unique) irreducible

component of (P,X )/W×(P2g,X2g)r that contains the image of (P ′,X ′). We are done if we

can choose the splittings such that σ0 maps to C(H02(g+1), P2g) under each of the morphisms

(P ′,X ′)→ (P2g,X2g).

If V = 1, then this is easy to achieve, since C(H02(g+1), P2g) is a half-line in a vector

space of dimension 1, and we may replace any splitting µ by −µ. Otherwise by 8.11 thefact that λ µ Ψ defines a polarization of Hodge structures on V means that λ µ isstrictly positive on B(X ′0, P ′), or equivalently that µ(B(X ′0, P ′)) is contained in (σ0)0. Byfunctoriality (8.11) it follows that B(X 0

2g)∩(σ0)0 6= ∅, where we denote the image of σ0 again

by σ0. But by 8.12 B(X 02g) = B(H0

2(g+1), P2g) ⊂ C(H02(g+1), P2g) r 0, and both σ0 and

C(H02(g+1), P2g) are half-lines in a one dimensional vector space, so σ0 = C(H0

2(g+1), P2g) asdesired. q.e.d.

9.2. A certain set of splittings: Let (P,X ) be mixed Shimura data, and π0 :(P ′,X ′) → (P,X ) a ((P0,X0) → (Gm,Q,H0))-torsor. As in the assertion of 9.1 (a) assumethat there exists a (not necessarily P -equivariant) splitting µ : U ′ → U0, such that ifΨ : V × V → U ′ is the pairing induced by the commutator in W ′, then λ µ Ψ defines apolarization of Hodge structures on V2g for every point in X ′0. Under this assumption, forevery irreducible component (P(0),X(0)) of (P,X ), the assertion in 9.1 (b) holds. We fix suchdata for most of this chapter. We shall eventually construct cone decompositions for thepair (P ′,X ′) → (P,X ), that satisfy all requirements of 8.13. For this we first concentrateon condition 8.13 (b). The main problem is to produce a sufficient supply of splittings thatsatisfy this condition.

Consider a rational boundary component (P ′1,X ′1) of (P ′,X ′), let (P1,X1) ∼= (P ′1,X ′1)/U0

be the corresponding rational boundary component of (P,X ), andQ′ the associated parabolicsubgroup of P ′. Let X ′0 be a connected component of X ′ that is mapped to X ′1, and let X 0

be the corresponding connected component of X . Let (P ′(0),X′(0)) be the unique irreducible

component of (P ′,X ′) with X ′0 ⊂ X(0).

144

Consider (P , X ), (P∗, X∗), and an embedding (P ′(0),X′(0)) → (P , X ) as in 9.1 (b), then

by functoriality we get a rational boundary component (P1, X1) of (P , X ), and connectedcomponents X 0

1 , X 0, and X 0∗ . Consider a homomorphism µ : U1 → U0 such that λ µ is

strictly positive on C(X 0∗ , P1) r 0, and which restricts to the identity on U0 ⊂ U ′ → U1.

Then its restriction to U ′1 is a splitting of the short exact sequence 0 → U0 → U ′1 →U1 → 0. The corresponding splitting map e : U1 → U ′1 is determined by the equationµ|U ′1 = id− e π0.

Let E(X ′0, P ′1) be the set of all splittings e : U1 → U ′1 that arise in this way for somechoice of (P , X ) and (P∗, X∗). Strictly speaking this is an abuse of notation, since this setdepends not only on X ′0 and P ′1, but also on the torsor structure on (P ′1,X ′1) → (P,X ).But since this is fixed, there should be no confusion. Clearly the definition of E(X ′0, P ′1)is invariant under P ′(Q). In particular the group StabQ′(Q)(X ′0) acts on E(X ′0, P ′1). Weshall denote this operation by e 7→ eq := (u 7→ q−1 · e(q · u)) for any q ∈ StabQ′(Q)(X ′0).Note that StabP ′(Q)(X ′0) acts on U0 through positive scalars, since we have a morhism(P ′,X ′)→ (P0,X0). This implies that for every u′ ∈ U ′(R)(−1)

λ (id− eq π0)(u′) = λ(q−1 · (id− e π0)(q · u′))

is a positive multiple of λ (id− e π0)(q · u′).

9.3 Proposition: (a) E(X ′0, P ′1) is an nonempty open subset of the space of allsplittings U1 → U ′1 (open with respect to the archimedian topology on the hyperplane ofall splittings in Hom(U1, U

′1)).

(b) For every e ∈ E(X ′0, P ′1) the linear map λ(id−eπ0) is strictly positive on B(X ′0, P ′1),i.e. satisfies condition 8.13 (b).

Proof. (a) Since by 4.15 (d) C(X 0∗ , P1) is a nondegenerate selfadjoint cone, the set of

all λ µ that are strictly positive on C(X 0∗ , P1) r 0 is just the set of rational points in

the interior of the dual cone, which is a nonempty subset of the dual space of U1(Q)(−1).

Since by assumption the image of σ0 lies in C(X 0∗ , P1), and by injectivity is not zero, the

restriction of µ to σ0 is the multiplication by some positive scalar αıQ>0. Thus α−1 · µsatisfies the requirements of 9.2, and runs through a nonempty open subset of splittingsU1 → U0. Since the restriction map Hom(U1, U0)→ Hom(U ′1, U0) is surjective, the assertionfollows.

(b) By the functoriality 8.11, B(X ′0, P ′1) is mapped to B(X 0, P1) = B(X 0∗ , P1). By 8.12,

the latter is contained in C(X 0∗ , P1) r 0. Since by assumption λ µ is strictly positive on

C(X 0∗ , P1)r 0, it follows that λ µ|U ′1 = λ (id− e π0) is strictly positive on B(X ′0, P ′1),

as desired. q.e.d.

We shall need the following information about the behavior of E(X ′0, P ′1) under restric-tion to another rational boundary component.

9.4. Proposition: Let (P ′2,X ′2) be a rational boundary component between (P ′1,X ′1)and (P ′,X ′).(a) The restriction map Hom(U ′1, U0) → Hom(U ′2, U0) induces a surjection E(X ′0, P ′1) →E(X ′0, P ′2).

145

(b) Let e ∈ E(X ′0, P ′1), A ⊂ C∗(X ′0, P ′1) r C∗(X ′0, P ′2) a finite subset, and α ∈ Q. Thenthere exists e′ ∈ E(X ′0, P ′1) such that e|U2

= e′|U2, λ (id − e′ π0) ≥ λ (id − e π0) on

C∗(X ′0, P ′1), and λ (id− e′ π0)(u′) ≥ α for every u′ ∈ A.

Proof. (a) We first show that every e ∈ E(X ′0, P ′1) restricts to an element of E(X ′0, P ′2).Suppose that e comes from a splitting µ : U1 → U0, then e|U2

comes from the restriction

of µ to U2. since by 4.21 (b) C(X 0∗ , P2) is contained in C(X 0

∗ , P1), the map µ|U2satisfies

the assumptions in 9.2, as desired. For the surjectivity we have to extend a homomorphism

µ2 : U2 → U0, such that λ µ2 is strictly positive on C(X 0∗ , P2) r 0 to a homomorphism

µ : U1 → U0 such that λ µ is strictly positive on C(X 0∗ , P1) r 0. This is possible by the

following lemma 9.5, applied to (P∗, X∗) in place of (P,X ), (P1, X1) in place of (P1,X1), theanalogous rational boundary component in place of (P2,X2), and λ µ2, λ µ in place ofµ2, resp. µ.

(b) Since C(X 02 /W2, P1/(P1∩W2)) is a nongenrate selfadjoint cone, there exists a homo-

morphism ν : U1/U1∩W2 → U0 such that λν is strictly positive on C(X 02 /W2, P1/(P1 ∩ W2))r

0. Note that by 4.21 (a) C(X 02 /W2, P1/(P1 ∩ W2)) is the image of C(X 0

∗ , P1), or also ofC(X 0, P1), under the canonical projection U1 → U1/U1∩W2. Thus, lifting ν to a homomor-

phism ν : U1 → U0, it follows that λ ν is nonnegative on C(X 0∗ , P1), and strictly positive

on C∗(X 0, P1)r(U1∩W2)(R)(−1). Now suppose that e comes from a splitting µ : U1 → U0

as in 9.2. Then for every β ∈ Q≥0 the splitting µ+β · ν also satisfies the requirement of 9.2,and the splitting e′ : U ′1 → U0 associated to µ+ β · ν has the first two of the three desiredproperties. By 4.23, the image of A is contained in C∗(X 0, P1) r C∗(X 0, P2). By 4.22 (c),this set is equal to C∗(X 0, P1) r (U1 ∩ W2)(R)(−1), hence λ ν is strictly positive on theimage of A. Thus third property also holds, provided that β is sufficiently large. q.e.d.

9.5. Lemma: Let (P,X ) be pure Shimura data, (P2,X2) a rational boundarycomponent of (P,X ), and (P1,X1) a rational boundary component of (P2,X2). Let X 0

be a connected component of X that is mapped to X1. Then for every linear formµ2 : U2(Q)(−1) → Q, which is strictly positive on C(X 0, P2) r 0, there exists a lin-ear form µ : U1(Q)(−1) → Q, which is strictly positive on C(X 0, P1) r 0 such thatµ2 = µ|U2(Q)(−1).

Proof. Since the positivity is an open condition, it is easy to see that it suffices toprove the same assertion with linear forms defined over R. Next note that since 4.21 (b)C(X 0, P2) is contained in C(X 0, P1), the restriction of any µ : U1(R)(−1) → R, that isstrictly positive on C(X 0, P1)r0, is strictly positive on C(X 0, P2)r0. Thus the desiredassertion holds for some µ2. Now the set of µ2’s in question is the interior of the dual coneof C(X 0, P2). Let Qi be the parabolic subgroup of P associated to (Pi,Xi) for i = 1, 2.We are going to show that (Q1 ∩Q2)(R)0 acts transitively on C(X 0, P2). The same groupthen also acts transitively on the set of µ2’s, and since the restriction map µ 7→ µ|U2(R)(−1)

is equivariant, the desired assertion holds for every µ2.

Since by 2.14 (a) Pi acts through a nontrivial scalar character on Ui, the image ofPi(R)0 in GL(Ui)(R) consists of all positive scalars. Clearly P1 ⊂ Q1∩P2 ⊂ Q1∩Q2, so theimage of (Q1 ∩Q2)(R)0 in GL(U2)(R) contains all positive scalars. On the other hand, by4.19 (b) we have Q2 = (Q1 ∩Q2) ·P2. Since P2(R)0 acts through positive scalars on U2(R),

146

the image of (Q1 ∩ Q2)(R)0 in GL(U2)(R) concides with that of Q2(R)0. Now by 4.15 (c)C(X 0, P2) is an orbit under Q2(R)0, so the same follows for the group (Q1 ∩ Q2)(R)0, asdesired. q.e.d.

9.6. An almost arithmetic subgroup: Let ρ : Q′ → PGL(U ′1) be the homomor-phism induced by the natural projective representation on U ′1. We shall be concerned withan arbitrary subgroup

Γ ⊂ StabQ′(Q)(X ′0)

whose image in ρ(Q′)(Q) is an arithmetic subgroup. For example if K ′f is an open compactsubgroup of P ′(Af ), then

Γ := StabQ′(Q)(X ′0) ∩K ′f · P ′1(Af )

is such a group. Indeed, this is a subgroup since Q′ normalizes P ′1, and its image in ρ(Q′)(Q)is an arithmetic subgroup, since by 2.14 (a) P ′1 acts through scalars on U ′1. We can alsotake any arithmetic subgroup of StabQ′(Q)(X ′0). Moreover let (P ′(0),X

′(0)) be the unique

irreducible component of (P ′,X ′) with X ′0 ⊂ X ′(0). Then by the condition 2.1 (viii) every

sufficiently small arithmetic subgroup of P ′(Q) is contained in (P ′(0)(Q) · Z(P ′)(Q). Since

Z(P ′) acts trivially on U ′1, we can also take any arithmetic subgroup of Stab(P ′(0)∩Q′)(Q)(X ′0).

Recall (9.2) that StabP ′(Q)(X ′0) acts through positive scalars on U0. Thus wheneveran element γ ∈ Γ acts through a scalar on U ′1, this scalar must be positive, since it is soon U0. Thus for all purposes, or objects, which are invariant under positive scalars, we cantreat Γ like an arithmetic subgroup of ρ(Q′)(Q). Among such objects are in particular allconvex rational polyhedral cones. Our aim is to construct Γ-invariant rational partial conedecompositons of C∗(X ′0, P ′1) and of C∗(X 0, P1), that satisfy certain additional conditions.

9.7. Proposition: Let e ∈ E(X ′0, P ′1) and u′ ∈ C∗(X ′0, P ′1) ∩ U ′1(Q)(−1). Then theset

γ ∈ Γ | λ (id− eγ π0)(u′) ≤ 0

is a finite union of StabΓ(R≥0 · u′)-cosets.

Proof. By the remarks in 9.6 we may without loss of generality assume that Γ is anarithmetic subgroup of Stab(P ′

(0)∩Q′)(Q)(X ′0). Recall that by the remark at the end of 9.2 we

may write λ (id− e π0)(γ ·u′) instead of λ (id− eγ π0)(u′). Suppose that e comes froma splitting µ : U1 → U0 as in 9.2, and let u be the image of u′ in C∗(X 0, P1) ∩ U1(Q)(−1).By the injectivity of U ′1 → U1 we have to show that the set

γ · u ∈ Γ such that λ µ(γ · u) ≤ 0

is finite. We shall a fortiori show that for all u ∈ C∗(X 0, P1) ∩ U1(Q)(−1) and α ∈ Q theset

γ · u ∈ Γ such that λ µ(γ · u) ≤ α

is finite.

147

Since Γ acts on U1 through P (Q), by 2.14 (a) it acts through scalars on U . But by9.2 it acts through positive scalars on U0, so since it is an arithmetic subgroup, it actstrivially on U . This shows that the assertion is invariant under shifting u by elements ofU(Q)(−1), if at the same time α is changed by a constant. Since by 4.22 (b) C∗(X 0, P1) =C∗(X 0

∗ , P1) + U(R)(−1), we may therefore assume that u ∈ C∗(X 0∗ , P1) ∩ U(Q)(−1). Fix a

Γ-invariant lattice U1(Z)(−1) that contains u. Since the set u ∈ C(X 0∗ , P1) | µ(u) ≤ α is

compact (cf. [AMRT] ch.II §5.3 p.137), the set γ · u | γ ∈ Γ, λ µ(γ · u) ≤ α is a boundedsubset of a lattice. It is therefore finite, as desired. q.e.d.

9.8. Definition of certain “locally polyhedral” subsets: Consider a nonemptycollection e1, . . . , em of splittings in E(X ′0, P ′1). Define

Σ :=⋂γ∈Γ

m⋂i=1

σ0 + eγi (C∗(X 0, P1))

= u′ ∈ C∗(X ′0, P ′1) | ∀γ ∈ Γ ∀1 ≤ i ≤ m λ (id− eγi π0)(u′) ≥ 0.

Clearly Σ is a Γ-invariant convex cone, with Σ = Σ + σ0, and relatively closed insideC∗(X ′0, P ′1) (Caution: in general it is not closed in U ′1(R)(−1)). In the following we shallalways assume:

Σ contains no nontrivial linear subspace.

Note that every nontrivial subspace of U ′1(R)(−1) that is contained in C∗(X ′0, P ′1), alreadylies in U ′1(R)(−1). By definition

U ′1(R)(−1) ∩ Σ ⊂ u′ ∈ U ′(R)(−1)∀1 ≤ i ≤ m λ (id− ei π0)(u′) ≥ 0,

so the assumption certainly holds if the id− ei π0 generate the vector space Hom(U ′, U0).Since these are splittings, the differences ei − ej factor through homomorphisms U → U0,and the assumption is verified if the ei − ej generate the Q-vector space Hom(U,U0).

We shall show (in 9.12) that every such Σ gives rise to rational partial polyhedraldecompositions satisfying the conditions 8.13 (a) and (b), and the condition (*) in 8.5 exceptfor the integrality of the splittings. The set Σ plays the role that Γ-polyhedral cocores play in[AMRT] ch.II §5.3. The term “locally polyhedral” is justified by the following proposition.

9.9. Proposition: For any convex rational polyhedral cone σ′ ⊂ C∗(X ′0, P ′1), theintersection σ′ ∩ Σ is again a convex rational polyhedral cone.

Proof. (Compare [AMRT] ch.II §5.4 p.140f) Every convex rational polyhedral cone iscontained in a finite union of simplicial convex rational polyhedral cones of top dimension.Thus we may fix u′1, . . . , u

′n ∈ C∗(X ′0, P ′1)∩U ′1(Q)(−1) that form a basis of U ′1(Q)(−1), and

assume that σ′ = Σnj=1αj · u′j | ∀1 ≤ j ≤ n αj ∈ R≥0. Choose a lattice U ′1(Z)(−1) that

contains all u′j . Let Γ′ ⊂ Γ be the subgroup of all elements that stabilize this lattice, thenby 9.6 Γ′ is, up to scalars, of finite index in Γ. Since eγi = ei for every scalar γ, the seteγi | 1 ≤ i ≤ m, γ ∈ Γ is the union of finitely many Γ′-orbits. Thus there exists a positiveinteger N so that

λ (id− eγi π0)(U ′1(Z)(−1)) ⊂ 1

N· Z

148

for all 1 ≤ i ≤ m and γ ∈ Γ. Now for all i and j, proposition 9.7 implies that the setλ (id − eγi π0)(u′j) | γ ∈ Γ is bounded below. Thus there exists an integer M so thatfor all 1 ≤ i ≤ m, 1 ≤ j ≤ n, and γ ∈ Γ

ai,j,γ := N · λ (id− eγi π0)(u′j) ∈M + Z≥0.

By the definitions of σ′ and of Σ we get

σ′ ∩ Σ = n∑j=1

αj · u′j | ∀1 ≤ j ≤ n αj ∈ R≥0 and ∀γ ∈ Γ ∀1 ≤ i ≤ mn∑j=1

ai,j,γ · αj ≥ 0.

Let S be the set of all n-tuples (ai,j,γ)1 ≤ j ≤ n for 1 ≤ i ≤ m and γ ∈ Γ, then we can alsowrite

σ′ ∩ Σ = n∑j=1

αj · u′j | ∀1 ≤ j ≤ n αj ∈ R≥0 and ∀s ∈ Sn∑j=1

sj · αj ≥ 0.

Now S is a subset of (M + Z≥0)n, so by the lemma [AMRT] ch.II §5.4 p.140 there exists afinite subset S0 ⊂ S, such that

∀s ∈ S∃s′ ∈ S so that ∀1 ≤ j ≤ n sj ≥ s′j .

This implies that

σ′ ∩ Σ = n∑j=1

αj · u′j | ∀1 ≤ j ≤ n αj ∈ R≥0 and ∀s ∈ S0

n∑j=1

sj · αj ≥ 0,

which is a convex rational polyhedral cone, as desired. q.e.d.

9.10. Proposition: There exist finitely many u′1, . . . , u′n ∈ C∗(X ′0, P ′1) ∩ U ′1(Q)(−1)

such that Σ is the convex closure of the set⋃γ∈Γ

n⋃j=1

R≥0 · γ · u′j .

Proof. By 6.19 (b) there exists a convex rational polyhedral cone σ′ ⊂ C∗(X ′0, P ′1)so that Γ · σ′ = C∗(X ′0, P ′1). Choose the u′j such that σ′ ∩ Σ is the convex closure of⋃nj=1 R

≥0 · u′j , this is possible by 9.9. Then Σ = Γ · (σ′ ∩ Σ) is contained in the convex

closure of⋃γ∈Γ

⋃nj=1 R

≥0 · γ · u′j . But Σ itself is convex, and contains all R≥0 · γ · u′j , so wehave equality. q.e.d.

9.11. Proposition: for all 1 ≤ i ≤ m and γ ∈ Γ, the subset eγi (U1(R)(−1)) ∩ Σ is aconvex rational polyhedral cone.

Proof. By Γ-invariance we may assume γ = 1. Then the set in question can be writtenas u′ ∈ Σ | λ(id−eiπ0)(u′) = 0. Let u′1, . . . , u

′n be as in 9.10, then λ(id−eiπ0)(γ′·u′j) ≥

149

0 for all 1 ≤ j ≤ n and γ′ ∈ Γ. Thus 9.10 implies that ei(U1(R)(−1) ∩ Σ is the convexclosure of the union of all R≥0 · γ′ · u′j with λ (id− ei π0)(γ′ · u′j) = 0. Since

λ (id− ei π0)(γ′ · u′j) = λ(γ′ · (id− eγ′

i π0)(u′j)),

this equation is equivalent to λ (id − eγ′

i π0)(u′j) = 0. But by 9.7 there are, modulo

StabΓ(R≥0 · u′j), only finitely pairs (γ′, j) with this property. Thus our set is the convex

closure of the union of finitely many R≥0 · γ′ · u′j , as desired. q.e.d.

9.12. Proposition: Let ei and Σ be as in 9.8. Let Σ+Σ be the set of all faces of all

cones eγi (U1(R)(−1) ∩ Σ. Let ΣΣ := π0(σ′) | σ′ ∈ Σ+Σ, and Σ′Σ := σ′, σ′ + σ0 | σ′ ∈ Σ+

Σ.Then Σ+

Σ and Σ′Σ are rational partial polyhedral decompositions of C∗(X ′0, P ′1), and ΣΣ is acomplete rational polyhedral decomposition of C∗(X 0, P1). All of them are Γ-invariant andfinite modulo Γ. They satisfy the line bundle condition of 5.11 (except for the integrality)i.e. Σ+

Σ = eσ(σ) | σ ∈ ΣΣ for certain splittings eσ : U1 → U ′1. Moreover they satisfyconditions 8.13 (a) and (b).

Proof. We first show that Σ+Σ is a rational partial cone decomposition. By 9.11 Σ+

Σ

is a nonempty set of convex rational polyhedral cones. Of the conditions 5.1 (i)–(iii) thefirst holds by definition of Σ+

Σ . The last one holds since Σ does not contain a nontrivial

linear subspace. For the second one let σ := eγi (U1(R)(−1)) ∩Σ, σ′ := eγ′

i (U1(R)(−1)) ∩Σ,τ a face of σ, and τ ′ a face of σ′. We can write τ = u′ ∈ σ | ∀ν `ν(u′) = 0 withfinitely many linear forms `ν : U ′(Q)(−1) → Q that are nonnegative on σ, and likewiseτ ′ = u′ ∈ σ′ | ∀ν ′ `′ν′(u′) = 0. Then

τ ∩ τ ′ = u′ ∈ σ ∩ σ′ | ∀ν `ν(u′) = 0 and ∀ν ′ `′ν′(u′) = 0,

which by definition is a face of σ∩σ′. But this cone can be written as u′ ∈ σ | λ(id−eγ′

i′ π0)(u′) = 0, and λ (id− eγ

′

i′ π0) is nonnegative on σ, so σ ∩ σ′ is a face of σ. thus τ ∩ τ ′is a face of σ, as desired. This proves that Σ+

Σ is a rational partial cone decomposition ofC∗(X ′0, P ′1).

¿From this it easily follows that ΣΣ and Σ′Σ are rational partial cone decompositions.For all γ and i we have

(eγi (U1(R)(−1)) ∩ Σ) + σ0 =

= u′ ∈ Σ | ∀γ′ ∈ Γ ∀1 ≤ i′ ≤ m λ (id− eγ′

i′ π0)(u′) ≥ λ (id− eγi π0)(u′)

so 9.7 implies that Σ is the union of all these, i.e. that |Σ′Σ| = Σ. Again by 9.7 we haveΣ + U0(R)(−1) = C∗(X ′0, P ′1), whence π0(Σ) = C∗(X 0, P1), and ΣΣ is a complete rationalpolyhedral decomposition of C∗(X 0, P1). The Γ-invariance and finiteness modulo Γ, as wellas the line bundle condition of 5.11 and conditions 8.13 (a) and (b) are clear from thedefinitions. q.e.d.

9.13. Corollary: Let (P ′,X ′)→ (P,X ) be as in 9.2, K ′f ⊂ P ′(Af ) an open compactsubgroup, and Kf := π0(K ′f ). Let Σ′ be a K ′f -admissible partial cone decomposition

150

for (P ′,X ′), and Σ a Kf -admissible partial cone decomposition for (P,X ). Assume thatfor every rational boundary component (P ′1,X ′1) of (P ′,X ′), every connected componentX ′0 of X ′ that maps into X ′1, and every p′f ∈ P ′(Af ), we have Σ′(X ′0, P ′1, p′f ) = Σ′Σ and

Σ(X 0, P1, π0, (p′f )) = ΣΣ for some Σ ⊂ C∗(X ′0, P ′1) as in 9.8. Then after possibly replacing

K ′f by K ′f ·KUf for some open compact subgroup KU

f of U0(Af ), the conditions 8.5 (*) and

8.13 (a) and (b) hold. In particular MKf (P,X ,Σ)(C) is projective.

Proof. Modulo the left action of P ′(Q) and the right action of K ′f there are only finitely

many (X ′0, P ′1, p′f ). Since by 9.12 Σ′Σ is finite modulo Γ, it follows that Σ′ is finite. The

same for Σ, and since by 9.12 |ΣΣ| = C∗(X 0, P1), it follows that Σ is complete. Moreover by9.12 the pair (Σ′,Σ) satisfies the conditions 8.13 (a) and (b), plus the line bundle condition(*) of 8.5 except possibly for the integrality of the splitting of the short exact sequence0→ U0(Q)∩ΓU ′ → ΓU ′ → ΓU → 0. But as in the proof of 8.8, if we replace K ′f by K ′f ·KU

f

for a large open compact subgroup KUf of U0(Af ), then U0(Q)∩ ΓU ′ is replaced by a larger

lattice. thus for any fixed σ ∈ Σ we can choose KUf such that the corresponding splitting

becomes integral. By the finiteness of Σ, and since this integrality condition is invariantunder the operations 6.4 (ii) and (iv), we can even choose KU

f so that every such splitting

becomes integral. If we now replace K ′f by K ′f ·KUf , then the other conditions still hold,

and all splittings are integral, as desired. q.e.d.

The following fact was implicit in the assumptions of the preciding corollary.

9.14. Proposition: Let Γ be as in 9.6, Σ as in 9.8, and ΣΣ, Σ+Σ , Σ′Σ as in 9.12.

Let (P ′2,X ′2) be a rational boundary component between (P ′1,X ′1) and (P ′,X ′), and Γ2 :=StabΓ(C∗(X ′0, P ′2)). Then Σ2 := C∗(X ′0, P ′2) ∩ Σ can be described as in 9.8, with (P ′2,X ′2)in place of (P ′1,X ′1), and Γ2 in place of Γ. Moreover Σ+

Σ2= σ′ ∈ Σ+

Σ | σ′ ⊂ C∗(X ′0, P ′2),and likewise for ΣΣ2 and Σ′Σ2

.

Proof. Let us first show that Γ2 is of the form described in 9.6 with (P ′2,X ′2) inplace of (P ′1,X ′1). Let Q′2 be the parabolic subgroup of P ′ associated to (P ′2,X ′2), thenΓ2 = Q′2(Q) ∩ Γ ⊂ StabQ′2(Q)(X ′0). By 9.6 the image of Γ2 in GL(U ′2) is up to scalarscommensurable with the image of an arithmetic subgroup of (Q′1∩Q′2)(Q). But by 4.19 (b)Q′2 = (Q′∩Q′2) ·P ′2, so since P ′2 acts through scalars on U ′2, the image of Γ2 in GL(U ′2) is upto scalars commensurable with the image of an arithmetic subgroup of (Q′2)(Q), as desired.

According to 6.19 (b) choose a convex rational polyhedral cone σ ⊂ C∗(X ′0, P ′2) suchthat Γ2 · σ = C∗(X ′0, P ′2), such a cone must contain U ′(R)(−1). By 9.9 σ ∩ Σ is again aconvex rational polyhedral cone. Since by definition

σ ∩ Σ = u′ ∈ σ | ∀γ ∈ Γ ∀1 ≤ i ≤ m λ (id− eγi π0)(u′) ≥ 0,

there exists a finite set S of pairs (γ, i) such that

σ ∩ Σ = u′ ∈ σ | ∀(γ, i) ∈ S λ (id− eγi π0)(u′) ≥ 0.

Write e′1, . . . , e′n := eγi |U ′2 | (γ, i) ∈ S. We have

σ ∩ Σ = u′ ∈ σ | ∀1 ≤ j ≤ n λ (id− e′j π0)(u′) ≥ 0= u′ ∈ σ | ∀1 ≤ j ≤ n ∀γ ∈ Γ2 λ (id− e′γj π0)(u′) ≥ 0,

151

hence

Σ2 = Γ2 · (σ ∩ Σ) = u′ ∈ C∗(X ′0, P ′2) | ∀1 ≤ j ≤ n ∀γ ∈ Γ2 λ (id− e′γj π0)(u′) ≥ 0,

as in 9.8. The assertions about ΣΣ2 , Σ+Σ2

, and Σ′Σ2are obvious. q.e.d.

The following construction is a kind of inverse to 9.14, and will be needed in theconstrution of cone decompositions as in 9.13.

9.15. Proposition: Let Γ be as in 9.6, and ∆ ⊂ C∗(X ′0, P ′1) r C(X ′0, P ′1) a subset,such that for every rational boundary component (P ′2,X ′2) between (P ′1,X ′1) and (P ′,X ′),different from (P ′1,X ′1), the intersection C∗(X ′0, P ′1)∩∆ is of the form described in 9.8 with(P ′2,X ′2) in place of (P ′1,X ′1) and some group of Γ2. Then there exists a Γ-invariant subsetΣ ⊂ C∗(X ′0, P ′1) as in 9.8, such that Σ r C(X ′0, P ′1) = ∆.

Proof. In the case (P ′1,X ′1) = (P ′,X ′) the assertion is just the existence of some Σas in 9.8. But by 9.3 (a) there exist splittings ei satisfying the condition in 9.8, so thisis clear. Now assume that (P ′1,X ′1) is a proper rational boundary component of (P ′,X ′).There are only finitely many Γ-conjugacy classes of rational boundary components between(P ′1,X ′1) and (P ′,X ′). Let (P ′ν ,X ′ν) be representatives for these conjugacy classes. LetΓν := StabΓ(C∗(X ′0, P ′ν)), by the proof of 9.14 this group is a valid choice as in 9.6 for(P ′ν ,X ′ν) in place of (P ′1,X ′1). Thus by assumption there exist finitely many eν,i ∈ E(X ′0, P ′ν)such that

(C∗(X ′0, P ′ν) ∩∆ = u′ ∈ C∗(X ′0, P ′ν) | ∀γ ∈ Γν ∀i λ (id− eγν,i π0)(u′) ≥ 0.

Also by 9.10 there exist finitely many u′ν,j ∈ C∗(X ′0, P ′ν) ∩ U ′ν(Q)(−1) such that ∆ ∩C∗(X ′0, P ′ν) is the convex closure of the set⋃

γ∈Γν

⋃j

R≥0 · γ · u′ν,j .

Let us fix ν and i for a moment. By 9.4 (a) there exists eν,i ∈ E(X ′0, P ′1) such thateν,i = eν,i|Uν . We would be happy if we had

λ (id− eγν,i π0)(u′µ,j) ≥ 0

for all µ, j, and all γ ∈ Γ. Since we have only finitely many u′µ,j , by 9.7 this inequalityfails at most if γ · u′µ,j is contained in one of finitely many half-lines. Since u′µ,j ∈ ∆

these points can not lie in C∗(X ′0, P ′ν). Thus by 9.4 (b) there exists ˜eν,i ∈ E(X ′0, P ′1) suchthat eν,i = ˜eν,i|Uν , that λ (id − ˜eν,i π0) ≥ λ (id − eν,i π0) on C∗(X ′0, P ′1), and that

λ (id− ˜eν,i π0) is nonnegative on each of these finitely many half-lines. Hence the desiredinequalities hold for ˜eν,i in place of eν,i.

Now we define Σ as in 9.8 using the splittings ˜eν,i. To prove Σ r C(X ′0, P ′1) = ∆, byΓ-invariance it suffices to prove C∗(X ′0, P ′ν) ∩ Σ = C∗(X ′0, P ′ν) ∩ ∆ for every ν. But bydefinition

C∗(X ′0, P ′ν) ∩ Σ = u′ ∈ C∗(X ′0, P ′ν) | ∀γ ∈ Γ ∀µ ∀i λ (id− eγν,i π0)(u′) ≥ 0⊂ u′ ∈ C∗(X ′0, P ′ν) | ∀γ ∈ Γν ∀i λ (id− eγν,i π0)(u′) ≥ 0= C∗(X ′0, P ′ν) ∩∆,

152

so one inclusion is clear. For the other inclusion it suffices to prove that all u′µ,j ∈ Σ, and

this follows from the choice of the ˜eν,i. Finally by assumption (P ′,X ′) is one of the (P ′ν ,X ′ν),so the equality U ′(R)(−1) ∩ Σ = C∗(X ′0, P ′) ∩ Σ = C∗(X ′0, P ′) ∩ ∆ implies in particularthat Σ, like ∆, contains no nontrivial linear subspace. q.e.d.

9.16. Proposition: Let (P ′,X ′) → (P,X ) be as in 9.2, K ′f ⊂ P ′(Af ) an opencompact subgroup, and Kf := π0(K ′f ). Then there exist admissible cone decompositionsΣ′ and Σ as in 9.13.

Proof. By induction we shall construct admissible cone decompositions Σ′1⊂6= Σ′2

⊂6= . . .

and Σ1⊂6= Σ2

⊂6= . . . with the following property: For every rational boundary compo-

nent (P ′1,X ′1) of (P ′,X ′), every connected component X ′0 of X ′, and every p′f ∈ P ′(Af )

either |Σ′i(X ′0, P ′1, p′f )| ⊂ C∗(X ′0, P ′1) rC(X ′0, P ′1) and |Σi(X 0, P1, π0(p′f ))| ⊂ C∗(X 0, P1) rC(X 0, P1), or there exists Σ as in 9.8 with Σ′i(X ′0, P ′1, p′f ) = Σ′Σ and Σi(X 0, P1, π0(p′f )) =ΣΣ. Since modulo the left action of P ′(Q) and the right action of K ′f there are only finitely

many triples (X ′0, P ′1, p′f ), this sequence stops at some place, at which Σ′i and Σi have thedesired properties.

Suppose that either i = 0, or Σ′i and Σi are already constructed. We want to constructΣ′i+1 and Σi+1. Choose (X ′0, P ′1, p′f ) such that ∆ := |Σ′i(X ′0, P ′1, p′f )| has the propertyrequired in 9.15. Let

Γ := StabQ′(Q)(X ′0) ∩ p′f ·K ′f · p′−1f · P ′1(Af ),

then by the K ′f -admissibility of Σ′i the set ∆ is invariant under Γ. If we let Σ be as in

9.15, then by 9.14 the cone decomposition Σ′i+1(X ′0, P ′1, p′f ) := Σ′Σ contains Σ′i(X ′0, P ′1, p′f ),

and Σi+1(X 0, P1, π0(p′f )) := ΣΣ contains Σi(X 0, P1, π0(p′f )). By the following lemma,

Σ′i+1(X ′0, P ′1, p′f ) ∪ Σ′i and Σi+1(X 0, P1, π0(p′f )) ∪ Σi extend to a K ′f -, resp. Kf -admissiblecone decompositions. By constructuion they have the desired properties. q.e.d.

9.17. Lemma: Let (P,X ) be mixed Shimura data, Kf ⊂ P (Af ) an open compactsubgroup, and Σ a Kf -admissible partial cone decompostion for (P,X ). Let (P1,X1) be arational boundary component of (P,X ), X 0 be a connected component of X that maps toX1, and pf ∈ P (Af ). Let Q be a parabolic subgroup of P associated to (P1,X1), and

Γ1 := StabQ′(Q)(X 0) ∩ pf ·Kf · p−1f · P1(Af ).

Let Σ0 be a rational partial polyhedral decomposition of C∗(X 0, P1), containing Σ(X 0, P1, pf ),such that σ0 ⊂ C(X 0, P1) for every σ ∈ Σ0rΣ(X 0, P1, pf ). Let T (X 0, P1, pf ) := Σ(X 0, P1, pf )∪Σ0. Then Σ ∪ T (X 0, P1, pf ) extends to a Kf -admissible partial cone decomposition for(P,X ) if and only if Σ0 is Γ1-invariant.

Proof. Since the conditions 6.4 (ii)-(iv) already hold for Σ, it is easily checked thatthey hold for T if and only if for all p, p′ ∈ P (Q), p1,f , p′1,f ∈ P1(Af ), kf , k′f ∈ Kf , we have

p · p1,f · Σ0 · kf = p′ · p′1,f · Σ0 · k′f , if p · X 0 = p′ · X 0, p · p1,f · pf · kf = p′ · p′1,f · pf · k′f ,and int(p)(P1) = int(p′)(P1). Here Σ0 is identified with the corresponding decomposition

153

of C∗(X 0, P1)×pf. In terms of the original decomposition of C∗(X 0, P1) this means thatp · Σ0 = p′ · Σ0 for all p, p′ ∈ P (Q) such that p′−1 · p · X 0 = X 0, p′−1 · p ∈ P (Af ) · pf ·Kf ·p−1f ·Pf (Af ) and int(p′−1 · p)(P1) = P1. Writing q = p′−1 · p this is equivalent to q ·Σ0 = Σ0

for all q ∈ StabQ′(Q)(X 0)∩P1(Af ) · pf ·Kf · p−1f ·P1(Af ). But this set is just Γ1, since Q(Q)

normalizes P1(Af ). q.e.d.

The following analog of 5.22 permits to construct refinements of cone decompositionsΣ as in 9.13.

9.18. Propositions: Let Γ be as in 9.6, and Σ as in 9.8. Consider a Γ-invariantfunction f : C∗(X 0, P1)→ U0(R)(−1), such that the restriction of λ f to every σ ∈ ΣΣ isa piecewise linear convex rational function, with respect to the rational structure given byU1(Q)(−1). For every ε ∈ Q>0 define

T : = u′ ∈ C∗(X ′0, P ′1) | u′ + ε · f π0(u′) ∈ Σ= u′ + u0 | u′ ∈ Σ, u0 ∈ U0(R)(−1) with λ(u0 + ε · f π0(u′)) ≥ 0.

If ε is sufficiently small, then T is again as in 9.8, and

ΣT =⋃σ∈ΣΣ

Σ(λ f|σ).

Proof. Choose representatives σi for the finitely many Γ-conjugacy classes in ΣΣ withR · σi = U1(R)(−1), and let ei ∈ E(X ′0, P ′1) be the associated splittings. Let τi,µ be thecones in Σ(λ f|σi) with R · τi,µ = U1(R)(−1) and `i,µ : U1 → U0 the unique linear mapssuch that f = `i,µ on τi,µ. Then by definition

T ∩ π−10 (τi,µ) = u′ ∈ π−1

0 (τi,µ) | λ (id(ei − ε · `i,µ) π0)(u′) ≥ 0.

If ε is sufficiently small, then by 9.3 (a) ei − ε · `i,µ ∈ E(X ′0, P ′1). To finish we need thatλ (id − (ei − ε · `i,µ) π0) is strictly positive on T r π−1

0 (τi,µ). In fact, it then followsthat T is as in 9.8 with respect to all ei − ε · `i,µ, and all τi,µ are in ΣT . This implies thatall Σ(λ f|σi) are contained in ΣT , and since both ΣT and

⋃σ∈ΣΣ

Σ(λ f|σ) are complete

rational polyhedral decompositions of C∗(X 0, P1), the desired equality follows.

Let R≥0 · uj,ν be all one dimensional cones in Σ(λ f|σj ). Since

T ∩ π−10 (R≥0 · uj,ν) = σ0 + R≥0 · (ej − ε · f)(uj,ν),

we have to show that for all γ ∈ Γ

λ (id− (ei − ε · `i,µ) π0)(γ · (ej − ε · f)(uj,ν)) > 0,

unless γ · uj,ν ∈ τi,µ. Fix i, j, µ, ν, and ε0 > 0 such that eτi − ε · `i,ν ∈ E(X ′0, P ′1) for allε0 > ε ∈ Q>0 then by 9.7 each of the four inequalities

λ (id− (ei − ε1 · `i,µ) π0)(γ · (ej − ε2 · f)(uj,ν)) > 0

with ε1, ε2,∈ 0, ε0 fails at most if γ ·uj,ν lies in one of finitely many half-lines. This showsthat there exists a finite number of γk ∈ Γ, such that for every ε0 > ε ∈ Q>0 the inequality

154

above fails at most if γ · uj,ν ∈ R≥0 · γk · uj,ν for some k. Thus we are reduced to showingthe inequality for some fixed γ · uj,ν 6∈ τi,µ. If γ · uj,ν 6∈ σi, then

λ (id− ei π0)(γ · (ej(uj,ν)) > 0

by the definition of ΣΣ, so the inequality certainly holds if ε is sufficiently small. Otherwiseγ · uj,ν ∈ σi implies that R≥0 · γ · uj,ν ∈ Σ(λ f|σi), so we may assume that j = i and γ = 1.Then we have

λ (id− (ei − ε · `i,µ) π0) (ei − ε · f)(uj,ν) = ε · λ (`i,ν − f)(ui,ν),

so the assertion is equivalent to λ (`i,µ − f)(ui,ν) > 0. But by the definition of τi,µ and`i,µ thi s inequality holds on all of σi − τi,µ, as desired. q.e.d.9.19. Proposition:

Let (P ′,X ′)→ (P,X ), K ′f , Kf , Σ′ and Σ be as in 9.13. Let Σ be a refinement of Σ. Thenthere exist T ′ and T satisfying the same conditions of 9.13, such that T is a refinement ofΣ.

Proof. The essential points of the proof are 5.21 and 9.18. To be able to use 9.18 weneed a function f : C(P,X )× P (Af )→ U0(R)(−1) with the following properties:

(i) The restrcition of λ f to every σ ∈ Σ is a piecewise linear convex rational function.

(ii) For every σ ∈ Σ, Σ(λ f|σ) is a refinement of τ ∈ Σ | τ ⊂ σ.(iii) f is invariant under the three actions 6.4 (ii)–(iv).

Suppose that such a function is given. For every (X ′0, P ′1, p′f ) let Σ := |Σ′(X ′0, P ′1, p′f )|,define T as in 9.18, and let T ′(X ′0, P ′1, p′f ) := Σ′T and T ′(X 0, P1, π0(p′f )) := ΣT . By 9.18 we

can make ε so small that any finite number of T ′(X ′0, P ′1, p′f ) and T (X 0, P1, π0(p′f )) satisfyall our requirements. Since by assumption this construction of T ′ and T is invariant underthe actions in 6.4 (ii)–(iv), and there are only finitely many triples (X ′0, P ′1, p′f ) modulo thisaction, there exists ε so that this works for all such triples, as desired.

We construct the desired function by induction over the classes of cones in Σ modulothe three operations 6.4 (ii)–(iv). if it has not yet been defined on all of these, choose(X ′0, P ′1, p′f ) and σ ∈ Σ(X 0, P1, π0(p′f )) with σ0 ⊂ C(X 0, P1), such that f is already definedon every proper face of σ, but not on σ itself. By 5.21 there exists a piecewise linear convexrational function g : σ → R, which coincides with λ f on every proper face of σ, suchthat Σ(g) is a refinement of τ ∈ Σ | τ ⊂ σ.

Let us check what the invariance conditions say about the desired extension of f . LetΓ1 := StabQ′(Q)(X ′0) ∩ P ′1(Af ) · p′f ·K ′f · p

′−1f . It is easy to see that any extension of f to σ

can be extended invariantly to the whole conjugacy class of σ if and only if f|σ is invariantunder the action of StabΓ1(σ). Let aγ be the scalar through which an element γ ∈ Γ1

acts on U0, by 9.2 this scalar is positive. Since λ f|σ is to be a piecewise linear convexrational function, we have λ fγ(u) = λ(a−1

γ · f(γ · u)) = λ f(a−1γ · γ · u) for all u ∈ σ, so

we want λ f|σ to be invariant under the twisted action (γ, u) 7→ a−1γ · γ · u of StabΓ1(σ).

Since P ′1 acts through scalars on U ′1, it acts trivially under this twisted action. But from6.19 (a) it follows that StabΓ1(σ) is finite modulo P ′1(Q) ∩ Γ1, so all we need is that λ f|σis invariant under a finite group of automorphisms of U1(Q)(−1) that fixes σ. Now the

155

function g : σ → R above is not necessarily invariant. But by assumption it coincides withλf on σrσ0, so at least on this subset it is invariant. Thus if we replace g by the averageof all its conjugates under that finite group, then it is invariant while still satisfying theother conditions. If we now define f to be equal to λ−1 g on σ, then we can extend it tothe whole conjugacy class of σ, as desired. q.e.d.

9.20. Proposition: In 9.19 we can achieve in addition that Σ is smooth with respectto Kf , and that the condition 7.12 (*) is satisfied. Then in particular MKf (P,X ,Σ)(C) issmooth, projective, and the completement MKf (P,X ,Σ)(C) rMKf (P,X )(C) is a union ofsmooth divisors with only normal crossings.

Proof. (For the smoothness compare [N] thm. 7.20, thm. 7.26.) If in the proof of 9.19we construct f using the strenghtness 5.23 of 5.21, then the resulting cone decomposition issmooth. To prove that in addition the condition 7.12 (*) can be satisfied, we may thereforeassume that Σ = Σ, and is already smooth. We want to apply the same procedure as inthe proof of 9.19. Using 5.25 we can choose the new function f such that the associatedrefinement T of Σ is the barycentric subdivision of Σ with respect to the lattices givenin 6.4. By 5.24 T is again smooth, and it remains to show that it satisfies the condition7.12(*).

For this choose τ ∈ T (X 0, P2, pf ) as in 7.12, let τ ′ be a face of τ that lies in T (X 0, P1, pf ),and suppose that p ·p1,f · τ ′ ·kf is also a face of τ . We have to show that p ·p1,f · τ ′ ·kf = τ ′.Choose σ ∈ Σ(X 0, P2, pf ) so that τ0 ⊂ σ0, and σ′ ∈ Σ(X 0, P1, pf ) so that τ ′0 ⊂ σ′0. Then byassumption (p·p1,f ·σ′ ·kf )0 ⊃ (p·p1,f ·τ ′ ·kf )0 ⊂ τ ⊂ σ, which implies (p·p1,f ·σ′ ·kf )0∩σ 6= ∅,so p · p1,f · σ′ · kf is a face of σ. Now we have an isomorphism p · p1,f · σ′ · kf ∼= σ′, andby assumption both p · p1,f · τ ′ · kf and τ ′ are faces of τ , which occurs in the barycentricsubdivision of σ. As explained in 5.24 it follows that p · p1,f · τ ′ · kf = τ ′, as desired. q.e.d.

9.21. Theorem: Let (P,X ) be arbitrary mixed Shimura data, and Kf ⊂ P (Af )a neat open compact subgroup. There exists a Kf -admissible complete cone decomposi-tion Σ for (P,X ) such that MKf (P,X ,Σ)(C) is smooth, projective, and the complementMKf (P,X ,Σ)(C)rMKf (P,X )(C) is a union of smooth divisors with only normal crossings.Moreover if T is any Kf -admissible complete cone decomposition for (P,X ), then Σ canbe chosen to be a refinement of T . Some ample line bundle on MKf (P,X ,Σ)(C) can bedescribed by algebraic constructions involving ω[dlog] and line bundles as in 8.13.

Proof. We shall reduce the assertion to 9.20. The problem is to relate (P,X ) withmixed Shimura data as in 9.2. First let (P , X ) := (P,X )× (Gm,Q,H0). Consider the opencompact subgroup Z× of Gm(Af ), then

M (Z)×(Gm,Q,H0)(C) = Q×\H0 × (Gm(Af )/(Z)×) ∼= ±1\H0

is just a single point. Let Kf := Kf × (Z)×, then there is an obvious one-to-one relationbetween all Kf -admissible partial cone decompositions Σ for (P,X ) and all Kf -admissiblepartial cone decompositions Σ for (P , X ), such that the projection (P , X )→ (P,X ) inducesisomorphisms

M Kf (P , X , Σ)(C) ∼= MKf (P,X ,Σ)(C)×M (Z)×(Gm,Q,H0)(C)

∼−−→MKf (P,X ,Σ)(C).

156

This reduces the assertion to that for (P , X ) and Kf .

Next let (P(0), X(0)) → (P , X ) be an irreducible component. Since the other projection

defines a morphism (P(0), X(0))→ (Gm,Q,H0), we can find a ((P0,X0)→ (Gm,Q,H0))-torsor

(P ′(0), X′(0)) → (P(0), X(0)) as in 9.1 (a). Let P∗ := P(0) · Z(P )0 ⊂ P , and (P∗,X∗) →

(P , X ) be the corresponding embedding as in 2.13. Since the extension class of the torsor(P ′(0), X

′(0))→ (P(0), X(0)) is fixed by Z(P ), we can extend it to a torsor (P ′∗,X ′∗)→ (P∗,X∗).

This torsor satisfies the requirements of 9.2. Now by 7.10 we have an isomorphism∐ν

∆ν\MKνf (P∗,X∗,Σν)(C)

∐[ ·pνf ][φ]

−−−−−−−−→M Kf (P , X ,Σ)(C)

with finitely many pνf ∈ P (Af ), Kνf := P∗(Af ) ∩ pνf · Kf · (pνf )−1, Σν := ([ ·pνf ]∗Σ)|(P∗,X∗),

and∆ν := (StabP (Q)(X∗) ∩ (P∗(Af ) · pνf · Kf · (pνf )−1))/P∗(Q).

We claim that in our situation we have ∆ν = 1. In fact, since P = P × Gm,Q, we have1×Gm,Q ⊂ Z(P )0 ⊂ P∗. Thus the image of P∗(Af ) ·pνf ·Kf ·(pνf )−1 in (P/P∗)(Af ) is equal

to that of Kf ⊂ Kf = Kf × (Z)×, which by 0.6 is neat. As in the proof of 7.10 it followsthat ∆ν is contained in a neat arithmetic subgroup of (P/P∗)(Q), which by 2.1 (viii) mustbe trivial.

Now for any fixed collection of pνf , as before there is an obvious one -to-one relation

between Kf -admissible cone decompositions Σ for (P , X ), and collections of Kνf - admissible

partial cone decomposition Σν for (P∗,X∗). Thus the assertion is reduced to (P∗,X∗). Butin that case it follows from 9.20. q.e.d.

9.22. Functoriality of cone decompositions: Consider a map [φ] : MK1f (P1,X1)(C)→

MK1f (P2,X2)(C) of 3.4 (b). Let Σi be a Ki

f -admissible cone decomposition of every (Pi,Xi).Then we define another K1

f -admissible partial cone decomposition T1 for (P1,X1) as follows.Every rational boundary component (P11,X11) of (P1,X1) determines a rational boundarycomponent (P21,X21) of (P2,X2), and every connected component X 0

1 of X1 maps to aunique connected component X 0

2 of X2. The homomorphism φ : U11 → U21 then mapsC∗(X 0

1 , P11) to C∗(X 02 , P21). For all such (P1,X1) and X 0

1 , and every pf ∈ P1(Af ), we define

T1(X 01 , P11, p1,f ) := σ1 ∩ φ−1(σ2) | σ1 ∈ Σ1(X 0

1 , P11, p1,f ), σ2 ∈ Σ2(X 02 , P21, φ(pf )).

It is clear form the definition that T1 is a K1f -admissible partial cone decomposition for

(P1,X1). If Σ1 and Σ2 are finite or complete, then T1 has the same property. MoreoverT1 = Σ1 if and only if the condition in 6.25 (b) holds for the pair (Σ1,Σ2). In general by6.25 (b) we have the following maps

MK1f (P1,X1,Σ1)(C)←MK1

f (P1,X1, T1)(C) −→MK2f (P2,X2,Σ2)(C).

A particular case is the identity map on (P1,X1), where T1 is the coarsest common refine-ment of Σ1 and Σ2. If (P2,X2) = (P1,X1) and pf ∈ P1(Af ), then applying this to Σ1 and

157

[ ·pf ]∗Σ2 yields the analogous statements for the morphism [ ·pf ] of 3.4 (a). The samearguments work for any finite number of morphisms of 3.4 (a) and (b).

9.23. Corollary: In 9.21 suppose that in addition a finite number of mapsMKf (P,X )(C)→MKi

f (Pi,Xi)(C) of 3.4 (a) and (b) are given, and for every i a complete K − f i-admissiblecone decomposition Σi for (Pi,Xi). Then one can choose Σ such that all the mapsMKf (P,X ,Σ)(C)→MKi

f (Pi,Xi,Σi)(C) (of 6.25 (a) and (b)) are defined.

Proof. By 9.22 there exists a complete Kf -admissible cone decomposition T for (P,X )such that the conditions in 6.25 (a) and (b) are verified for all the morphismsMKf (P,X , T )(C)→MKi

f (Pi,Xi,Σi)(C). If in 9.21 Σ is a refinement of T , then the same holds for Σ in place ofT , as desired. q.e.d.

9.24. Proposition: For every mixed Shimura data (P,X ), and every open compactsubgroup Kf ⊂ P (Af ), the mixed Shimura variety MKf (P,X )(C) possesses a canonicalstructure of a normal quasiprojective algebraic variety over C, such that all the maps of3.4 become algebraic morphisms. Whenever Σ is a complete Kf -admissible cone decom-position for (P,X ) such that MKf (P,X ,Σ)(C) is projective, then the open embeddingMKf (P,X )(C) →MKf (P,X ,Σ)(C) is algebraic.

Proof. Let Σ be a complete Kf -admissible cone decomposition for (P,X ) such thatMKf (P,X ,Σ)(C) is projective. Since by 6.25 (a) the complement of MKf (P,X )(C) inMKf (P,X ,Σ)(C) is a closed analytic set, by Chow’s theorem (see [S] §19 prop.13) it isa Zariski-closed subset. Thus its complement MKf (P,X )(C) inherits the structure of aquasiprojective variety over C.

To prove the functoriality of this algebraic structure consider other mixed Shimuradata (P ∗,X ∗), an open compact subgroup K∗f ⊂ P ∗(Af ), and a complete K∗f -admissible

cone decomposition for (P ∗,X ∗), such that MK∗f (P ∗,X ∗,Σ∗)(C) is projective. Considera map MKf (P,X )(C) → MK∗f (P ∗,X ∗)(C) of 3.4 (a) or (b). Choose a neat open normalsubgroup K+

f ⊂ Kf , then by 9.23 there exists a refinement T of Σ, such that the map

MK+f (P,X , T )(C)→MK∗f (P ∗,X ∗,Σ∗)(C) of 6.25 (a) or (b) is defined. Then the maps

MKf (P,X ,Σ)(C)[id]←−−−MK+

f (P,X , T )(C)→MK∗f (P ∗,X ∗,Σ∗)(C)

are holomorphic maps between projective complex spaces, hence algebraic. In particularthe maps

MKf (P,X )(C)[id]←−−−MK+

f (P,X )(C)→MK∗f (P ∗,X ∗)(C)

are algebraic with respect to the induced algebraic structures. Recall that by 6.25 (a)

MKf (P,X )(C) is as complex space the quotient of MK+f (P,X )(C) by the finite group of

Kf/K+f . Since the map on the right hand side factors through MKf (P,X )(C), the map

MKf (P,X )(C)→MK∗f (P ∗,X ∗)(C) is also algebraic, as desired. q.e.d.

A special case of the functoriality is that the algebraic structure on MKf (P,X )(C) isindependant of the choice of Σ. Thus the assertions are already proved whenever there existsa complete Kf -admissible cone decomposition for (P,X ) such that MKf (P,X ,Σ)(C) is

158

projective. By 9.21 this is the case if Kf is neat. For arbitrary Kf choose a neat open normal

subgroup K+f ⊂ Kf . By the functoriality for MK+

f (P,X )(C) the action of Kf/K+f respects

the canonical algebraic structure on MK+f (P,X )(C). Since this is a quasiprojective variety,

and by [SGA1] exp.VIII cor. 7.7 the quotient of every quasiprojective algebraic variety by afinite group exists and is again quasiprojective, we get a canonical quasiprojective algebraicstructure on MKf (P,X )(C). The functoriality for this algebraic structure follows by thesame arguments as above, in particular it does not depend on the choice of K+

f . q.e.d.

9.25. Definition: For every mixed Shimura data (P,X ) and every open compactsubgroup Kf ⊂ P (Af ) let

MKfC (P,X )

be the normal algebraic variety over C associated by 9.24 to the normal complex spaceMKf (P,X )(C). Let Σ be a Kf -admissible partial cone decomposition for (P,X ). If thereexists an algebraic structure on MKf (P,X ,Σ)(C) extending the above algebraic structureon MKf (P,X )(C), we let

MKfC (P,X ,Σ)

be the associated normal algebraic variety over C. Furthermore if P is reductive, thenwe denote the normal projective variety corresponding to the Baily-Borel compactificationMKf (P,X )∗(C) by

MKfC (P,X )∗.

In all these cases the variety is, strictly speaking, defined by a universal property, so itreally consists of a complex varietyX together with an isomorphismX(C) ∼= MKf (P,X )(C),resp. X(C) ∼= MKf (P,X ,Σ)(C), resp. X(C) ∼= MKf (P,X )∗(C). We have to prove that it is

unique up to canonical isomorphism. For MKfC (P,X ) this follows from 9.24, for M

KfC (P,X )∗

by projectivity. For MKfC (P,X ,Σ) recall that MKf (P,X )(C) is dense in MKf (P,X ,Σ)(C),

hence Zariski-dense with respect to any algebraic structure. It is easily seen that an analyticmap between two normal complex varieties is an algebraic morphism if it is an algebraic

morphism on a Zariski-open dense subset. Thus MKfC (P,X ,Σ) is well-defined whenever

such an algebraic structure exists.

By 9.24 we already know that all the maps 3.4 (a) and (b) correspond to unique alge-

braic morphisms between different MKfC (P,X )∗. By the argument above, the same holds for

the maps 6.25 (a) and (b), if MKfC (P,X ,Σ) etc. exists. For the Baily-Borel compactifica-

tion the same follows by projectivity. Moreover for arbitrary P , by the argument above the

map [π]∗ defined in 6.24 corresponds to a morphism MKfC (P,X ,Σ)→M

π(Kf )C ((P,X )/W )∗

whenever MKfC (P,X ,Σ) exists. In all these cases we shall use the same symbols to denote

the corresponding algebraic morphisms.

The assertion of 6.25 (functoriality, open embeddings, isomorphisms, quotients by finitegroups), 6.26 (smoothness), 8.6 (line bundle structure), 8.13–14 (ampleness) are equallyvalid for these algebraic varieties, since these are properties that translate directly betweenthe algebraic and the analytic category, once all the objects and maps are algebraic.

159

9.26. Remark: ¿From the proof of 9.24 it follows that the system of algebraicstructures on all MKf (P,X )(C) is uniquely determined by the properties expressed in 9.24.One may ask whether these algebraic structures can be characterized uniquely withoutreference to toroidal compactifications. As it turns out, they are already characterizeduniquely by the functoriality with respect to all the maps 3.4.

Let us briefly indicate why. If P is reductive and possesses no almost direct fac-tor to type SL2,Q, then the boundary of MKf (P,X )(C) in the Baily-Borel compactifi-cation has codimension ≥ 2, so it possesses at most one algebraic structure. For P =GL2,Q the uniqueness follows via an embedding (GL2,Q,H2) → (CSP4,Q,H4), as in theproof of 8.4. Thus we have uniqueness whenever P is reductive. Next, if U = 1, thenMKf (P,X )(C) → Mπ(Kf )((P,X )/W )(C) is proper, so again the uniqueness follows byfunctoriality. Finally observe that every torus possesses a unique algebraic structure.Thus for arbitrary P , and sufficiently small Kf , we may use the torsor structure onMKf (P,X )(C)→Mπ′(Kf )((P,X )/U)(C) to get the result.

9.27. Remarks about the existence of MKfC (P,X ,Σ): (a) By the definition in

9.25, MKfC (P,X ,Σ) exists whenever MKf (P,X ,Σ)(C) is projective, in particular in the

situations of 8.13, 9.13, or 9.21. If MKfC (P,X ,Σ) exists, then the same follows for every

Kf -admissible partial cone decomposition contained in Σ. If Σ is the union of any (finite

or infinite) collection Σi, such that MKfC (P,X ,Σi) exists, then by gluing it follows that

MKfC (P,X ,Σ) exists. These three facts, together with 9.16–21, shows that M

KfC (P,X ,Σ)

exists for a large class of cone decompositions.

(b) In the situation of 8.13 (e.g. 9.13) not only MKfC (P,X ,Σ), but also the “line bundle”

MK′fC (P ′,X ′,Σ′) exists as an algebraic variety and is quasiprojective. In fact, by ampleness

one can construct a canonical ample line bundle on the associated P1(C)-bundle obtainedby adding the ∞-section. This, too, can be expressed in the formalism of chapter 8.

(c) The knowledge so far about the existence of MKfC (P,X ,Σ) is probably sufficient for

most purposes. Nevertheless it seems somewhat unsatisfactory to stop here and not try tofind out more about the algebracity of our toroidal compactifications. Therefore we includemore results in this direction, although the proof of 9.29 is rather technical. See also 9.35below.

First we show (using 9.20) that MKfC (P,X ,Σ) exists whenever Σ is sufficiently “fine”.

9.28. Proposition: Let (P ′,X ′) → (P,X ), K ′f , Kf , Σ′, and Σ be as in 9.13.Assume that the condition 7.12 (*) is satisfied for Σ. Let T be a Kf -admissible partialcone decomposition for (P,X ) such that every τ ∈ T is contained in some σ ∈ Σ. Then

MKfC (P,X , T ) exists. In fact there exists a collection of T ′i and T i as in 9.13, and Ti ⊂ T i,

such that T is the union of all Ti.Proof. It suffices to prove that for every convex rational polyhedral cone τ contained

in some σ ∈ Σ there exist T ′ and T as in 9.13 such that τ ∈ T . Using the same procedureas in the proof of 9.19 it suffices to construct the function f such that τ ∈ Σ(λf|σ). To dothis choose a piecewise linear convex rational function g : σ → R such that τ ∈ Σ(g). Toshow that such a function exists, take any linear rational function ` : σ → R that is strictly

160

positive on σr0. Then let g be the smallest nonnegative convex function σ → R≥0 whichis ≥ ` on τ . It is clear that g = ` on τ , and g > ` on σ r τ , hence τ ∈ Σ(g), as desired.

Now define f|σ := λ−1 g : σ → U0(R)(−1). The condition 7.12 (*) implies that no twodifferent faces of σ are equivalent under the actions 6.4 (ii)–(iv). This in turn implies thatwhenever these actions induce an automorphism of a face ρ of σ, then this automorphismmaps every face of ρ to itself. But by 6.19 (a) and 9.6 some power of this automorphismacts like a scalar on ρ. This shows that the automorphisms acts by the same scalar on everyone dimensional face of ρ, hence it acts like a scalar on ρ. Thus there is no obstruction toextending f|σ invariantly to all conjugates of σ under the actions 6.4 (ii)–(iv). Finally by

the proof of 9.19 (with Σ = Σ) it can be extended to all of C(P,X )× P (Af )→ U0(R)(−1)in the desired way. q.e.d.

9.29. Proposition: Let (P ′,X ′)→ (P,X ) be as in 9.2, (P1,X1) a rational boundarycomponent of (P,X ), and X 0 a connected component of X that maps into X1. Let pf ∈P (Af ), and σ a convex rational polyhedral cone in C(X 0, P1) that does not contain anontrivial linear subspace. Then there exists an open compact subgroup K ′f ⊂ P ′(Af ) and

Σ and Σ′ as in 9.13, such that σ ∈ Σ(X 0, P1, pf ).

Remark. Incidentally, this shows that the cones in a complete admissible decomposi-tion can be arbitrarily large.

Proof. Using the operation 6.5 (a) we may assume that pf = 1. Without loss ofgenerality we may assume that R · σ = U1(R)(−1), since otherwise we may (repeatedly)replace σ by σ + R≥0 · u′ for arbitrary u′ ∈ C∗(X 0, P1) ∩ U1(Q)(−1) r R · σ. Let (P ′1,X ′1)be the rational boundary component of (P ′,X ′) with (P1,X1) = (P ′1,X ′1)/U0, and X ′0 theconnected componentof X ′ corresponding to X 0. We want to apply the same inductiveprocedure as in the proof of 9.16. Suppose that for some K ′f we have constructed a K ′f -

admissible partial cone decomposition Σ′ for (P ′,X ′), such that for some Σ ⊂ C∗(X ′0, P ′1)as in 9.8 Σ′(X ′0, P ′1, 1) = Σ′Σ, and σ ∈ ΣΣ. Then as in the proof of 9.16 it can be extended tothe desired Σ′. We therefore have to construct Σ ⊂ C∗(X ′0, P ′1) as in 9.8 such that σ ∈ ΣΣ

and Σ′(X ′0, P ′1, 1) = Σ′Σ extends to a K ′f -admissible partial cone decomposition. While thefirst property is not difficult to achieve, the second one makes the proof complicated.

Fix an arbitrary e1 ∈ E(X ′0, P ′1). By 9.3 (a) we may fix e2, . . . , em ∈ E(X ′0, P ′1) suchthat the λ (e1 − ei) ∈ Hom(U1(R)(−1),R) span the dual cone σ, in other words

σ = u ∈ C∗(X 0, P1) | ∀2 ≤ i ≤ m λ (e1 − ei)(u) ≥ 0.

Let u1, . . . , un ∈ C∗(X 0, P1) ∩ U1(Q)(−1) such that σ =∑n

j=1 R≥0 · uj . Suppose that K ′f

has been chosen, and let

Γ1 := StabQ′(Q)(X ′0) ∩K ′f · P ′1(Af ).

Define Σ0 as in 9.8 using this group and the sections e1, . . . , em. Since σ does not contain anontrivial linear subspace, its dual cone generates the dual of U1(R)(−1), so by the choiceof the ei the assumption in 9.8 holds. Suppose that all ei(uj) lie in Σ0. Then e1(σ) ⊂ Σ0,and the choice of e2, . . . , em implies that λ (id− e1 π) is strictly positive on Σ0 r e1(σ).Thus σ ∈ ΣΣ0 , which is one of the desired properties. But Σ0 is not yet the right subset,

161

since Σ′Σ0does not necessarily extend to a K ′f -admissible partial cone decomposition. To

remedy this we shall construct the “right” Σ as a Γ1-invariant subset of Σ0 that containsall the e1(uj). By the same argument it then follows that σ ∈ ΣΣ, and it remains to chooseΣ such that Σ′Σ extends.

Let us check what this extendability condition amounts to. By lemma 9.32 below Σ′Σ ex-tends to a K ′f -admissible partial cone decomposition if and only if for every rational bound-

ary component (P ′2,X ′2) between (P ′1,X ′1) and (P ′,X ′), and every q′ ∈ StabQ′(Q)(X ′0)∩K ′f ·P ′2(Af ), we have

q′ · τ | τ ∈ Σ′Σ, τ ⊂ C∗(X ′0, P ′2) = τ ∈ Σ′Σ | τ ⊂ q′ · C∗(X ′0, P ′2).

By the definition of Σ′Σ this can be translated into the following equality in terms of Σ:

q′ · (Σ ∩ C∗(X ′0, P ′2)) = Σ ∩ q′ · C∗(X ′0, P ′2).

Together with the condition e1(uj) ∈ Σ this has the following consequence. Every uj lies inC(X 0, Pj) for a unique rational boundary component (P ′j ,X ′j) between (P ′1,X ′1) and (P ′,X ′).Thus e1(uj) ∈ Σ implies q′ · e1(uj) ∈ Σ for all j and q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′j(Af ).

We shall construct Σ by induction over the (P ′2,X ′2) for which the above equality holds.

Consider an increasing sequence ∅ = A0⊂6= A1

⊂6= . . . of Γ1-invariant collections of rational

boundry components between (P ′1,X ′1) and (P ′,X ′), such that whenever (P ′2,X ′2) ∈ Aν , thenso is every rational boundary component between (P ′2,X ′2) and (P ′,X ′). We shall constructthese Aν and a sequence of Γ1-invariant subsets Σ0 ⊃ Σ1 ⊃ . . . as in 9.8, such that the aboveequality holds for all Σν and (P ′2,X ′2) ∈ Aν . Furthermore we require that q′ · e1(uj) ∈ Σν

for all ν, j, and q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′j(Af ). Since modulo Γ1 there are only finitelymany rational boundary components between (P ′1,X ′1) and (P ′,X ′), this sequence stops atsome point, where Σ := Σν has all the desired properties.

Of course this works only if K ′f is chosen in advance. So we first show that it can bechosen so that the conditions hold for Σ0. Since A0 = ∅, only the condition q′ · e1(uj) ∈ Σ0

matters. By the definition of Σ0 this is equivalent to

λ (id− eγi π0)(q′ · e1(uj)) ≥ 0

for all i, j, γ ∈ Γ1, and q′ ∈ StabQ′(Q)(X ′0) ∩ K ′f · P ′j(Af ). By the remark in 9.2 this isequivalent to λ (id − ei π0)(γ · q′ · e1(uj)) ≥ 0, so by lemma 9.30 below we are reducedto γ = 1. Since there are only finitely many i and j, the desired K ′f exists by lemma 9.31below. We fix such K ′f once and for all.

Next assume that Σν has been constructed. Fix a rational boundary component(P ′2,X ′2) between (P ′1,X ′1) and (P ′,X ′), such that (P ′2,X ′2) 6∈ Aν , but Aν contains everyboundary component between (P ′2,X ′2) and (P ′,X ′) that is different form (P ′2,X ′2). Let Aν+1

be the union of Aν with the set of all Γ1-conjugates of (P ′2,X ′2). Let Q′2 be the parabolicsubgroup of P ′ associated to (P ′2,X ′2), and define Γ2 := Stab(Q′∩Q′2)(Q)(X ′0) ∩K ′f · P ′2(Af ),

this group contains StabΓ1(C∗(X ′0, P ′2)) as a subgroup of finite index. Thus the set

Σ′ :=⋂γ∈Γ2

γ · (Σν ∩ C∗(X ′0, P ′2))

162

is of the form of 9.8 for (P ′2,X ′2) in place of (P ′1,X ′1). We shall construct a Γ1-invariantsubset Σν+1 ⊂ Σν as in 9.8, such that

(a) q′ · e1(uj) ∈∑

ν+1 for all j and all q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′j(Af ),

(b)∑

ν+1 ∩C∗(X ′0, P ′3) =∑

ν ∩C∗(X ′0, P ′3) for every (P ′3X ′3) ∈ Aν , and

(c)∑

ν+1 ∩q′ · C∗(X ′0, P ′2) = q′ · Σ′ for every q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′2(Af ).

Then by (c) and by Γ− 1-invariance the extendability condition holds for all Γ1-conjugatesof (P ′2,X ′2), and by (b) and the assumption about Σν it holds for all (P ′3X ′3) ∈ Aν , so itholds for all (P ′3X ′3) ∈ Aν+1, as desired.

Using the fact that∑

ν+1 is Γ1-invariant we can reduce this infinity of conditions toa finite number. First by lemma 9.30 below it suffices to require (c) only for a finitesubset R ⊂ StabQ′(Q)(X ′0) ∩ K ′f · P ′2(Af ). For the same reason it suffices to require (a)

only for a finite subset Rj ⊂ StabQ′(Q)(X ′0) ∩ K ′f · P ′j(Af ). For (b) we are reduced to a

finite number of (P ′3X ′3) ∈ Aν . By 9.14 every Σν ∩ C∗(X ′0, P ′3) is of the form in 9.8 withrespect to the group StabΓ1(C∗(X ′0, P ′3)), so by 9.10 this set is the convex closure of theunion of all StabΓ1(C∗(X ′0, P ′3))-orbits of finitely many rational half-lines. Therefore byStabΓ1(C∗(X ′0, P ′3))-variance the condition (b) is equivalent to U ⊂ Σν+1 for some finitesubset

U ⊂∑ν

∩U ′1(Q)(−1) ∩⋃

(P ′3X ′3)∈Aν

C∗(X ′0, P ′3).

Now for every q′ ∈ R let Γ2,q′ := StabΓ1(C∗(X ′0, q′ · P ′2 · q′−1)), and choose finitelymany eq′,i ∈ E(X ′0, q′ · P ′2 · q′−1) so that q′ · Σ′ is of the form in 9.8 with this group andthese splittings. Lift every eq′,i to an element eq′,i ∈ E(X ′0, P ′1), and define Σν+1 as in9.8 with the group Γ1, and as splittings these eq′,i and those that define Σν . It remainsto show that the conditions (a)–(c) hold for a suitable choice of the liftings eq′,i. Let usrewrite these conditions in terms of the eq′,i. For (c) note that for every q′ ∈ R the inclusionΣν+1 ∩ q′ · C∗(X ′0, P ′2) ⊂ q′ · Σ′ holds by construction. For the other inclusion we can asabove write q′ ·Σ′ as the convex closure of the union of all StabΓ1(q′ ·C∗(X ′0, P ′2))-orbits offinitely many rational half-lines, so it suffices to show that V ⊂ Σν+1 for some finite subset

V ⊂⋃q′∈R

q′ · Σ′ ∩ U ′1(Q)(−1).

Altogether it follows that the conditions (a)–(c) hold if and only if Σν+1 contains the finitesubset

W := q′ · e1(uj) | 1 ≤ j ≤ n, q′ ∈ Rj ∪ U ∪ V ⊂ Σν ∩ U ′1(Q)(−1).

By the definition of Σν+1 this means that for all q′ ∈ R, i, u′ ∈ W and γ ∈ Γ1 we need

λ (id− eq′,i π0)(γ · u′) ≥ 0.

By the argument in the proof of 9.15 one can choose the liftings eq′,i such that this inequalityfails at most if γ ·u′ ∈ q′·C∗(X ′0, P ′2). Thus it remains to prove Γ1·W∩q′·C∗(X ′0, P ′2) ⊂ q′·Σ′for every q′ ∈ R.

Translated back into the original formulation of these conditions we have to prove forall q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′2(Af ):

163

(a′) For all j and q′′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′j(Af ):

q′′ · e1(uj) ∈ q′ · C∗(X ′0, P ′2)⇒ q′′ · e1(uj) ∈ q′ · Σ′.

(b′) Σν ∩ C∗(X ′0, P ′3) ∩ q′ · C∗(X ′0, P ′2) ⊂ q′ · Σ′ for every (P ′3X ′3) ∈ Aν .

(c′) q′′ · Σ′ ∩ q′ · C∗(X ′0, P ′2) ⊂ q′ · Σ′ for all q′′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′2(Af ).

Since q′ · γ2 ∈ StabQ′(Q)(X ′0) ∩ K ′f · P ′2(Af ) for every γ2 ∈ Γ2, by the definition of Σ′ wemay replace the inclusions ⊂ q′ ·Σ′ by the inclusions ⊂ q′ ·Σν . Now in (a′) the assumptionimplies that q′′ ·C(X ′0, P ′j)∩ q′ ·C∗(X ′0, P ′2) 6= ∅, so int(q′)((P ′2,X ′2)) is a rational boundary

component of int(q′′)((P ′j ,X ′j)), and in particular int(q′′−1 · q′)((P ′2,X ′2)) ⊂ (P ′j ,X ′j). Let

p′2,f ∈ P ′2(Af ) such that q′ ∈ K ′f · p′2,f , then it follows that

q′−1 = p′2,f · q′−1 · q′′ · int(q′′−1 · q′)(p′−1

2,f )

∈ K ′f ·K ′f · P ′j(Af ) · int(q′′−1 · q′)((P ′2,X ′2)) = K ′f · P ′j(Af ),

whence q′−1 · q′′ ∈ StabQ′(Q)(X ′0) ∩ K ′f · P ′j(Af ). Thus without loss of generality we may

assume q′ = 1, so we are reduced to proving q′·e1(uj) ∈ Σν for all j, and q′ ∈ StabQ′(Q)(X ′0)∩K ′f · P ′j(Af ). But this is one of the assumptions about Σν .

In (b′) it suffices to consider only those (P ′3X ′3) ∈ Aν for which int(q′)((P ′2,X ′2)) is arational boundary component of (P ′3X ′3). As above it follows that q′−1 ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′3(Af ), and the inclusion

Σν ∩ C∗(X ′0, P ′3) ⊂ q′ · Σν

⇔ q′−1 · (Σν ∩ C∗(X ′0, P ′3)) ⊂ Σν

follows from the invariance condition for (P ′3X ′3) and Σν . Finally for (c′) the left handside is always contained in q′′ · C∗(X ′0, P ′3) for a rational boundary component (P ′3X ′3)between (P ′2X ′2) and (P ′X ′), such that int(q′)((P ′2,X ′2)) is a rational boundary componentof int(q′′)((P ′3X ′3)). Thus as above we are reduced to prove the inclusion

q′′ · (Σ′ ∩ C∗(X ′0, P ′3)) ⊂ Σν

for all q′′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′3(Af ) such that q′′ · C∗(X ′0, P ′3) ⊂ C∗(X ′0, P ′2). In thecase (P ′3X ′3) = (P ′2X ′2) we must have q′′ ∈ Γ2, so this follows from the definition of Σ′.Otherwise we have (P ′3X ′3) ∈ Aν , so this follows from the invariance condition for (P ′3X ′3)and Σν . Thus the conditions (a′)–(c′) are satisfied. q.e.d.

9.30. Lemma: Let (P,X ) be mixed Shimura data, (P2,X2) a rational boundarycomponent of (P,X ), and (P1,X1) a rational boundary component of (P2,X2). Let X 0 bea connected component of X , Q the parabolic subgroup of P associated to (P1,X1), andKf an open compact subgroup of P (Af ). Let

Γ1 := StabQ(Q)(X 0) ∩Kf · P1(Af ).

There exists a finite subset R ⊂ StabQ(Q)(X 0) ∩Kf · P2(Af ) such that

StabQ(Q)(X 0) ∩Kf · P2(Af ) = Γ1 ·R · Stab(P2∩Q)(Q)(X 0).

164

Proof. For abbreviation write A := StabQ(Q)(X 0)∩Kf ·P2(Af ). Then we have Γ1 ·A =A. Indeed, let q ∈ A and q′ ∈ Γ1, then obviously q′ · q ∈ StabQ(Q)(X 0), and

q′ · q ∈ Kf · P1(Af ) · q ⊂ Kf · (Kf · P2(Af )) · (q−1 · P1(Af ) · q) = Kf · P2(Af ),

since q normalizes P1(Af ). Clearly Stab(P2∩Q)(Q)(X 0) acts on the right hand side on A, andwe have to show that the double quotient Γ1\A/ Stab(P2∩Q)(Q)(X 0) is finite.

Since P2 ∩ Q is a parabolic subgroup of P2, for every open compact subgroup KP2f ⊂

P2(Af ) the double quotient KP2f \P2(Af )/(P2 ∩Q)(Q) is finite. In particular there exists a

finite subset R ⊂ P2(Af ) such that

P2(Af ) = (P2(Af ) ∩Kf ) ·R · Stab(P2∩Q)(Q)(X 0).

This implies

A = StabQ(Q)(X 0) ∩Kf · (P2(Af ) ∩Kf ) ·R · Stab(P2∩Q)(Q)(X 0)

= (StabQ(Q)(X 0) ∩Kf ·R) · Stab(P2∩Q)(Q)(X 0),

and that(StabQ(Q)(X 0) ∩Kf )\(StabQ(Q)(X 0) ∩Kf ·R)

is finite. Since StabQ(Q)(X 0) ∩Kf is contained in Γ1, we are done. q.e.d.

9.31. Lemma: Consider the situation of 9.2. Let (P ′2,X ′2) be a rational bound-ary component between (P ′1,X ′1) and (P ′,X ′). Let e ∈ E(X ′0, P ′1) und u′ ∈ C(X ′0, P ′2) ∩U ′2(Q)(−1) such that λ (id − e π0)(u′) ≥ 0. There exists an open compact subgroupK ′f ⊂ P ′(Af ) such that for all q′ ∈ StabQ′(Q)(X ′0) ∩K ′f · P ′2(Af )

λ (id− e π0)(q′ · u′) ≥ 0.

Proof. If P ′2,X ′2) = (P ′,X ′), then the inequalities always hold, since in this caseStabQ′(Q)(X ′0) ∩K ′f · P ′2(Af ) = StabQ(Q)(X ′0) acts by positive scalars on U ′. Otherwise fix

an arbitrary K ′f . We first show that there are finitely many half-lines R≥0 ·u′i such that the

inequality fails at most if R≥0 · q′ · u′ = R≥0 · u′i for some i. To prove this let

Γ1 := StabQ′(Q)(X ′0) ∩K ′f · P ′1(Af ),

and a fix subset R = q′1, . . . , q′n ⊂ StabQ(Q)(X 0) as in 9.30. Then for any q′ = γ · q′j · p2 ∈StabQ′(Q)(X ′0) ∩K ′f · P ′2(Af ) with p2 ∈ StabP ′2(Q)(X ′0) and γ ∈ Γ1, the element q′ · u′ is a

positive multiple of γ ·q′j ·u′. By 9.7 there are at most finitely many half- lines R≥0 ·γ ·q′j ·u′,for which the inequality fails, as desired.

Next suppose that R · u′i = R · u′ for some i. Then u′i is a negative multiple of u′,since by assumption the inequality fails for u′i. Since both u′ and u′i lie in the convex setC(X ′0, P ′2), this contains 0, in contradiction to the assumption (P ′2,X ′2) 6= (P ′,X ′). ThusR ·u′i 6= R ·u′ for all i. Now consider an open compact subgroup K ′′f ⊂ K ′f . If the inequality

fails for some q′ ∈ StabQ′(Q)(X ′0) ∩ K ′′f · P ′2(Af ), then R≥0 · q′ · u′ = R≥0 · u′i for some i.

165

But for every i, by 7.14 this cannot happen if K ′′f is sufficiently small. Since there are onlyfinitely many i, we are done. q.e.d.

9.32. Lemma: Let (P,X ) be mixed Shimura data, (P1,X1) a rational boundarycomponent of (P,X ), and X 0 a connected component of X that maps to X1. Let Q bethe parabolic subgroup of P associated to (P1,X1). Furthermore let Kf ⊂ P (Af ) be anopen compact subgroup, and Σ0 a rational partial polyhedral decomposition of C∗(X 0, P1).There exists a Kf -admissible partial cone decomposition Σ for (P,X ) with Σ(X 0, P1, 1) =Σ0 if and only if the following condition holds:

For every rational boundary component (P2,X2) between (P1,X1) and (P,X ), andevery q ∈ StabQ(Q)(X 0) ∩Kf · P2(Af )

q · σ | σ ∈ Σ0, σ ⊂ C∗(X 0, P2) = σ ∈ Σ0 | σ ⊂ q · C∗(X 0, P2).

Proof. If Σ is to exist, then for every (P2,X2) above we must have Σ(X 0, P2, 1) :=σ ∈ Σ0 | σ ⊂ C∗(X 0, P2). By the conditions 6.4 (ii)–(iv) we must also have

Σ(p · X 0, p · P2 · p−1, p·2,f ·kf ) = p · σ | σ ∈ Σ0, σ ⊂ C∗(X 0, P2)

for all p ∈ P (Q), p2,f ∈ P2(Af ), and kf ∈ Kf . Thus a necessary condition is that the righthand side does not change if p, p2,f , kf are replaced by other elements such that p · X 0,p ·P2 ·p−1, p ·p2,f ·kf do not change. It is quite obvious that this condition is also sufficient.Now this condition holds if and only if for all (P2,X2) and p ∈ StabP (Q)(X 0)∩Kf ·P2(Af ),if (P1,X1) is a boundary component of int(p)((P2,X2)), then

p · σ | σ ∈ Σ0, σ ⊂ C∗(X 0, P2) = σ ∈ Σ0 | σ ⊂ p · C∗(X 0, P2).

For every such p, both (P1,X1) and int(p−1)((P1,X1)) are rational boundary components of(P2,X2), so there exists p2 ∈ StabP2(Q)(X 0) such that int(p−1)((P1,X1)) = int(p−1

2 )((P1,X1)).

In other words q := p · p−12 normalizes (P1,X1), hence lies in Q(Q). But p2 · C∗(X 0, P2) =

C∗(X 0, P2), and acts by scalars on U2(R)(−1). If (P2,X2) is a proper boundary component,then this scalar is positive, and in the above equation we may replace p by q, as in thedesired assertion. Otherwise we anyway have Q = P , and may even take p2 = 1. q.e.d.

9.33. Corollary (of 9.29): Consider mixed Shimura data (P,X ), an open compactsubgroup Kf ⊂ P (Af ), and a finite Kf -admissible cone decomposition Σ for (P,X ). Then

MK+f

C (P,X ,Σ) exists for every sufficiently small open compact subgroup K+f ⊂ Kf . In

fact MK+f

C (P,X ,Σ) is covered by quasiprojective MK+f

C (P,X ,Σi), for which an ample linebundle can be described as in 8.14.

Remark. Except for the last sentence this can be proved with less pain than taken9.29, if one disregards the invariance condtion 6.4 (iv).

Proof. Since we may choose K+f neat, as in the proof of 9.21 we can reduce to the case

where (P,X ) is as in 9.2. Let σi be (finitely many) representatives in Σ for the two actions6.4 (ii)–(iii) of P (Q) and Kf . By 9.29 for every i there exists an open compact subgroup

166

Kif ⊂ P (Af ), and a Ki

f - admissible partial cone decompostion Σi for (P,X ), that contains

σ, and such that MKif

C (P,X ,Σi) exists. Since we have only finitely many i, we can findan open compact subgroup K∗f , contained in the intersection of all Ki

f , that is a normal

subgroup of Kf . Then every MK∗fC (P,X ,Σi) exists, and so does M

K∗fC (P,X , [ ·kf ]∗Σi) for

every kf ∈ Kf modulo K∗f . Now we may replace every Σi by the smallest K∗f -admissiblecone decompostion that contains σi. Then Σi is contained in Σ, and by construction Σ is

the union of all [ ·kf ]∗Σi. Thus by gluing MK∗fC (P,X ,Σi) exists, as desired. q.e.d.

9.34. Corollary: (a) “MKfC (P,X ,Σ)” always exists as an algebraic space.

(b) Suppose that MKfC (P,X ,Σ) exists. If Σ is finite, resp. complete, then M

KfC (P,X ,Σ) is

of finite type, resp. proper, over C.

Proof. (a) In the category of algebraic spaces every quotient of a variety by a finite

group exists (see [K] introduction). Thus by 9.33, “MKfC (P,X ,Σ)” exists as the quotient of

MK+f

C (P,X ,Σ) by Kf/K+f , whenever Σ is finite. By gluing along open subspaces of finite

type, the same follows for arbitrary Σ.

(b) Both assertions are invariant under quotients by finite groups. Thus the firstassertion follows from 9.34, and the second follows from this and the compactness result6.27. q.e.d.

9.35. Remarks: (a) From now on we implicitly assume the existence of MKfC (P,X ,Σ)

as a scheme, whenever we write these symbols. Since by 9.28 and 9.33, it exists wheneverΣ is sufficiently fine or Kf sufficiently small, this will not be a real restriction in theapplications we have in mind. In any case, mutatis mutandum our assertions will also bevalid in the category of algebraic spaces, without this restriction.

(b) Counterexample: MKfC (P,X ,Σ) does not always exist. It is a nice exercise to realize

the example [H2] app.B ex 3.4.2 in the toroidal compactification for the mixed Shimura data(P,X ) = (P2g,X2g) of 2.25 with g = 1, or for (P,X ) = (SP4,Q,H4) of 2.7.

Finally, we consider the stratification of chapter 7.

9.36. Proposition: The stratifications in 7.2, 7.3, and 7.10 are algebraic, and theassertion of 7.17 (a) holds algebraically.

Proof. By [S] §19 prop. 15 the map in 7.17 (a) is algebraic, since both sides arecompact. This shows the second assertion. In particular this yields an algebraic morphism.

Stab∆1([σ])\Mπ[σ](K

1f )

C (P1,[σ],X1,[σ]) −→MKfC (P,X ,Σ),

which proves the other assertions whenever Σ is complete. By 9.33 the general case followsfrom this, since the assertions are invariant under replacing the open compact subgroup bya smaller one. q.e.d.

167

9.37. Proposition: The isomorphism of 7.17(b) induces a canonical isomorphism

between the formal completion of MKfC (P,X ,Σ) and of Stab∆1([σ])\M

K1f

C (P1,X1,Σ1) along

Stab∆1([σ])\Mπ[σ](K

1f )

C (P1,[σ],X1,[σ],Σ1,[σ]).

Proof. This is a special case of the following lemma. q.e.d.

9.38. Lemma: Let X be a proper algebraic variety over C, Y ⊂ X a closed subvariety,and XY the formal completion of X along Y . Let X1 be another algebraic variety over C,Y1 ⊂ X1 a closed subvariety, and X1,Y1 the formal completion of X1 along Y1. Suppose thatwe are given open analytic neighborhoods Y (C) ⊂ U ⊂ X(C) and Y1(C) ⊂ U1 ⊂ X1(C),and an analytic isomorphism U ∼= U1 that restricts to an isomorphism Y (C) ∼= Y1(C):

Y (C) ⊂ U ⊂ X(C)yo yoY1(C) ⊂ U1 ⊂ X1(C)

Then this isomorphism corresponds to a canonical isomorphism XY ∼= X1,Y1 :

Y ⊂ XYyo yoY1 ⊂ X1,Y1

Proof. The isomorphism U ∼= U1 induces an isomorphism between the nonreducedcomplex spaces (Y (C),OanX /(IanY )n) and (Y1(C),OanX1

/(IanY1)n). Since X is proper over C,

it follows that Y (C), and hence Y1(C) is compact. Thus by [S] §19 prop. 15 the ana-lytic isomorphism corresponds to a unique algebraic isomorphism between the infinitesimalneighborhoods (Y (OX/(IY )n) and (Y1,OX1/(JY1)n). By taking inverse limits the assertionfollows. (Compare also [H1] ch.VI ex. 2.9) q.e.d.

9.39. Examples: (a) As pointed out in 9.8, our sets Σ play the role of the Γ-polyhedral cocores in [AMRT] ch.II §5.3. In the case where P is reductive, the precisecorrespondence is the following. As explained in 8.15 (a), we have a canonical splittingof U ′1, whence a canonical decomposition C∗(X ′0, P ′1) ∼= R · σ0 × C∗(X 0, P1). Fix someu0 ∈ (σ0)0∩U ′(Q)(−1). The correspondence between Γ-polyhedral cocoresK ⊂ C∗(X 0, P1),and our sets Σ, can then be given by

K = u ∈ C∗(X 0, P1) | (−u0, u) ∈ Σ, and

Σ = convex closure of σ0 ∪ −u0 ×K.

(b) We include an example not subsumed by (a), which at the same time illustratesthe remark 8.15 (b). Let (P ′,X ′) be the mixed Shimura data (P2g,X2g) defined in 2.25

168

for g := 1, and (P,X ) := (P ′,X ′)/U ′. Let K ′f ⊂ P ′(Af ) be an open compact subgroup

whose image in (P ′/W ′)(Af ) ∼= GL2(Af ) is conjugate to GL2(Z). Then modulo the actionsof K ′f and of P ′(Q), there is precisely one conjugacy class of proper rational boundary

components. Describing it as in 4.27, we have U ′1∼= (Ga,Q)⊕3,

λ(C∗(X ′0, P ′1) ∼= (x, y, z) ∈ R3 | z > 0 or y = z = 0,

and Γ1 := StabQ′(Q)(X ′0) ∩ (K ′f · P ′1(Af ) acts on this through positive scalars and theunipotnet transformations

(x, y, z) 7→ (x+ 2ny + n2z, y + nz, z)

for all n ∈ Z.

A K ′f -admissible cone decomposition as in 9.13 now corresponds to a subset Σ ⊂C∗(X ′0, P ′1) as in 9.8 with respect to the above unipotent transformations. In terms of theabove identification, 9.1 shows that σ0 = R≥0 ·(1, 0, 0). Clearly Σ = σ0∪R>0 ·(x, y, 1) ∈ Σfor such Σ, so we can describe it in terms of its intersection with the plane z = 1. Thismust be the convex closure of Γ1 · (xi + R≥0, yi, 1) | 1 ≤ i ≤ n for some xi, yi ∈ Q. In thespecial case n = 1 and y1 = 0, we get

A little calculation shows that the associated splittings lie in E(X ′0, P ′1) if and only ifx1 < 0. In this case the assumptions of 9.13 are verified.

Coming back to 8.15 (b), we now see a geometric reason for the necessarity of quadraticterms in the action of Γ1: the strict convexity condition. At the same time, it prohibits theexistence of a splitting of the unipotnet extension (P ′,X ′)→ (P,X ).

169

Chapter 10

Moduli schemes of abelian varieties

In this chapter we describe an interpretion of some mixed Shimura varieties as modulischemes, together with the universal family, of abelian varieties or of similar algebraicobjects. Given this interpretation, we show that some of the maps 3.4 and 7.17 can beexpressed in modular terms: so that, in particular, they are defined over Q.

We first describe a well-known, suitable rigidified, moduli functor of polarized abelianvarieties with level structure. It is representable by a smooth scheme Md over Q (10.1–6).Certain mixed Shimura varieties associated to the mixed Shimura data (P2g,X2g) definedin 2.25 naturally carry the structure required by this functor (10.7–8). This yields isomor-phisms between these mixed Shimura varieties on one side, and Md,C, together with itsuniversal family, on the other side (10.9-10). We show that some of the maps of 3.4 (a) canbe described in terms of modular data (10.11-14).

In 10.15–16 we do the same for the mixed Shimura data (P0,X0) defined in 2.24. Thisis more or less a special case of (P2g,X2g), where the moduli functor degenerates to one forroots of unity.

The rest of the chapter deals with one special case of the above, but takes into accountthe toroidal compactification. We describe the Tate-curve (10.17–18), and the modulischeme for generalized elliptic curves with d-structure (10.19-20), in terms of toroidal com-pactifications of mixed Shimura varieties. The main result 10.22 is a modular interpretationfor the isomorphism 7.17/9.37.

10.1. Abelian schemes: Let S be a scheme over Q, and A → S an abelian scheme(see [M3] ch.6). We denote the group operation on A by (a, b) 7→ a+ b. For every positiveinteger d let [d] : A→ A be the morphism a 7→ d · a = a+ . . .+ a (d times). The subgroupscheme ker([d]) ⊂ A is finite and etale over S, since d is invertible over S. Let X → A bea Gm-torsor. For every abelian scheme T → S let AT be the abelian scheme A ×S T overT , and XT the Gm-torsor X ×S T over AT . For every section a ∈ A(T ) let Ta : AT → ATbe the morphism b 7→ b+ a. Consider the functor

T 7→ H(X)(T ) := a ∈ A(T ) | T ∗aXT∼= XT .

The condition T ∗aXT∼= XT is equivalent to the existence of an isomorphism f : XT → XT

170

of schemes over T such that the two diagrams

XTf−−→ XT Gm ×XT

id×f−−−−→ Gm ×XT

↓ ↓ and ↓ ↓AT

Ta−−−→ AT XTf−−→ XT

commute, where the vertical arrows in the diagram on the right hand side represent theaction of Gm. This functor is represented by a closed subgroup scheme H(X) ⊂ A, whichis smooth over S. If X is relatively ample with respect to A→ S, then H(X) is finite andetale over S.

10.2. The canonical pairing: Let a, b ∈ H(X)(T ) be sections and f resp. gcorresponding isomorphisms XT → XT . Then f g f−1 g−1 is an automorphism of theGm-torsor XT over AT , so there exists a unique section α ∈ Gm(T ) such that f gf−1g−1

is equal to the multiplication by α. Since f and g commute with the Gm-action, α dependsonly on a and b. thus the map

H(X)(T )×H(X)(T )→ Gm(T ), (a, b) 7→ e(X)(a, b) := α

is well-defined. This morphism of functors is represented by a unique morphism of schemese(X) : H(X) ×S H(X) → Gm,S . It is easily seen that this is an alternating bilinear form.If X is relatively ample with respect to A → S, then this is a perfect duality (see [M2]§1thm. 1).

Let A = A/Γ be an abelian variety over C, where A is the universal covering of A in thetopological sense, and E : Γ×Γ→ 2π

√−1 · Z the alternating form that correponds to the

Chern class of X (compare 3.18–19 ). Let Γ⊥ := γ ∈ Γ⊗T | ∀γ′ ∈ Γ E(γ, γ′) ∈ 2π√−1·Z,

then by [M1] §9 prop. p.84 (iii) we have H(X) = Γ⊥/Γ. The canonical pairing e(X) isaccording to [M1] §24 p.236 given by

(Γ⊥/Γ)× (Γ⊥/Γ)→ C×, ([γ], [γ′]) 7→ exp(E(γ, γ′)).

10.3. Level structure: Fix integers g ≥ 1 and d ≥ 1, and free Z/dZ-modules V [d] ofrank 2g and U [d] of rank 1. Fix also a perfect alternating paring Ψ[d] : V [d]×V [d]→ U [d].We consider abelian schemes A→ S of constant relative dimension g, together with a Gm-torsor X → A such that H(X) = A[d] for the above d. Consider the constant finite groupschemes V [d]× S and U [d]× S over S. A symplectic d-structure on X → A → S consistsof an isomorphism λ : V [d]× S → A[d] and a monomorphism µ : U [d]× S → Gm,S , suchthat the following diagram commutes:

(V [d]× S)×S (V [d]× S)Ψ[d]−−−−→ U [d]× S

λ×λ ↓ ∩↓ µ

A[d]×S A[d]e(X)−−−−→ Gm,S

Locally in the etale topology on S there always exists a symplectic d-structure.

171

10.4. Normalization of Gm-torsor: Let [−1] : A→ A be the inverse a 7→ −a, andi : A[2] → A the inclusion. The equation [−1] i = i yields a canonical i∗[−1]∗X ∼= i∗X.The Gm-torsor X is called totally symmetric if there exists an isomorphism [−1]∗X ∼= Xthat agrees with this canonical isomorphism on A[2]. In other words X is totally symmetricif there exists an isomorphism of schemes f : X → X such that the three diagrams

Xf−−→ X Gm ×X id×f−−−−→ Gm ×X X ×A A[2]

↓ ↓ ↓ ↓ pr1 pr1A

[−1]−−−−→ A Xf−−→ X X

f−−→ X

are commutative. If S is the spectrum of an algebraically closed field, the remark in [M2]§2 p.307 says that every totally symmetric Gm-torsor is uniquely determined by its Chernclass.

Let e : S → A be the zero section. A normalized Gm-torsor is a Gm-torsor X → Atogether with an isomorphism e∗X ∼= Gm,S , i.e. with a trivialization along the zero section.The trivialization is uniquely determined by its restriction to the “identity”-section in Gm,S .Thus it is equivalent to a morphism of schemes τ : S → X such that the diagram

Xτ ↓

S e−−→ A

commutes. In the following we shall be interested in normalized totally symmetric Gm-torsors (X, τ) on A → S. By the above remark such an object uniquely determined up toisomorphism by the relative Chern class of X.

10.4. A moduli problem: Let g, d, V [d], U [d] and Ψ[d] be as in 10.3. We supposethat d is even and at least 4. For every scheme S over Spec(Q) let

Md(S) :=

isomorphy classes of quintuples (A,X, τ, λ, µ), where:A→ S is an abelian scheme of relative dimension g,(X, τ) is a relatively ample normalized totally symmetric

Gm-torsor on A→ S such that H(X) = A[d], and(λ, µ) a symplectic d-structure on X → A→ S.

In the category of complex spaces we define Md(S) analogously. The following theorem iswell-known.

10.6. Theorem: The functor Md on the category of schemes is representable by asmooth quasiprojective scheme Md over Q.

Proof. The functor Ag,dg ,d of [M3] thm. 7.9 is representable by a quasiprojectivescheme over Spec(Z). Our functor Md is isomorphic to an open and closed subfunctorof Ag,dg ,d × Spec(Q), hence Md is representable by a quasiprojective scheme Md over Q.Since d ≥ 4 an object (A,X, τ, λ, µ) possesses no nontrivial automorphism, hence Md issmooth. q.e.d.

172

10.7. Construction of the moduli scheme as Shimura variety: Consider themixed Shimura data (P,X ) = (P2g,X2g) defined in 2.25. We shall describe a naturalstructure as in 10.5 on certain mixed Shimura varieties associated to (P2g,X2g).

As explained in 2.15, we may assume that W = U × V with the group operation(u, v) · (u′, v′) := (u + u′ + 1

2 · Ψ(v, v′), v + v′) and the G = CSP2g,Q-action g((u, v)) =(g(u), g(v)). Choose Z-structures on U and V such that Ψ induces a unimodular pairingV (Z)× V (Z)→ U(Z). Since G acts faithfully on V this also gives a Z-structure on G. Forevery even positive integer d we define the open compact subgroups

Kf = Kf (d) := g ∈ G(Z) | g ≡ 1 mod d,

KWf = KW

f (d) := (d · U(Z))× (d · V (Z)).

Since d is even, the definition of the group operation on W implies that KWf is indeed a

subgroup of W (Af ). Clearly it is normalized by Kf . Thus KPf := KP

f (d) := KWf oKf is an

open compact subgroup of P (Af ). ¿From now on we assume d ≥ 4. Let KUf := U(Af )∩KW

f

and KVf := KW

f /KUf , and write

M = M(d) := MKfC (CSP2g,Q,H2g),

MV = MV (d) := MKVf oKf

C (V o CSP2g,Q,H2g), and

MW= MW (d):= MKPf

C (P2g,X2g),

(see 9.25). The projections

(P2g,X2g) −→ (V2g o CSP2g,Q,H′2g) −→ (CSP2g,Q,H2g).

induces holomorphic maps MW →MV →M .

By 3.14, MV → M is a family of abelian varieties of relative dimension g. Rememberfrom 2.25 the structure on (P2g,X2g)→ (V2g oCSP2g,Q,H′2g) as a ((P0,X0)→ (Gm,Q,H0))-torsor. The image of Kf in Gm(Af ) is

K∗f = K∗f (d) := t ∈ Gm(Af ) | t ≡ 1 mod d,

so by 3.12 (b), MW → MV is a torsor under the group scheme MKUf oK∗f

C (P0,X0) →M

K∗fC (Gm,Q,H0). As in 3.17, under the isomorphism

MKUf oK∗f

C (P0,X0) ∼−−→ Gm ×MK∗fC (Gm,Q,H0)

defined in 3.16, this turns MW → MV into a Gm-torsor. By 3.21 its inverse is relativelyample with respect to MV → M . From now on we replace this Gm-action by its twistunder the automorphism Gm → Gm, z 7→ z−1. This corresponds to replacing the Gm-torsor MW →MV by its inverse, but as a scheme the “new” MW is equal to the “old” one.With this new structure, it is relatively ample.

Let e : (CSP2g,Q,H2g) → (V2g o CSP2g,Q,H′2g) and τ : (CSP2g,Q,H2g) → (P2g,X2g)be the morphism that corresponds to the splitting CSP2g,Q,H2g → V2g o CSP2g,Q and

173

CSP2g,Q,H2g → P2g. They induce morphisms [e] and [τ ] such that the following diagramcommutes.

M

[τ ] ↓ [e] id

MW[π′]−−−→ MV

[π′′]−−−−→ M

Since [e] is the zero section, this defines a normalization of the Gm-torsor MW →MV .

Define V [d] := V (Z)/d · V (Z) ∼= V (Z)/KVf , and U [d] := U(Z)/d · U(Z) ∼= U(Z)/KU

f .Since Ψ : V (Z) × V (Z) → U(Z) is unimodular, it induces a perfect alternating pairingΨ[d] : V [d]× V [d]→ U [d]. The group V (Z) normalizes KV

f oKf in V (Af ) oG(Af ). Thus

3.15 gives an isomorphism λ : V [d]×M →MV [d] of group schemes over M . Likewise U(Z)

normalizes KUf oK∗f in P0(Af ), so 3.15 yields a monomorphism U [d]×M

K∗fC (Gm,Q,H0) →

MKUf oK∗f

C (P0,X0). Under the above identifications we get a monomorphism µ : U [d]×M →Gm ×M .

10.8. Proposition: The quintuple (MW ,MV , [τ ], λ, µ) is an element of Md(M), andhence corresponds to a unique morphism M →Md,C.

Proof. We have to show:

(a) MW →MV is totally symmetric.

(b) H(MW ) = MV [d].

(c) The pair (λ, µ) is a symplectic d-structure on MW →MV →M .

It suffices to verify these assertions fibrewise and in the category of complex spaces. Fixa point [(y, gf )] ∈M(C) = MKf (CSP2g,Q,H2g)(C). Since CSP2g(Af )G(Af ) = G(Q) ·G(Z),we may assume that (y, gf ) ∈ H2g×G(Z). Let ΓV := d·V (Z) and ΓW := (d·U(Z))×(d·V (Z)),then the fibres of MW and MV over [(y, gf )] are isomorphic to X := ΓW \W (R) · U(C),respectively A := ΓV \V (R).

For (a) we consider the homomorphism W → W, (u, v) 7→ (u,−v). It maps KWf to

itself and commutes with the action of G, so it induces an automorphism ı of the Shimuradata (P2g,X2g), and hence an automorphism [ı] of MW . It operates in X by the sameformula (u, v) 7→ (u,−v), and it commutes with the action of U(C) on X and thereforewith its structure as C×-torsor. Every point in A[2] is of the form [v] fur v ∈ d

2 · V (Z). Forsuch v we have (0, 2v) ∈ ΓW , hence for any u ∈ U(C)

(0, 2v) · (u,−v) = (u+1

2·Ψ(2v,−v), 2v − v) = (u, v),

since Ψ(v, v) = 0. This shows [ı]([(u, v)]) = [(u,−v)] = [(u, v)], so [ı] is equal to the identifyover A[2]. By 10.4 this proves (a).

Since we have inverted the Gm-torsor MW →MV , 3.19 implies that the Chern class ofX → A now corresponds to the alternating pairing +Ψ : ΓV × ΓV → ΓU := U(Q) ∩ ΓW .Via the isomorphism 3.16, we get the equation E = 1

d · λψ(y) Ψ. It is verfied at once that

(ΓV )⊥ = V (Z), whence K(X) = V (Z)/ΓV = A[d], which implies (b).

174

By 10.2 this also shows that e(X) : A[d]×A[d]→ C× is given by

([v], [v′]) 7→ exp(1

d· λψ(y) Ψ(v, v′)).

By 3.15 λ is given by

V [d] ∼= V (Z)/d · V (Z) → U(Z)/d · V (Z) ∼= V (Z)/d · V (Z) = A[d],[vf ] 7→ [gf (vf )],

and by 3.15 and 3.16, µ is given by

U [d] = U(Z)/d · U(Z) → U(Z)/d · U(Z) = U(Z)/d · U(Z) → C×,[uf ] 7→ [−gf (uf )] = [u] 7→ exp(−1

d · λψ(y)(u)).

The commutativity of the diagram 10.3 is therefore equivalent to the identify

Ψ(−gf (vf ),−gf (v′f )) ≡ Ψ(vf , v′f ) mod d · U(Z).

This congruence follows from the bilinearity and the G-equivariance of the pairing Ψ. Thisproves (c). q.e.d.

10.9. Proposition: The morphism M →Md,C of 10.8 is an isomorphism.

Proof. Both are smooth varieties over C, so it suffices to prove the bijectivity ontheir C-valued points. Consider (A,X, τ, λ, µ) ∈ Md(C). Write A = A/Γ, where A is theuniversal covering of A, and let E : Γ × Γ → 2π

√−1 · Z be the pairing associated to X.

Let λ0 be an isomorphism Z → 2π√−1 · Z. Then Γ⊥ = 1

d · Γ is in Γ ⊗ Q, and the pairingΓ⊥ × Γ⊥ → Z induced by d2 · λ−1

0 E is unimodular. Thus there exists an isomorphismφ : Γ→ ΓV such that this pairing corresponds to the pairing Ψ : V (Z)×V (Z)→ U(Z) = Z.The complex structure on A ∼= V (R)/ΓV then corresponds to a unique action of S on V (R).By [M1]§2 cor. p.18 we have E(

√−1 ·v,

√−1 ·v′) = E(v, v′) for all v, v′ ∈ V (R), hence this

action factors through a unique homomorphism h : S → GR. Since X is ample, −λ0 Ψis a polarization of the Hodge structure on V defined by h, hence h ∈ H2g. The givenisomorphism λ : V [d]→ A[d] = V [d] is symplectic, so there exists gf ∈ G(Z) such that λ isinduced by conjugation with gf . We have already chosen an isomorphism between A and thefibre of MV →M over the point [(h, gf )]. The Chern classes of the two normalized totallysymmetric C×-torsors correspond to each other, so by 10.4 they are isomorphic. Since µis uniquely determined by λ and the commutativity of the diagram 10.3, the d-structuresalso correspond. Thus (A,X, τ, λ, µ) is in the image of the morphism M →Md,C, and thesurjectivity is proved.

Every other isomorphism Γ→ ΓV , such that E corresponds to the pairing Ψ : V (Z)×V (Z) → U(Z) = Z, must be the form int(g) φ for some g ∈ G(Z). The homomorphism hmust then be replaced by int(g)h and gf by g ·gf modKf . But [(int(g)h, g ·gf )] = [(h, gf )]in M , this proves the injectivity. q.e.d.

10.10. Corollary: Let Xd → Ad → Md be the universal family, these are smoothquasiprojective schemes over Q. The data in 10.8 determines isomorphisms MW (d) →Xd,C, MV (d)→ Ad,C and M(d)→Md,C, compatible with all described structures.

175

Proof. This follows from 10.9 and the universal property of Md. q.e.d.

We now study the relation of the modular interpretation of MW with some of the mapsof 3.4 (a).

10.11. Proposition: Let d ≥ 4 be an even integer. Any element wf ∈ KWf (2)

normalizes KPf (d), and the automorphism of the scheme Xd,C corresponding to [ ·wf ] :

MW →MW can be described uniquely in terms of modular data.

Proof. Let (uf , vf ) = wf , and consider w′f = (u′f , v′f ) ∈ KW

f (d). We have [wf , w′f ] =

Ψ(vf , v′f ) ∈ U(Af ), and since vf ∈ 2 · V (Z) and v′f ∈ d · V (Z), it follows that Ψ(vf , v

′f ) ∈

2d · U(Z) ⊂ KUf (d). Thus wf normalizes KW

f (d). Next let gf ∈ Kf (d), then

[wf , gf ] = wf · gf (wf )−1 = (uf , vf ) · (gf (uf ), gf (vf ))−1

= (uf , vf ) · (−gf (uf ),−gf (vf ))

= (uf − gf (uf )− 1

2·Ψ(vf , gf (vf )), vf − gf (vf )).

Since gf ≡ 1(mod d) and vf ∈ 2 · V (Z), we have vf − gf (vf ) ∈ 2d · V (Z) and uf − gf (uf ) ∈2d · U(Z). Since Ψ is alternating, we have

1

2·Ψ(vf , gf (vf )) =

1

2·Ψ(vf , gf (vf )− vf ) ∈ 1

2· 2 · 2d · U(Z) ⊂ d · U(Z).

Thus [wf , gf ] ∈ KWf (d), which proves that wf normalizes KP

f (d).

For the second assertion we write wf = (uf , 0) · (0, 2vf ) with uf ∈ 2 · U(Z) and vf ∈V (Z). If vf = 0, then [ ·wf ] operates on the C×-torsor MW → MV by multiplication byµ(−[uf ]) ∈ C×, so this map is given in modular terms. From now on we assume uf = 0.

As in the proof of 10.8 consider a fibre X := ΓW \W (R)·U(C) of MW →M over [(y, gf )],where gf ∈ G(Z). By 3.15 [ ·wf ] operates on X by [w′] 7→ [w−1 ·w′] for any w ∈W (Q) suchthat g−1

f (w) ≡ wf modKWf . Let v ∈ V (Z) such that g−1

f (v) ≡ vf mod d · V (Z), then wecan for instance choose w = (0, 2v). The isomorphism [ ·wf ] then induces the translation[v′] 7→ [v′ − 2v] on A := ΓV \V (R), and since U is in the center of W , it commutes with theaction of C×. Now [v] ∈ H(X), and every holomorphic isomorphism of C×-torsors X → Xover the translation T[−v] is of the form f : [w′] 7→ [(u, v)−1 · w′] for some u ∈ U(C).The isomorphism [ı] : X → X, [(u′, v′)] 7→ [(u′,−v′)] in the proof of 10.8 is the uniqueisomorphism X → X over the inversion map on A, which induces the identity map overthe zero element of A. For arbitrary u ∈ U(C) we have

[ı]((u, v)) · (u, v)−1 = (u,−v) · (−u,−v) = (0,−2v) = (0, 2v)−1.

Therefore [ı] f−1 [ı] f = [ ·wf ] on X, and this isomorphism does not depend on thechoice of u. This shows that [ ·wf ] is uniquely determined by the modular data (compare[M2] §2 p.319 remark 3). q.e.d.

10.12. Lemma: Let gf ∈ G(Af ). Then for every sufficiently divisible even integerd ≥ 4 there exists an even integer d′ ≥ 4 such that

KPf (d′) ⊂ gf ·KP

f (d) · g−1f ⊂ K

Pf (2).

176

Proof. This follows immediately from the fact that the KPf (d) are cofinal in the system

of all open compact subgroups of P (Af ). q.e.d.

10.13. Proposition: Let gf , d and d′ as in 10.12, and define d′′ := d · d′. Themorphism Xd′′,C → Xd,C corresponding to the map [ ·gf ] : MW (d′′) → MW (d) can bedescribed uniquely in terms of modular data.

Proof. By the inclusions KPf (d′′) ⊂ KP

f (d′) ⊂ gf · KPf (d) · g−1

f the map [ ·gf ] :MW (d′′) → MW (d) is defined. The fibre over a point [(y, g′f )] ∈ M(d′′)(C) is mapped

to the fibre over [(y, g′f · gf )] ∈ M(d)(C). Let Γ′′W := W (Q) ∩ g′f (KWf (d′′)) and ΓW :=

W (Q)∩ (g′f · gf )(KWf (d)), then with the identification of 3.13, [ ·gf ] is on these fibres given

byX ′′ := Γ′′W \W (R) · U(C) −→ X := ΓW \W (R) · U(C), [w] 7→ [w].

Consider the isomorphism

gf (KWf (d))/KW

f (d′′)g′f ( )∼−−→(g′f · gf )(KW

f (d))/g′f (KWf (d′′)) ∼= ΓW /Γ

′′W .

Let [wf ] ∈ gf (KWf (d))/KW

F (d′′), and [w] its image in ΓW /Γ′′W . Since by assumption

gf (KWf (d)) ⊂ KW

f (2), it follows, as in the proof of 10.11, that [ ·wf ] is on X ′′ given

by ([w′] 7→ [w−1 ·w′]). Hence [ ·gf ] induces an isomorphism X ′′/gf (KWf (d))→ X. By 10.11

this quotient is uniquely determined by the modular data on X ′′. It remains to show thatthe other structures on X are determined by those X ′′.

According to 3.16 we have the identification

(U(Q) ∩ Γ′′W )\U(C)→ C×, [u] 7→ exp(1

d′′· λψ(y)(u)),

and since we have inverted the C×-torsor MW →MV , this group operates on X ′′ by [w] 7→[u−1 · w]. The analogous formula holds for X, and the identification lie in a commutaivediagram

(U(Q) ∩ Γ′′W )\U(C) ∼−−→ C×

[u] 7→ [u] ↓ ↓ z 7→zd′

(U(Q) ∩ ΓW )\U(C) ∼−−→ C×,

which is made such that the projection X ′′ → X is equivariant under the two actions.Hence the C×-action on X comes from that on X ′′. Since gf ∈ G(Af ), the map [ ·gf ]commutes with the section [τ ], so the normalization of the C×-torsor X ′′ induces that ofX. Let ΓV and Γ′′V be the images of ΓW and Γ′′W in V (Q), and let A := ΓV \V (R) andA′′ := Γ′′V \V (R). It remains to show that the symplectic d-structure on A comes from thesymplectic d′′-structure on A′′.

The symplectic d′′-structure on A′′ is uniquely determined by the isomorphism λ :V [d′′]→ A′′[d′′]. Under the canonical identification

A′′[d′′] = (1

d′′· Γ′′V )/Γ′′V

∼= g′f (V (Z))/g′f (d′′ · V (Z))

177

this isomorphism is by 3.15 of the form

V [d′′] = V (Z)/d′′ · V (Z)→ g′f (V (Z))/g′f (d′′ · V (Z)), [vf ] 7→ [−g′f (vf )].

The analogous formula hold for A. Since d′′ = d · d′, and by assumption d′ · V (Z) ⊂gf (d · V (Z)), the subgroup 1

d · Γ′′V is already contained in ΓV . Thus the isogeny A′′ → A

factors through the endomorphism [d] : A′′ → A′′. So let us consider the new isogeny

A′′ = Γ′′V \V (R)→ A = ΓV \V (R), [v] 7→ [1

d· v].

The inverse image of A[d] under this map is ΓV /Γ′′V = (g′f · gf )(d · V (Z))/g′f (d′′ · V (Z)).

Since by assumption gf (d · V (Z)) ⊂ 2 · V (Z), we have (g′f · gf )(d · V (Z)) ⊂ g′f (V (Z)), sothis inverse image lies in A′′[d′′]. For the d′′-structure on A′′ and the d-structure on A wetherefore have the commutative diagram

[d · gf (vf )]∈V [d′′] −→ A′′[d′′]→A′′3[v]↑ ↑ ↓ ↓v[f ]∈V (Z) id−−→ V [d] −→A[d] →A 3[1

d · v].

This implies the assertion. q.e.d.

10.14. Corollary: The automorphism of Xd,C in 10.11, and the morphism Xd′′,C →Xd,C in 10.13, both descend to morphisms of schemes over Q.

Proof. By 10.11, 10.13, and the universal property of Xd →Md. q.e.d.

Remark. P2g(Af ) is generated by KWf (2) together with CSP2g(Af ). Thus, for our

purposes, the corollary holds for sufficiently many maps.

We now want to obtain the same results for the mixed Shimura data (P0,X0) definedin 2.24. This is, mutatis mutandum, a special case of the above.

10.15. A moduli scheme for roots of unity: For any positive integer d, letµd,Q ⊂ Gm,Q be the kernel of the homomorphism t → td. Let M0

d ⊂ Gm,Q be the reducedclosed subscheme µd,Q −

⋃d′|d,d′<d µd′,Q of all primitive dth roots of unity. For any scheme

S over Q, it is equivalent to give an S-valued point on M0d, or to give an isomorphism

(Z/d · Z)× S ∼= µd,S . In other words, M0d represents the functor

Isom((Z/d · Z)× Spec(Q), µd,Q).

Consider the mixed Shimura data (P0,X0) defined in 2.24. Let

KUf (d) := d · Z ⊂ Af = U0(Af ) and

Kf (d) := tf ∈ Gm(Z) | tf ≡ 1 mod d,

then KPf (d) := KU

f (d) oKf (d) is an open compact subgroup of P0(Af ). Every KPf (d) is a

normal subgroup of KPf (1). Put

M0(d) := MKf (d)C (Gm,Q,H0) and

M0U (d) := M

KPf (d)

C (P0,X0).

178

The canonical isomorphism M0U (d)(C) ∼= C× ×M0(d)(C) defined in 3.6 gives an isomor-

phism M0U (d) ∼−−→ Gm ×M0(d). Since KU

f (1)/KUf (d) = Z/d · Z ∼= Z/d · Z, 3.15 defines a

monomorphism (Z/d · Z)×M0(d) →M0U (d) ∼−−→ Gm ×M0(d), whence an isomorphism

(Z/d · Z)×M0(d) ∼−−→ µd ×M0(d).

By the universal property of M0d, this defines a morphism M0(d)→M0

d,C. Defining X0d :=

Gm,Q ×M0d, we in turn get a morphism M0

U (d)→ X0d,C.

10.16. Proposition: (a) The morphisms M0(d) → M0d,C and M0

U (d) → X0d,C are

isomorphisms.

(b) Let pf ∈ KUf (1) ∪ Gm(Af ), and d′, d such that the map [ ·pf ] : M0

U (d′) → M0U (d) of

3.4 (a) is defined. Then with the identifications in (a) this map corresponds to a morphismX0d′ → X0

d over Q.

Proof. (a) Let e : (Gm,Q,H0) → (P0,X0) be the given section, then the isomorphismM0U (d)(C) ∼−−→ C× ×M0(d)(C) of 3.16 is given by

[(u · e(y), e(tf ))] 7→ (exp(1

d· λy(u)), [(y, tf )])

for u ∈ U(C) and t ∈ H0, and tf ∈ Gm(Z). Using 3.15 it follows that the isomorphism(Z/d · Z)×M0(d) ∼−−→ µd ×M0(d) is given by

([u], [(y, tf )]) 7→ (exp(1

d· λy(u))tf , [(y, tf )])

for [u] ∈ Z/d · Z, y ∈ H0, and tf ∈ Gm(Z). Clearly every isomorphism Z/d · Z ∼= µd(C)occurs in precisely on fibre over M0(d)(C), so the map M0(d)(C) → M0

d(C) is a bijection.Since both are normal reduced schemes, M0(d) → M0

d,C must be an isomorphism. Thisimplies (a).

(b) If pf = uf ∈ Z = KUf (1), then by the definition 3.15 the map [ ·uf ] : M0

U (d) →M0U (d) corresponds to the automorphism (t, ζ) 7→ (t·ζuf , ζ) of Gm,Q×M0

d. Next if pf = e(tf )with tf ∈ Gm(Z), then the formulas in (a) show that the map [ ·e(tf )] : M0

U (d) → M0U (d)

corresponds to the automorphism (t, ζ) 7→ (t, ζtf ) of Gm,Q×M0d. It remains to consider the

case pf = e(t) with t ∈ Q>0 ⊂ Gm(Q). Since e(t) ·KPf (d) · e(t)−1 = KP

f (d)(t · d), d′ must be

a multiple of t · d. On C× ×M0(d)(C) the map [ ·e(t)] : M0U (d′)→M0

U (d) is given by

(z, [(y, tf )])=[(λ−1y (d′ · log(z)) · e(y), e(tf ))]

7→ [(λ−1y (d′ · log(z)) · e(y), e(tf ) · e(t))]

= [(e(t)−1 · λ−1y (d′ · log(z)) · e(y), e(tf ))]

= [(t−1(λ−1y (d′ · log(z))) · e(y), e(tf ))]

=(exp(1

d· λy(

1

t· λ−1

y (d′ · log(z)))), [(y, tf )])

= (zd′′, [(y, tf )]),

179

where d′′ = d′

d·t ∈ Z. Thus it corresponds to the morphism

Gm,Q ×M0d′ → Gm,Q ×M0

d, (t, ζ) 7→ (td′′, ζd

′/d).

In each of the three cases the morphism X0d′ → X0

d is defined over Q, as desired. q.e.d.

For the rest of this chapter we shall do something similar for the compactification of amixed Shimura variety. In a very special case, we shall give a modular interpretation forthe isomorphism 7.17. The following construction is well-known.

10.17. The Tate curve as a torus embedding: Consider the projection pr1 :G2m,Q → Gm,Q, we may consider this as the (relative) commutative group scheme obtained

from Gm,Q → Spec(Q) by base change ( ) ×Q Gm,Q. The torus embedding associated toσ := R≥0 ⊂ R = Y∗(Gm,Q)R is just Gm,Q → A1

Q. Let T be the partial cone decomposition ofR2 = Y∗(G2

m,Q)R that consists of the cones R≥0 · (1, n) + R≥0 · (1, n + 1) for all n ∈ Z, andtheir faces; as illustrated in the diagram:

Let T r ⊂ T be the subset of all cones of dimension ≤ 1. Write Z := (G2m,Q)T and

Zr := (G2m,Q)T r

By the criterion 5.4 for the functoriality of torus embeddings, the projection G2m,Q →

Gm,Q extends to a morphism Z → A1Q. The inverse image of Gm,Q ⊂ A1

Q is just G2m,Q, but

the fibre over 0 is a infinite sequence of projective lines, with the point 0 of each componentglued to the point ∞ of the next one. It is easily checked that the morphism Z → A1

Q isflat. The open subscheme, on which it is smooth, is just Zr. The structure of commutativegroup scheme on pr1 : G2

m,Q → Gm,Q is given by the identity section, the inversion, and thegroup operation. These are

Gm,Q → G2m,Q, s 7→ (s, 1),

G2m,Q → G2

m,Q, (s, t) 7→ (s, t−1),G2m,Q ×Gm,Q G2

m,Q∼= G3

m,Q → G2m,Q, (s, t1, t2) 7→ (s, t1 · t2)

respectively. We claim that they extend to morphisms

A1Q → Zr, Zr → Zr, and Zr ×A1

QZr → Zr and Zr ×A1

QZ → Z

respectively. In fact, for the first two morphisms this is a direct application of 5.4. It iseasily checked that the embedding

G3m,Q∼= G2

m,Q ×Gm,Q G2m,Q → Zr ×A1

QZ

is the torus embedding with respect to the partial cone decomposition of R3, consisting ofthe cones

R≥0 · (1,m, n) + R≥0 · (1,m, n+ 1)

180

for all m,n ∈ Z, and their faces. Since such a cone maps to the cone

R≥0 · (1,m, n) + R≥0 · (1,m+ n+ 1) ∈ T ,

by functoriality we get a morphism Zr ×A1QZ → Z. The same argument applies to Zr ×A1

Q

Zr → Zr. Since G2m,Q is dense in Zr, the group axioms for pr1 : G2

m,Q → Gm,Q extend, soZr → A1

Q is a commutative group scheme. Likewise, the morphism Zr ×A1QZ → Z defines

a group action. (Zr → A1Q is the Neron-model of Gm, but we will not need this fact.)

Consider the action of a ∈ Z on G2m,Q through (s, t) 7→ (s, sa · t). Clearly T is invariant

under these substitutions, and Z cyclically and transitively permutes the irreducible com-ponents of the special fibre. The identity component of the special fibre is again Gm,Q, sowe get a canonical group isomorphism between the special fibre of Zr → A1

Q and Z×Gm,Q.

Let Z be the formal completion of the special fibre in Z. For any positive integer dwe can form the quotient Z/d · Z. This is formally proper and flat of relative dimension 1over the completion of A1

Q in 0, so the formal scheme is algebraizable, yielding a properflat scheme E → SpecQ[[s]]. Letting Er ⊂ E be the open subscheme which is smooth overSpecQ[[s]], we obtain the structure of a commutative group scheme on Er → SpecQ[[s]],and the action Er ×SpecQ[[s]] E → E. This is a generalized elliptic curve in the sense of[DR] II.1.12. It is easy to show that E → SpecQ[[s]] is the universal deformation of theclosed fibre as a generalized elliptic curve (compare [DR] III.1.2).

The kernel of the multiplication by d on Z/d · Z is the formal scheme associated to(Z/d · Z)× µd, so Er[d] is canonically isomorphic to

(Z/d · Z)× µd × SpecQ[[s]]→ SpecQ[[s]].

A d-structure on E → SpecQ[[s]], in the sense of [DR] IV.2.3, is an isomorphism (Z/d ·Z)2 ∼= Er[d]. Thus giving a d-structure is in our case equivalent to giving an isomorphismZ/d · Z ∼= µd. In particular, if M0

d is as in 10.15, there is a canonical d-structure onE ×M0

d → SpecQ[[s]] ×M0d. Again, this is the universal deformation of the closed fibre

as a generalized elliptic curve with d-structure.

10.18. The Tate curve in terms of mixed Shimura varieties: Consider themixed Shimura data (P0,X0) defined as in 2.24. Let (P0, X0) be the fibre product of (P0,X0)with itself over (Gm,Q,H0); we have canonically P0

∼= (U0 ×U0) oGm,Q. Consider the opencompact subgroups

KU0f := d · Z ⊂ Af = U0(Af ),

KGmf := t ∈ Z× | tf ≡ 1 mod d,

KP0f := KU0

f oKGmf ⊂ P0(Af ), and

KP0f := (KU0

f ×KU0f ) oKGm

f ⊂ P0(Af ).

With the identification of 10.15 we have canonical isomorphisms

M0U := M

KP0f

C (P0,X0) ∼= Gm ×M0d,C,

M0U := M

KP0f

C (P0, X0) ∼= G2m ×M0

d,C.

181

By 6.8–9, the partial cone decompositions T r ⊂ T of 10.17 correspond to unique KP0f -

admissible partial cone decompositions Σr0 ⊂ Σ0 for (P0, X0). Thus if Σ0 denotes the cone

decomposition for (P0,X0) considered in 6.9, we get canonical isomorphisms

M0,r

U := MKP0f

C (P0, X0,Σr0) ∼= Zr ×M0

d,C

∩ ∩ ∩

M0

U := MKP0f

C (P0, X0,Σ0) ∼= Z ×M0d,C

↓ ↓ ↓

M0U := M

KP0f

C (P0,X0,Σ0) ∼= A1 ×M0d,C.

Recall (2.21,3.12 (a)) that the group structure on M0U →M0

U is given in terms of maps3.4 (b). We can directly translate 10.17: the condition of 6.25 (b) holds for each of the

necessary maps. The action of Z on Z ×Md,C corresponds to the action on M0

U induced

by the maps [φa], where φa is the automorphism of (P0, X0) given by

P0 = (U0 × U0) o Gm,Q 3 (u1, u2, t) 7→ (u1, a · u1 + u2, t).

By 10.15 the canonical isomorphism Z/d · Z ∼= µd over Md,C can be expressed in terms of

maps 3.4 (a). Thus 10.17 shows that, over the formal completion of M0U along the boundary

M0UrM0

U , we have a canonical generalized elliptic curve with d-structure, and this structurecan be defined purely in terms of maps 6.25.

We now “glue” the objects just constructed into the boundary of other objects.

10.19. A certain toroidal compactification: Consider the Shimura data (GL2Q,H2)defined in 2.7, and its unipotent extention (P,X ) := (V2×GL2Q,H′2), defined in 2.25. HereV2 is just the standard 2-dimensional representation of GL2Q. For every positive integer dwe let

Kf = Kf (d) := g ∈ GL2(Z) | g ≡ 1 mod d,KVf = KV

f (d) := d · Z2, and

KPf = KP

f (d) := KVf oKf ,

and defineM := M

KfC (GL2,Q,H2),

MV := MKPf

C (V2 o GL2,Q,H′2).

As explained in 4.15, we identify the mixed Shimura data (P0,X0) with a rational boundarycomponent of (GL2,QH2). Every proper rational boundary component is GL2(Q)-conjugateto this one. The rational boundry component is (V2 o GL2,Q,H′2) associated to (P0,X0) isisomorphic to (P0, X0). Embedding all our groups into GL3,Q we have

P ∼=

∗ ∗ ∗∗ ∗ ∗0 0 1

, P0∼=

∗ ∗ ∗0 1 00 0 1

, Q ∼=

∗ ∗ ∗∗ ∗ ∗0 0 1

,

182

where Q is the normalizer of P0. We identify the unipotent radical of P0 with G2a,Q by

(x, y)←→

1 x y0 1 00 0 1

.

The cone decomposition of 10.18 correspond canonically to KPf (1)-admissible cone

decompositions for (P,X ). In fact, since P (Af ) = Q(Q) · KPf (1), there are precisely two

conjugacy classes of (P1, pf ) modulo the actions 6.4 (ii) and (iii). One of these correspondsto P1 = P , for which there is nothing to specify. Since the other is represented by (P0, 1), anycone decomposition Σ for (P,X ) is determined by Σ|(P0,X0). By 4.26, the cone C∗(X 0, P0)

corresponds to 0∪R>0×R under the above identification. Moreover, Γ := StabQ(Q)(X 0)∩P0(Af ) ·KP

f (1) acts on this through

(x, y) 7→ b · (x,±y + ax) for a ∈ Z and b ∈ Q>0.

Thus there are unique KPf (1)-admissible partial cone decompositions Σr ⊂ Σ for (P,X )

with Σr|(P0,X0)

= Σr0 and Σ|(P0,X0) = Σ0, and the latter is complete. There exists a unique

complete Kf (1)-admissible cone decomposition for (GL2,QH2), which we denote by Σ. Ofcourse, all these cone decompositions remain admissible for KP

f = K0f (d), resp. Kf =

Kf (d). We define

MV := MKPf

C (V2 o GL2,Q,H′2,Σ),

MrV := M

KPf

C (V2 o GL2,Q,H′2,Σr), and

M := MKfC (GL2,Q,H2,Σ) = M

KfC (GL2,Q,H2)∗.

(By 9.39 (b) we know that MV is a projective variety for every d, but we shall not needthis fact.)

10.20. The moduli scheme of generalized elliptic curves in terms of mixedShimura varieties: ¿From now on we assume that d ≥ 3. Then the fibres of MV → M

are elliptic curves. Consider the boundary stratum MKGmf

C (Gm,Q,H0) of M , associated to(P0,X0) and pf = 1 (see 6.3). This is canonically identified with the unique proper boundary

stratum of M0U , and 9.37 yields an isomorphim between its formal neighborhoods in M and

M0U respectively. Likewise, for the rational boundary component (P0, X0) of (P,X ), and

pf = 1, the group ∆1 of 7.3 acts on M0

U just like the group d ·Z in 10.18. Thus, by 7.17, the

inverse image of MKGmf

C (Gm,Q,H0) of MV →M is isomorphic to (M0

U rM0U )/d ·Z, and 9.37

yields an isomorphim between its formal neighborhood in MV and the quotient by d · Z of

the formal neighborhood of M0

U r M0U in M

0

U .

Recall (10.7–8) that the structure of MV →M as an elliptic curve with d-structure canbe defined purely in terms of the maps 3.4. As in 10.18, we find that this structure extendsand makes MV →M into a generalized elliptic curve with d-structure.

183

Now letMd be the moduli scheme of generalized elliptic curves over Q with d-structure.By [DR] IV.2.9, it is smooth, projective, of dimension 1 over Q. Consider the moduli schemeMd in 10.6 for g = 1. One easily checks that, for g = 1, the data (A,X, τ, λ, µ) in 10.5is determined up to unique isomorphism by (A, λ). Thus the forgetful functor induces anopen embedding M →Md, and the image is Zariski-dense. By the universal property ofMd, the structures defined above induce a morphism M →Md,C. This morphism extends

the isomorphism M ∼−−→Md,C of 10.9. It is easy to see that M →Md,C is an isomorphism

(for instance because M is also proper and smooth). If Ed →Md is the universal family,then this also induces an isomorphism MV

∼= Ed,C.

10.21 Summary of 10.17–20: To tie everthing up, consider the generalized ellipticcurve with d-structure, constructed in 10.17. By the universal property ofMd, it defines amorphism SpecQ[[s]]×M0

d →Md. Since the former is a universal deformation, this givesan isomorphism of the completion of Md along M0

d →Md.

In summary, the different objects and morphisms we have constructed form the follow-ing diagram.

Here FC() denotes formal completion along a closed subscheme that was specified above. Alldiagonal arrows are isomorphisms. These, and the arrows in the foreground, are all definedin terms of modular data, and the latter descend to Q. The arrows in the backgroundhave been defined in terms of maps 6.25. By the universal property of the moduli schemesinvolved, the whole diagram is commutative! We have thus proved the following proposition.

10.22 Proposition: With the identification M ∼=Md,C of 10.20, and M0U∼= A1×M0

d,C

184

of 10.18, the isomorphism 9.37 of the formal completions of M and M0U corresponds to an

isomorphism of the formal completions of Md and A1 ×M0d,C, that is defined over Q.

185

Chapter 11

Canonical models

We introduce the concept of canonical models, and prove the existence and uniqueness ofcanonical models for all mixed Shimura varieties. The existence is proved by reduction tothe case of usual Shimura varieties, where it is known by [Mi1] thm. 7.2. A central role isplayed by the modular interpretation of certain mixed Shimura varieties, and by the maintheorem of Shimura and Taniyama about abelian varieties with complex multiplication. Inthe whole chapter, we closely follow [D1].

After defining canonical models (11.1–5), we study their functorial behavoir. In 11.9we show that if a canonical model exists for some mixed Shimura data, then this inducesa canonical model on any embedded mixed Shimura variety. This implies functoriality(11.10), and in particular uniqueness (11.11) of canonical models. We prove the existenceof canonical models for those mixed Shimura varieties, whose modular interpretation wasstudied in the previous chapter (11.14–16). The main theorem (11.18) on the existence ofcanonical models in general is then reduced to [Mi1]thm. 7.2.

11.1. The reflex field: Let (P,X ) be mixed Shimura data. For every x ∈ X considerthe homomorphism hx µ : Gm,C → SC → PC, where µ : Gm,C → SC is the homomorphismdefined in 1.3. Since the different hx are all conjugate under P (C), we get a conjugacy ofcocharacters of PC that depends only on P and X . The automorphism group of C actscontinuously on the (discrete) set of all such conjugacy classes, so the given conjugacy classis defined over some number field E ⊂ C.

Definition. The reflex field of (P,X ), denoted by E(P,X ), is the field of definition ofthe conjugacy class of hx µ for any x ∈ X .

11.2. Properties: (a) It is important that the reflex field is given as a subfield of C.

(b) For any morphism of mixed Shimura data (P1,X1) → (P2,X2) we have E(P1,X1) ⊃E(P2,X2). This follows at once from the fact that the map from the set of conjugacy classesof cocharacters of P1,C to the set of conjugacy classes of cocharacters of P2,C is invariantunder Aut(C).

(c) The set of all conjugacy classes of cocharcters of PC depends only on P/W . ThusE(P,X ) = E((P,X )/W ) = E((P, h(X ))/W ). In particular, E(P1,X1) = E(P2,X2) for anyunipotent extension (P1,X1)→ (P2,X2).

186

(d) If the central torus of G splits over a totally real field or a CM -field, then the reflexfield is totally real or a CM -field. This follows from (c) and [D1] 3.8.

11.3. The class of field isomorphism: Let E be a number field. Class field theorygives a canonical isomorphism

Gal(Eab/E) ∼= π0(Gm(AE)/Gm(E)).

In order to avoid a sign ambiguity we state it explicitly. Let v be a nonarchimedean primeof E, πv a local parameter in Ev, and E′ an abelian extension of E that is unramified at v.Then the composite map

π0(Gm(AE)/Gm(E)) ∼−−→ Gal(Eab/E) ı Gal(E′/E)

maps the idele (. . . , 1, π−1v , 1, . . .) to the Frobenius at v. This convention determines the

class field isomorphism uniquely.

In the case E = Q the inclusion Z× → Gm(Af ) → Gm(A) induce an isomorphism

Z× ∼= X×f /(Q×)+ ∼= (R×)0\A×/Q× ∼= π0(Gm(A)/Gm(Q)).

Its composite with the above class field isomorphism is precisely the cyclotomic character

Gal(Qab/Q) ∼−−→ Z×, σ 7→ aσ,

which is given by the condition: For any root of unity ζ ∈ Qab we have σ(ζ) = ζaσ .

11.4. The reciprocity law for a torus: Consider Shimura data (T,Y), where Tis a torus. Let E = E(T,Y) be its reflex field. By definition E is the field of definitionof the cocharacters hy µ : Gm,C → TC for an arbitrary y ∈ Y. Let µh : Gm,E → TE bethe corresponding homomorphism over E, and RE/Q(µh) : RE/QGm,E → RE/QTE its Weilrestriction. The composite map

RE/QGm,ERE/Q(µh)−−−−−−−→ RE/QTE

NormE/Q−−−−−−−→ T

induces a continuous homomorphism

π0(Gm(AE)/Gm(E)) −→ π0(T (A)/T (Q)).

Together with the class field isomorphism of 11.3

Gal(E/E) ı Gal(Eab/E) ∼−−→ π0(Gm(AE)/Gm(E))

we get a homomorphism

ψ : Gal(E/E) −→ π0(T (A)/T (Q)),

which is canonically associated to the Shimura data (T,Y). This homomorphism ψ isreciprocity law for (T,Y).

187

Consider the natural action T (A) from the left hand side on Y × T (Af ):

T (A) = T (R)× T (Af ) 3 (t∞, tf ) : Y × T (Af )→ Y × T (Af ),

(y, t′f ) 7→ (t∞ · y, tf · t′f ).

Let Kf ⊂ (Af ) be an open compact subgroup. Since T is abelian, this action induces atransitive action of T (A)/T (Q) on MKf (T,Y)(C) = T (Q)\Y × (T (Af )/Kf ). This is a finitediscrete set, so this action factors over π0(T (A)/T (Q)). We define an action of Gal(E/E)from the left hand side on MKf (T,Y)(C) by the formula

σ(x) := ψ(σ)(x).

This action defines descent data for MKfC (T,Y), yielding a model MKf (T,Y) over E, i.e. a

scheme MKf (T,Y) over E together with an isomorphism MKfC (T,Y) ∼= MKf (T,Y) ×E C,

unique up to canonical isomorphism. This is the canonical model of MKf (T,Y)(C).

Let tf ∈ T (Af ) and K ′f ⊂ Kf open compact subgroups of T (Af ). Then the map

[ ·tf ] : MK′fC (T,Y) → M

KfC (T,Y) is defined. Since T is abelian, this map is equivariant

under the above action of T (A), hence also under the action of Gal(E/E). Therefore it

corresponds to a morphism of the canonical models MK′f (T,Y)→MKf (T,Y), defined overE, which we again denote by [ ·tf ].

11.5. Definition: Let (P,X ) be arbitrary mixed Shimura data. A canonical modelfor (P,X ) consists of a scheme MKf (P,X ) over E(P,X ) for all open compact subgroups

Kf ⊂ P (Af ), together with an isomorphism MKfC (P,X ) ∼= MKf (P,X )×E(P,X ) C, such that

:

(a) Every morphism [ ·pf ]K′f ,Kf : MK′fC (P,X ) → M

KfC (P,X ) descends to a morphism

defined over E(P,X )

[ ·pf ]K′f ,Kf : MK′f (P,X )→MKf (P,X ).

(b) If φ : (T,Y) → (P,X ) is an embedding, where T is a torus, and KTf ⊂ T (Af )

and Kf ⊂ P (Af ) open compact subgroups such that φ(KTf ) ⊂ Kf , then the morphism

[φ]KTf ,Kf

: MKTf

C (T,Y)→MKfC (P,X ) descends to E(T,Y):

[φ]KTf ,Kf

: MKTf (T,Y)→MKf (P,X )×E(P,X ) E(T,Y).

In order to study the functorial behavior of canonical models, we first need some tech-nical facts.

11.6. Lemma: Let (P,X ) be mixed Shimura data. For every finite extension E′

of E(P,X ) in C there exists an embedding (T,Y) → (P,X ), with T a torus, such thatE(T,Y) ∩ E′ = E(P,X ).

Proof. For P reductive and h : X → Hom(SC, PC) injective this is the assertionof [D1] thm. 5.1. Thus for P = G reductive we already know it for (G, h(X )). Let

188

(φ, ψ) : (T,Y) → (G, h(X )) be an embedding, then Y consists of one element only. Choosex ∈ X such that h(x) ∈ ψ(Y). Let Y ′ be the T (R)-orbit in X that is generated by x, and letψ′ : Y ′ → X be the inclusion. Then (φ, ψ′) : (T,Y ′) → (G,X ) is an embedding of Shimuradata. Since by 11.2 E(G,H) = E(G, h(H)), the assertion follows for (G,X ). Finally letP = W oG be arbitrary and (P,X )/W = (G,H). Since by 11.2 E(P,X ) = E(G,H), andthere exists a section (G,H) → (P,X ), the assertion follows from that for (G,H). q.e.d.

11.7. Lemma: Let (φ, ψ) : (P ′,X ′)→ (P,X ) be a morphism of mixed Shimura data,and Kf ⊂ P (Af ) an open compact subgroup. Then the union of the images of every map

[ ·pf ] [φ] : MK′f (P ′,X ′)(C)→MKf (P,X )(C) for all K ′f and pf for which it is defined, is

Zariski-dense in MKfC (P,X ).

Proof. (Compare [D1] 5.2) The set in question is

P (Q)\(P (Q) · ψ(X ′))× (P (Af )/Kf ) ⊂ P (Q)\X × (P (Af )/Kf ),

and it suffices to show that it is not contained in any proper closed complex analytic subset.Since P (Q) is dense in P (R), the closure of P (Q) ·ψ(X ′) in X is equal to P (R) ·ψ(X ′). Thusit suffices to show that no P (R)-orbit in X is contained in a proper closed complex analyticsubset of X . Consider the projection X → Z := U(C)\X . Since P (R) acts transitively onZ, it suffices to prove the assertion in every fibre. But X → Z is a holomorphic U(C)-torsor,and U(R) is not contained in any proper closed complex analytic subset of U(C). q.e.d.

11.8. Lemma: Let E be a subfield of C, X a scheme over E, and YC a reduced closedsubscheme of X ×E C. Suppose that we are given a collection of overfields E ⊂ Ei ⊂ Cwhose intersection is E. Suppose that for every i we are given schemes Zi,α over Ei withmorphisms φi,α : Zi,α → X ×E Ei over Ei, such that YC is the Zariski-closure of⋃

α

φi,α(Zi,α × EαC).

Then YC = Y ×E C for a unique reduced closed subscheme Y ⊂ X defined over E.

Proof. (Compare [D1] 5.3) The data defines descent data for YC as closed subschemesof XC. By [SGA1]exp.VIII cor. 1.9 this descent data is effective. q.e.d.

A central role is played by the following proposition.

11.9. Proposition: Let (P1,X1) → (P2,X2) be an embedding of mixed Shimura data.If a canonical model exists for (P2,X2), then there exists a canonical model for (P1,X1),

such that every morphism [φ] : MK1f

C (P1,X1)→MK2f

C (P2,X2) descends to a morphism

MK1f (P1,X1)→MK2

f (P2,X2)×E(P2,X2) E(P1,X1)

defined over E(P1,X1).

Proof. (Compare[D1] 5.7) By 11.2 (b), E(P2,X2) is containd in E(P1,X1), so theassertion makes sense. Let us write E := E(P1,X1). Let K1

f ⊂ P1(Af ) be an open compact

189

subgroup, and K2f ⊂ P2(Af ) an open compact subgroup according to 3.8, such that [φ] :

MK1f

C (P1,X1) → MK2f (P2,X2) is a closed embedding. Let X := MK2

f (P2,X2) ×E(P2,X2) E,and YC ⊂ XC := X ×E C the image of [φ], this is a closed subscheme. For any embeddinge : (T,Y) → (P1,X1) with T a torus, and all pf ∈ P1(Af ), choose an open compact

subgroup KTf ⊂ T (Af ) such that [ ·pf ] [e] : M

KTf

C (T,Y)→MK1f

C (P1,X1) is defined. Then

[φ][ ·pf ][e] : MKTf

C (T,Y)→MK2f

C (P2,X2) is also defined and by 3.5 equal to [ ·φ(pf )][φe].By assumption this map descends to a morphism MKT

f (T,Y)→ X×EE(T,Y) defined overE(T,Y). Over C it factors through YC, and by 11.6 and 11.7 the conditions of 11.8 aresatisfied for the collection of all these morphisms. 11.8 now implies that YC descends to aunique reduced closed subscheme Y ⊂ X defined over E.

Let K2′f ⊃ K2

f be another open compact subgroup of P2(Af ). By 3.5 we have [φ]K1f ,K

2′f

=

[id]K2f ,K

2′f [φ]K1

f ,K2f, and by assumption [id]K2

f ,K2′f

descends to a morphism MK2f (P2,X2)→

MK2′f (P2,X2) defined over E(P2,X2). Therefore [φ]K1

f ,K2′f

: MK1f

C (P1,X1)→ MK2′f (P2,X2)

descends to the morphism over E

[id]K2f ,K

2′f [φ]K1

f ,K2f

: Y →MK2′f (P2,X2).

If this is also a closed embedding, and we define Y ′ ⊂MK2′f (P2,X2)×E(P2,X2)E in the same

way as Y ⊂ X, then we get a canonical isomorphism Y → Y ′ defined over E. Applying thisto the intersection of two K2

f shows that , up to canonical isomorphism, Y is independent

of the choice of K2f . Thus we may define MK1

f (P1,X1) := Y . The condition 11.5 (b) beingclear by construction, it remains to prove 11.5 (a).

So let pf ∈ P1(Af ), and let K1f and K1′

f be open compact subgroups of P1(Af ) such

that [ ·pf ] : MK1′f

C (P1,X1) → MK1f

C (P1,X1) is defined. Let K2f ⊂ P2(Af ) be as above, and

K2′f ⊂ P2(Af ) likewise forK1′

f . SinceK1′f ⊂ pf ·K1

f ·p−1f → φ(pf )·K2

f ·φ(pf )−1 we may replace

K2′f by K2′

f ∩ φ(pf ) ·K2f · φ(pf )−1. Then the map [ ·φ(pf )] : MK2′

f (P2,X2)→MK2f

C (P2,X2)is also defined. By assumption it descends to a morphism of the canonical models, definedover E(P2,X2). Since by 3.5 (c) we have [φ] [ ·pf ] = [ ·φ(pf )] [φ], the morphism [ ·pf ]

descends to a morphism MK1′f (P1,X1)→MK1

f (P1,X1) defined over E, as desired. q.e.d.

Now we can prove functoriality for canonical models.

11.10. Proposition: Let φ : (P1,X1)→ (P2,X2) be a morphism of mixed Shimuradata, and K1

f ⊂ P1(Af ) and K2f ⊂ P2(Af ) open compact subgroups such that φ(K1

f ) ⊂K2F . If canonical subsets exist for (P1,X1) and (P2,X2), then the morphism [φ]K1

f ,K2f

:

MK1f

C (P1,X1)→MK2f

C (P2,X2) descends to a morphism defined over E(P1,X1):

[φ]K1f ,K

2f

: MK1f (P1,X1)→MK2

f (P2,X2)×E(P2,X2) E(P1,X1).

are defined over E(P1×P2,X1×X2) = E(P1,X1). Applying 11.9 to the morphism e yieldsa “new” canonical model for (P1,X1), such that all morphisms

[e] : MK1f (P1,X1)(new) →MK1

f×K2f (P1 × P2,X1,X2)

190

are defined over E(P1,X1). Its composite with the projection to the first factor is a mor-phism defined over E(P1,X1):

[pr1] [e] : MK1f (P1,X1)(new) →MK1

f (P1,X1).

Since this is an isomorphism after base change to C, it is an isomorphism itself. From

[pr2] [e] = [φ] [pr1] [e] it follows that [φ] descends to a morphism MK1f (P1,X1) →

MK2f (P2,X2)×E(P2,X2) E(P1,X1) defined over E(P1,X1), as desired. q.e.d.

11.11. Corollary: If a canonical model exists for Shimura data (P,X ), then it isuniquely determined up to isomorphism.

Proof. (Compare [D1] 5.5) Apply 11.10 to (P1,X1) = (P2,X2) = (P,X ). q.e.d.

11.12 Proposition: Let φ : (P1,X1) → (P,X ) be an improper rational boundarycomponent. If (P1,X1) possesses a canonical model, then so does (P,X ).

Proof. Since P1 is normal in P , E := E(P,X ) = E(P1,X1). Let Kf ⊂ P (Af ) be anopen compact subgroup. By 7.10 there exist finitely many pνf ∈ P (Af ), such that with

Kνf := P1(Af ) ∩ pνf ·Kf · (pνf )−1 and

∆ν := (StabP (Q)(X1) ∩ (P1(Af ) · pνf ·Kf · (pνf )−1))/P1(Q),

the maps 3.4 induce an isomorphism∐ν

∆ν\MK1f

C (P1,X1)

∐[ ·pνf ][φ]

−−−−−−−−→MKfC (P,X ).

Since ∆ν acts through maps 3.4, by 11.5 and 11.10 it acts on the canonical modelMKνf (P1,X1).

We may therefore define MKf (P,X ) as∐ν ∆ν\MKν

f (P1,X1) under this isomorphism.

To verify 11.5 (a) let K ′f be another open compact subgroup of P (Af ), and pf ∈P (Af ) such that K ′f ⊂ pf · Kf · p−1

f . We have to show that for every p′f ∈ P (Af ) with

K1f := P1(Af ) ∩ p′f · K ′f · p

′−1f the morphism [ ·pf ] [ ·p′f ] [φ] descends to a morphism

MK1f (P1,X1)→MKf (P,X ) defined over E. There exists a unique ν so that pνf and p′f · pf

represent the same class in (StabP (Q)(X1)·P1(Af ))\P (Af )/Kf . Writing p′f ·pf = p·p1,f ·pνf ·kfaccordingly, the properties 3.5 (a) through (e) imply

[ ·pf ] [ ·p′f ] [φ] = [ ·p′f · pf ] [φ] = [ ·kf ] [ ·pνf ] [ ·p1,f ] [ ·p] [φ]

= [ ·pνf ] [φ] [ ·p1,f ] [int(p−1)|(P1,X1)].

By 11.5 and 11.10 the morphisms [ ·p1,f ] and [int(p−1)|(P1,X1)] descend to E, and [ ·pνf ] [φ] does so by definition. This proves 11.5 (a); in particular the model MKf (P,X ) isindependent of the choice of the pνf .

To prove 11.5 (b) let ψ : (T,Y) → (P,X ) be an embedding with T a torus. By thesame arguments as above we may reduce to the case where (T,Y) is irreducible. Then ψ(T )is contained in P1. Fix p ∈ P (Q) such that p · ψ(Y) has a non-empty intersection with X1.

191

Since ψ(T (R)) ⊂ P1(R), it then follows that p ·ψ(Y) ⊂ X1, whence int(p) ψ = φ ψ1 for amorphism ψ1 : (T,Y) → (P1,X1). By 3.5 it follows that

[ψ] = [int(p)−1] [φ] [ψ1] = [ ·p] [φ] [ψ1].

Here [ ·p] [φ] descends to E by the above definition of MKf (P,X ), and [ψ1] descends toE(T,Y) by the assumption on (P1,X1). q.e.d.

In order to apply the results of the previous chapter, we need the following technicallemma.

11.13. Lemma: Let (P,X ) be mixed Shimura data, (G,H) := (P,X )/W , ande : (G,H) → (P,X ) a splitting. Let P ⊂ P (Af ) be a subset that generates P (Af ) asan abstract semigroup. Let K be a cofinal system of open compact subgroups of P (Af ).

Suppose that for every Kf ∈ K we are given a model MKf (P,X ) of MKfC (P,X ) over

E(P,X ). Assume that:

(a) For all pf ∈ P, Kf ∈ K, and all sufficiently small K ′f ∈ K (here the bound depends on

pf and Kf ), the morphism [ ·pf ]K′f ,Kf descends to a morphism MK′f (P,X )→ MKf (P,X )

defined over E(P,X ).

(b) For every embedding φ : (T,Y) → (G,H) with T a torus,every Kf ∈ K, and everyopen compact subgroup KT

f ⊂ T (Af ) such that eφ(KTf ) ⊂ Kf , the morphism [eφ]KT

f ,Kf

descends to a morphism MKTf (T,Y)→MKf (P,X )×E(P,X ) E(T,Y) defined over E(T,Y).

Then there exists a canonical model for (P,X ).

Proof. If condition (a) holds for pf , p′f ∈ P (Af ), then it clearly holds for pf · p′f as

well. Since by assumption P (Af ) = P ∪P · P ∪P · P · P ∪ . . ., (a) holds for all pf ∈ P (Af ),and from now on we may assume P = P (Af ).

Write E := E(P,X ). Let Kf ⊂ P (Af ) be an arbitrary open compact subgroup, andfix K ′f ∈ K with K ′f ⊂ Kf . We shall show that there exists a model MKf (P,X ) over E

of MKfC (P,X ), such that the morphism [id] : M

K′fC (P,X ) → M

KfC (P,X ) descends to a

morphism MK′f (P,X ) → MKf (P,X ). Let K ′′f :=⋂kf∈Kf kf · K

′f · k

−1f , and kνf a set of

representatives of Kf/K′′f . Fix some K ′′′f ∈ K with K ′′′f ⊂ K ′′f . Let Y be the image of the

morphism

MK′′f (P,X )π[ ·kνf ]−−−−−→

∏ν

MK′fC (P,X ),

with the structure of a reduced closed subscheme. By construction, over C this map factorsthrough a closed embedding

MK′′fC (P,X ) ∼= YC →

∏ν

MK′fC (P,X ),

in other words Y is a model of MK′′fC (P,X ) over E. This closed embedding is equivariant

under the following action of Kf/K′′f : on M

K′′fC (P,X ) through the automorphism [ ·kf ], and

on∏νM

K′fC (P,X ) through the obvious permutations. This shows that this action descends

192

to an action on Y . Our schemes being quasiprojective, we may form the quotient, which isthe desired scheme MKf (P,X ).

We now prove 11.5 for the models just constructed. For 11.5 (a) consider pf , Kf ,K ′f so that K ′f ⊂ pf · Kf · p−1

f . Let Kf , K ′f ∈ K so that Kf ⊂ Kf , K ′f ⊂ K ′f , and

the maps [id]Kf ,Kf and [id]K′f ,K′f

descend to morphisms M Kf (P,X ) → MKf (P,X ) and

M K′f (P,X ) → MK′f (P,X ). Let K ′′f ∈ K so that K ′′f ⊂ K ′f ∩ pf · Kf · p−1f . Then we have

the commutative diagram

M K′′f (P,X )[id]−−−→ M K′f (P,X )

[id]−−−→ MK′f (P,X )

[pf ] ↓ [pf ]

M Kf (P,X )[id]−−−→ MKf (P,X ),

where the solid arrows are morphisms over E, but the dotted arrow is a priori defined onlyover C. Since the upper line conists of finite flat coverings, the dotted arrow also descendsto E, as desired. This also proves that the model MKf (P,X ) is independent of the choiceof K ′f .

For 11.5 (b) let φ : (T,Y) → (P,X ) be an embedding with T a torus. By conjugacyof all splittings of P , there exists p ∈ P (Q) such that int(p) φ factors through G. Thusφ = int(p−1) e φ′ for some embedding φ′ : (T,Y) → (G,H). Having proved 11.5 (a), wemay assume p = 1. Let Kf ⊂ P (Af ) and KT

f ⊂ T (Af ) be open compact subgroups such

that φ(KTf ) ⊂ Kf . Let Kf ∈ K so that Kf ⊂ Kf and the map [id]Kf ,Kf descends to a

morphism M Kf (P,X ) → MKf (P,X ). With KTf := KT

f ∩ φ−1(Kf ) we get a commutativediagram

M KTf (T,Y)

[id]−−−→ MKTf (T,Y)

[φ] ↓ ↓ [φ]

M Kf (P,X )×E E(T,Y)[id]−−−→ MKf (P,X )×E E(T,Y),

and the assertion follows as above. q.e.d.

11.14. Proposition: There exists a canonical model for the mixed Shimura data(P0,X0) defined in 2.24. If Kf = KU

f oK∗f according to the decomposition P0 = U0 oGm,Q,descends to an isomorphism over Q

MKf (P0,X0) ∼= Gm,Q ×MK∗f (Gm,Q,H).

Proof. The reflex field of (P0,X0) is Q. We use the notations of 10.15. Clearly the setP := KU

f (1) ∪ Gm(Af ) generates P0(Af ) as a semi group, and the collection K = KPf (d) |

d ≥ 1 of open compact subgroups is cofinal. Letting MKPf (d)(P0,X0) := X0

d for every d ≥ 1,condition 11.13 (a) directly follows from 10.16 (b). Since by construction the zero sectiondescends to a morphism M0

d → X0d over Q, condition 11.13 (b) is equivalent to saying that

the identification in 10.16 (a) turns M0d into the canonical model of M

Kf (d)C (Gm,Q,H). This

is so by the explicit description of the reciprocity law for Gm,Q in 11.3, and the definitionof the identification in 10.15. The first assertion now follows from 11.13.

193

The second statement holds by construction if Kf = KPf (d) for some d. In general

some conjugate of Kf by an element of Gm(Q) ⊂ P0(Q) is of the form KUf (d) o K∗f for

some Kf (d) ⊂ K∗f ⊂ Kf (1), and by the equivariance of the isomorphism 3.16 it is enough

to prove the assertion for the latter group. But MKUf (d)oK∗f (P0,X0) is isomorphic to the

quotient of Xd by the right action of K∗f/Kf (d). As in the proof of 10.16 (b) one finds that

through the isomorphism X0d∼= Gm ×M0

d this group acts only on the second factor. Thus

X0d/K

∗f∼= Gm× (M0

d/K∗f ) ∼= Gm×MK∗f (Gm,Q,H), and this is just the isomorphism defined

in 3.16. q.e.d.

11.15. Proposition: There exists a canonical model for the mixed Shimura data(P0, h(X0)) defined in 2.24.

Proof. Let ı : (P0,X0)→ (P0,X0) be the automorphism that is the identity on P0 andon h(X0), but interchanges the two connected components of X0. For every open compact

subgroup KPf ⊂ P0(Af ) the automorphism [ı] of M

KPf

C (P0,X0) is defined, and by 11.10 and11.14 it descends to an automorphism of its canonical model over Q. It defines an action of

a group of order 2, and over C the quotient is canonically isomorphic to MKPf

C (P0, h(X0)).

We define MKPf (P0, h(X0)) as the quotient of MKP

f (P0,X0) by this action; this is possiblesince the scheme is affine.

We show that this defines a canonical model for (P0, h(X0)). The property 11.5 (a)follows directly from the corresponding property for (P0,X0), and the universal property ofthe quotient by a finite group. Let (φ, ψ) : (T,Y) → (P0, h(X0)) be an embedding with T atorus. As in the proof of 11.6, we can construct an embedding (φ, ψ′) : (T,Y ′) → (P0,X0)such that the composite (T,Y ′) → (P0,X0) → (P0, h(X0)) factors through (T,Y). Thecondition 11.5 (b) now follows as in the proof of 11.13. q.e.d.

11.16. Theorem: For every g ≥ 1, there exists a canonical model for the mixedShimura data (P2g,X2g) defined as in 2.25. For the open compact subgroup in 10.7, thiscanonical model is isomorphic to the universal family Xd over the moduli Md of 10.6.

Proof. The reflex field of (P2g,X2g) is Q. We use the notations of chapter 10, inparticular 10.7. Clearly the set P := KW

f (2)∪CSP2g(Af ) generates P2g(Af ) as a semigroup,

and the collection K = KPf (d) | 2|d ≥ 4 of open compact subgroups is cofinal. Let

MKPf (d)(P2g,X2g) := Xd for all even d ≥ 4, where Xd is as in 10.10. By 10.13, condition

11.13 (a) holds for pf = 1. This, together with 10.11, proves the same for all pf ∈ KWf (2).

Fix pf ∈ CSP2g(Af ) and an even integer d ≥ 4. Let d be a multiple of d, and 2|d′ ≥ 4,so that KP

f (d′) ⊂ pf · KPf (d) · p−1

f ⊂ KPf (2) as in 10.12. With d′′ := d′ · d, 10.13 implies

that both [ ·pf ]KPf (d′′),KP

f (d) and [id]KPf (d),KP

f (d) descends to morphisms over Q, so 11.13 (a)

holds for all pf ∈ P.

It remains to verify 11.13 (b). Since, in the notations of 10.7, the section M(d) →MW (d) corresponds to the morphism τ : Md → Xd, defined over Q, this is equivalent to say-

ing that the identification in 10.9 turnsMd into the canonical model ofMKf (d)C (CSP2g,Q,H2g).

This is a consequence of the main theorem of Shimura and Taniyama about abelian vari-eties with complex multiplications (see [D1] thm. 4.19, thm. 4.21). Note that the contrast

194

to [D1] we have a different sign convention at two places, namely on the one hand in thedefinition of the operation of the torus S in a Hodge structure (cf. [D1] 1.3), on the otherhand in the definition of the reciprocity law for a torus (cf. [D1] 3.9). These two signchanges cancel in translating [loc. cit.] thm. 4.19. q.e.d.

11.17. Theorem: Let (G,H) be Shimura data with G reductive and h : H →Hom(SC, GC) injective. Then there exists a canonical model for (G,H).

Proof. see [Mi1] thm. 7.2.

11.18. Theorem: There exists a canonical model for every mixed Shimura data.

Proof. By 11.12 it suffices to consider irreducible (P,X ). Suppose first that V = 1.Choose an embedding

(P,X ) → (T,Y)× (G,H)×n∏i=1

(P0, h(X0))

as in 2.26 (a). By 11.4, 11.17, and 11.15 each of the factors on the right hand side possessesa canonical model. Thus the assertion follows from 11.9.

Next suppose 2g := dim(V ) > 0. According to 2.26 (b) choose a morphism φ :(P,X )→ (Gm,Q,H), a (P0,X0)→ (Gm,Q,H)-torsor (P1,X1)→ (P,X ), and an embedding

(P1,X1) → (T,Y)× (G,H)×n∏i=1

(P2g,X2g).

By 11.4, 11.17 and 11.16 each factor on the right hand side possesses a canonical model.Thus, by 11.19, (P1,X1) also possesses a canonical model. It remains to push this down to(P,X ).

Let K1f ⊂ P1(Af ) be a neat open compact subgroup, and Kf its image in P (Af ). We

abbreviate M1 := MK1f (P1,X1), MC := M

KfC (P,X ), and M := Mπ(Kf )((P,X )/W ); the

latter exists by the first part of the present proof. The Gm-action on M1,C is given intermsof maps 3.4, so by 11.10 and 11.14 it induces a Gm,E-action on M1, where E = E(P,X ).Since M1,C → MC is a Gm-torsor, it is Zariski-locally trivial, so the descent data for M1,Crelative to E ⊂ C induces descent data for MC. Since by 11.10 the composite morphismM1,C → MC → MC descends to E, we may view this as descent data with respect toMC → M . If LC denotes the line bundle on MC that corresponds to the inverse of theGm-torsor M1,C → MC, then we have descent data for the pair (MC, LC). Recall that by3.21, LC is relatively ample on MC → MC. Thus by [SGA1] exp.VIII cor, 7.8 this descentdata is effective.

For all neat open compact subgroups Kf ⊂ P (Af ) we have thus defined a model of

MKfC (P,X ) over E. Letting K be the set of these Kf , and P := P (Af ), it remains to verify

the conditions 11.13. These follow easily from the construction, by the same arguments asin the proof of 11.13. (Condition 11.13 (a) is an application of the descent of morphisms[SGA1] exp.VIII thm. 5.2) q.e.d.

195

Chapter 12

Canonical model of thecompactification

In this chapter we formulate and prove the main result of this thesis: the extension of thecanonical model to both the Baily-Borel compactification and the toroidal compactifica-tions, together with an explicit description of the model induced on the boundary. Beforestating these theorems (12.3–5), we prove an assertion relating the reflex field of mixedShimura data with that of boundary component (12.1–2). For the structure of the proofssee 12.6–7; suffice it to say here that we are reaping the benefits of the consistent use ofthe adelic formalism, in particular in chapter 6. The proofs occupy 12.8–17. In 12.18–20,we apply the results to q-expansion of modular forms. We conclude with two examples(12.21-22).

12.1. Proposition: Let (P1,X1) be a rational boundary component of some mixedShimura data (P,X ). Then

E(P1,X1) = E(P,X )

Proof. Let x ∈ X , and x1 ∈ X1 the corresponding point. We claim: hx µ is conjugateto hx1 µ under P (C). To prove this by the definition 4.11 of hx1 , it suffices to show thatthe two cocharacters h0 µ and h∞ µ of H0,C are conjugate under H0(C), where H0, h0,h∞ are as in 4.3. But it is easily checked that both of these are conjugate to the map

Gm(C)→ H0(C), z 7→ (z,(z 00 1

)).

Let Q ⊂ P be the parabolic subgroup associated to (P1,X1). By the following lemma12.2, we have inclusions

Y \(P1) → Y \(Q) ⊃ Y \(Q)+ ∼−−→ Y \(P ).

By the claim above, and by 4.4, our conjugacy class [hx µ] = [hx1 µ] lies in each of thesesubsets. By the definition of the reflex field, the assertion follows. q.e.d.

12.2. Lemma: For any linear algebraic group G denote by Y \(G) the (discrete) setof G-conjugacy classes of cocharacters of G.

196

(a) If U is the unipotent radical of G, then Y \(G) ∼= Y \(G/U).

(b) If H is a connected normal subgroup of G, then Y \(H) → Y \(G).

(c) If G is reductive, P ⊂ G a parabolic subgroup, and U the unipotent radical of P , thenthe subset

Y \(P )+ := λ ∈ Y \(P ) | λ has nonnegative weights on U

maps isomorphically to Y \(G).

Proof. Assertion (a) follows from the conjugacy of all maximal tori. Assertion (b)holds, because G and H have the same image in Aut(H). For (c), fix a Borel subgroup Binside P . The fact that every weight is conjugate to a unique dominant weight, means thatevery element of Y \(G) possesses a unique representative in Y \(B)+. Applying this fact toB ⊂ P , the assertion follows. q.e.d.

We are now in the position to formulate our main results.

12.3. Main Theorem for the Baily-Borel compactification: Consider pureShimura data (P,X ) and an open compact subgroup Kf ⊂ P (Af ).

(a) The canonical modelMKf (P,X ) extends uniquely to a schemeMKf (P,X )∗ over E(P,X ),with an isomorphism

MKfC (P,X )∗ = MKf (P,X )∗ ×E(P,X ) C.

(b) All morphisms 3.4 correspond to morphisms of these schemes, defined over E(P,X ), inthe case 3.4 (a), and over E(P1,X1) in the case 3.4 (b) (compare 6.2, 9.25). The functorialityof these morphisms is preserved, as well as the assertions of 8.2 and 8.4 (ampleness).

(c) The stratification in 6.3, and the maps in 7.6, are defined over the common reflex fieldE(P,X ).

12.4. Main Theorem for the toroidal compactification: Let (P,X ) be mixedShimura data, Kf ⊂ P (Af ) an open compact subgroup, and Σ a Kf -admissible partial

cone decomposition for (P,X ). We assume that MKfC (P,X ,Σ) exists and can be covered

by quasiprojective MKfC (P,X ,Σi) for Σi ⊂ Σ, such that, on each M

KfC (P,X ,Σi), some

ample line bundle can be described as in terms of ω[dlog] and line bundles as in 8.13 (e.g.in the situation of 9.21, 9.28, or 9.33).

(a) The canonical model MKf (P,X ) extends uniquely to a scheme MKf (P,X ,Σ) overE(P,X ), with an isomorphism

MKfC (P,X ,Σ) = MKf (P,X ,Σ)×E(P,X ) C.

(b) All morphisms 6.25 (a) ∼= 11.5 (a) correspond to morphisms of these schemes, definedover E(P,X ). The same holds for the morphism [π]∗ in 6.24. All morphisms 6.25 (b) ∼=11.10 correspond to morphisms of these schemes, defined over E(P1,X1). The assertionsof 6.25 (functoriality, open embeddings, isomorphisms, quotients by finite groups), 6.26(smoothness), 8.6 (line bundle structure), and 8.13–14 (ampleness) are equally valid forthese schemes.

197

(c) The stratifications in 7.2, 7.3, and 7.10 (see 9.36) are defined over the common reflexfield E(P,X ), and the assertion of 7.17 (a) holds over E(P,X ). The isomorphism of 7.17 (b)(see 9.37) induces a canonical isomorphism between the formal completion of MKf (P,X ,Σ)

and of Stab∆1([σ])\MK1f (P1,X1,Σ1) along

Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X1,[σ],Σ1,[σ]).

12.5. Corollary: Without the assumption on (P,X ), Kf , and Σ, “MKf (P,X ,Σ)”always exists as an algebraic space. The remaining assertions of 12.4 carry over literally.

Proof. As in 9.34 (a), using 9.33 the assertion reduces to that in the theorem. q.e.d.

12.6. Beginning of the proof of 12.3 and 12.4: The central points in boththeorems (which imply the others) are 12.3 (a), 12.4 (a): the existence of an extension ofthe canonical models; and the last assertion of 12.4 (c): the isomorphism of formal schemesat the boundary. The remaining assertions say that a certain morphism descends to thereflex field. For each such morphism, by the effectivity of descent of morphisms ([SGA1]exp.VIII thm. 5.2), it suffices to show that there is canonical descent data. Since all ourschemes are normal, this is certainly so if the morphism descends on some open densesubscheme. Taking into account 9.25, 11.5 (a) and 11.10, this proves that part (b) of eithertheorem follows from part (a). In particular, this proves the uniqueness in (a). Likewise,in 12.4 (c) it suffices to prove the assertion for the formal neighborhood of the stratum

Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X1,[σ]) instead of its closure. This assertion, together with

12.3 (a), also implies 12.3 (c). Indeed, fix any complete Kf -admissible cone decompositionΣ for (P,X ), so that 12.4 holds for MKf (P,X ,Σ). The map of 6.3 lies in the commutativediagram

Stab∆1([σ])\Mπ[σ](K

1f )

C (P1,[σ],X1,[σ]) → MKfC (P,X ,Σ)

↓↓ ↓↓

∆1\MK1f

C ((P1,X1)/W1) → MKfC (P,X )∗

for some cone σ. By part (b) of both theorems, the two vertical maps descend to E(P,X ).By assumption, the upper inclusion descends to E, so the same follows for the lower one,as desired.

By similar arguments, we may assume that Kf is arbitrarily small. In fact, eithertheorem reduces to the quasiprojective case, in which quotients by finite groups exist.Thus part (a) of either theorem reduces to all sufficiently small open compact subgroups,and the rest follows from the universal property of quotients by finite groups.

12.7. Strategy of proof: Of the statements that remain to be proved, the existenceof MKf (P,X ,Σ) and the isomorphism of formal neighborhoods over the reflex field areintimately connected. In fact, we shall the construct MKf (P,X ,Σ) in such a way that 12.4(c) holds automatically. The existence of MKf (P,X )∗ for reductive P will be proved onthe way. The intuitive idea is the following.

198

In a way, we can view MKf (P,X ,Σ) as being glued together from formal neighborhoodsof torus embeddings along the unipotent fibre of other mixed Shimura varieties, much asin the construction of MKf (P,X ,Σ)(C) in chapter 6. Of course, one cannot glue formalschemes in the way that would be needed here, so we have to argue more indirectly. Inany case, we must show that the “gluing isomorphism” given by 9.37 descends to the reflexfield. To prove this we use an analog of “ special points”: namely certain embedded mixedShimura varieties, for which the assertions can be shown directly. The descent is theneffected by a density argument similar to 11.6–10.

This way of proving 12.4 (c) seems to assume 12.4 (a), but it proves this assertion aswell. In fact, when we have not yet shown the existence of MKf (P,X ,Σ), we can at least

interpret the argument as proving that the descent data for MKfC (P,X ) with respect to

E(P,X ) ⊂ C extends to descent data for MKfC (P,X ,Σ), and it remains to show that this

descent is effective. Here our assumption about the ample line bundle comes in. Since itcan be expressed purely in terms of the toroidal compactification of other mixed Shimuradata, we get compatible descent data for this ample line bundle as well. In such a situation,descent is always effective, and we are done.

In detail, there are the following steps. First, we prove 12.4 in the case of torusembeddings along the unipotent fibre (12.8). Next, using modular interpretation we dothe same for (P,X ) = (GL2,QH2) (see 12.9). Taken together, these special cases imply12.4 for certain (P,X ), and cone decompositions which are “supported” only on certainrational boundary components (see 12.10–11). In particular, if P is reductive, this holdsfor all boundary components of codimension 1 in the Baily-Borel compactification; and thisis sufficient to prove 12.3 (a) (see 12.12). In 12.13, we prove that the embedded mixedShimura varieties of the above special type satisfy the density condition alluded to before.Some descent questions are then dealt with the general context (12.14–16), before (in 12.17)we put everything together to finish the proof.

12.8. The case of a torus embedding along the unipotent fibre: We first prove

12.4 in the situation of 6.8. Recall thatMKfC (P,X )→M

π′(Kf )C ((P,X )/U) is a torsor under a

(relative) torus, namely either MK∗fC (P∗,X∗)→M

φ(Kf )C (Gm,Q,H0), or the same with X∗, H0

replaced by h(X∗), h(H0) respectively. Moreover there exists a unique K∗f -admissible cone

decomposition Σ∗ for (P∗,X∗), resp. for (P∗, h(X∗)), such thatMKfC (P,X ) →M

KfC (P,X ,Σ)

is the torus embedding associated to the torus embedding MK∗fC (P∗,X∗) →M

K∗fC (P∗,X∗,Σ∗)

(resp. with h(H∗) in place of X∗). By 11.2, the common reflex field of (P∗,X∗), (P∗, h(X∗)),(Gm,Q,H0), and (Gm,Q, h(H0)) is Q, which is of course contained in E(P,X ) = E(P ′,X ′).Thus, by 11.10, the torsor structure descends to the canonical model MKf (P,X ). Theassertion 12.4 (a) for MKf (P,X ,Σ) now follows if we know it for MK∗f (P∗,X∗,Σ∗), resp.for MK∗f (P∗, h(X∗),Σ∗). The first sentence of (c) is then true by the definition of thestratification in terms of the canonical projection (see 5.2). The last assertion of (c) is atautology.

It remains to prove 12.4 (a) for (P∗,X∗) and (P∗, h(X∗)). We begin with (P∗,X∗).

199

Letting ΓU := U(Q) ∩KUf , by 6.9 the map 3.16 induces canonical isomorphisms

MK∗fC (P∗,X∗) ∼−−→ Gm,C ⊗ ΓU ×M

φ(Kf )C (Gm,Q,H0)

∩ ∩ ‖

MK∗fC (P∗,X∗,Σ∗) ∼−−→ (Gm,C ⊗ ΓU )ΣU ×M

φ(Kf )C (Gm,Q,H0).

By 11.14, the first line descends to an isomorphism

MK∗f (P∗,X∗) ∼−−→ Gm,Q ⊗ ΓU ×Mφ(Kf )(Gm,Q,H0)

for the canonical models. Thus we may define MK∗f (P∗,X∗,Σ∗) as (Gm,Q ⊗ ΓU )ΣU ×Mφ(Kf )(Gm,Q,H0).

Finally we show how this implies the assertion for (P∗, h(X∗)). As in the proof of11.15, let ı be the automorphism of (P∗,X∗) that is the identity on P∗ and on h(X∗),but interchanges the two connected components of X∗. By assumption, this induces aninvolution [ı] on the canonical model MK∗f (P∗,X∗,Σ∗). We define MK∗f (P∗, h(X∗),Σ∗) asthe quotient of MK∗f (P∗,X∗,Σ∗) by [ı]; this quotient exists, since MK∗f (P∗,X∗,Σ∗) is coveredby affine MK∗f (P∗,X∗,Σi), each of which is stable under [ı]. Since MK∗f (P∗, h(X∗)) is the

quotient of MK∗f (P∗,X∗), and MK∗fC (P∗, h(X∗),Σ∗) the quotient of M

K∗fC (P∗,X∗,Σ∗), our

definition gives the desired extension of the canonical model.

12.9. The case P = GL2,Q: Next we prove 12.4 for (P,X ) = (GL2,Q,H2), asdefined in 2.7. In this case every Kf -admissible cone decomposition is contained in a

unique complete cone decomposition Σ, with MKfC (P,X ,Σ) ∼= M

KfC (P,X )∗. It suffices to

consider this Σ. By projectivity it suffices to prove the assertions for a cofinal system ofopen compact subgroups. So we may assume that Kf = Kf (d) for d ≥ 3, as in 10.19.Let Md be the moduli scheme of generalized elliptic curves over Q with d-structure, andMd ⊂Md the open subscheme parametrizing (honest) elliptic curves. By 10.20 and 11.16,we have compatible isomorphisms

MKfC (P,X ,Σ) ∼=Md,C and MKf (P,X ) ∼=Md.

To prove 12.4 (a), we just set MKf (P,X ,Σ) :=Md.

In 12.4 (c), the last statement implies the earlier ones. Since in our case all the pairs(pf (P1,X1)) consisting an element of P (Af ) and a proper rational boundary componentof (P,X ), are conjugate under the left action of GL2(Q) and the right action of GL2(Z)(which normalizes Kf ), it suffices to prove that assertion in any one instance. This instanceis provided by 10.22, taking into account 11.14 and 12.8.

Now we can prove the assertion for a certain type of boundary components.

12.10. The case of boundary components “of codimension 1” for reductiveP : Consider pure Shimura data (P,X ) and surjective homomorphism P ı PGL2,Q. Theinverse image of any rational Borel subgroup of PGL2,Q is an admissible Q-parabolic sub-group of P . We shall prove 12.4 in the case that Σ “is supported” only on those rational

200

boundary components associated to such parabolic subgroups. If (P1,X1) is such a rationalboundary component, then by assumption U1 has dimension 1, so as in 12.9, we have toprove the assertion for just one Σ, the maximal one with the above property. In other

words, we want to prove the assertion for MKfC (P,X )+, as defined in 8.2.

Let us first assume that the surjection P ı PGL2,Q lifts to a morphism (P,X ) →(GL2,Q,H2). Then PGL2,Q lifts to an almost direct factor of P der, isomorphic to SL2,Q. Let(P ′,X ′) := (P,X )/SL2,Q (see 2.9), then since SL2(R) is connected, we get an embedding(P,X ) → (P ′,X ′) × (GL2,Q,H2). Since E(GL2,Q,H2) = Q, we clearly have E(P,X ) =E(P ′,X ′), which is also reflex field of this product. Using 12.9, the assertion of 12.4 followsfor the product, with the product of trivial cone decomposition for (P ′,X ′), and the uniquemaximal one for (GL2,Q,H2). By descent of closed subschemes ([SGA1]exp.VIII cor. 1.9),resp. of morphisms, the assertions also follows for (P,X ).

In general case, let P be the difference kernel of the two maps P × GL2,Q ´ PGL2,Q.The inclusion P → P ×GL2,Q corresponds to an embedding (P , X ) → (P,X )× (GL2,QH2)for some Shimura data (P , X ). The kernel of the projection P ı P is isomorphic to Gm,Q,and (P,X ) ∼= (P , X )/Gm,Q. Clearly E(P , X ) → E(P,X ). By assumption, the desiredassertions hold for (P , X ). Let Σ, Σ be the respective partial cone decompositions satisfyingthe condition above. For all open compact subgroups Kf mapping to Kf , the morphism

M Kf (P , X , Σ)(C) → MKf (P,X ,Σ)(C) is finite; and even surjective, since H1(A,Gm) = 0.

Thus we may define MKf (P,X ,Σ) as a finite quotient of M Kf (P , X , Σ) which exists byquasi-projectivity. This proves 12.4 (a), and the compatibilities (c) follow by descent ofmorphisms.

As the last special case, we prove 12.4 for certain unipotent extensions of the Shimuradata of 12.10.

12.11. Generalization of 12.10: Let (P,X ) and Σ be as in 12.10, only this timeassume that (P,X ) is irreducible. Let (P ′,X ′) be a unipotent extension of (P,X ) whoseunipotent redical W ′ is pure of weight - 2, i.e. W ′ = U ′. Let Σ′ be a finite admissiblepartial cone decomposition “supported” only on the rational boundary components (P ′1,X ′1)associated to those (P1,X1) in 12.10. As the last special case, we prove 12.4 for all such Σ′

that are sufficiently fine, in the sense explained below, and a cofinal system of K ′f ⊂ P ′(Af ).

With (P∗,X∗) defined as in 12.8, (P ′,X ′) is (non-canonically) isomorphic to the fibreproduct (P∗, h(X∗)) ×(P0,h(X0)) (P,X ) (see 2.20). Let K∗f = KU

f o KTf ⊂ P∗(Af ) be neat,

Kf ⊂ P (Af ) mapping to KTf , and let K ′f ⊂ P ′(Af ) be their fibre product. By 3.11,

MK′fC (P ′,X ′) ∼= M

K∗fC (P∗, h(X∗))×

MKTf

C (Gm,Q,h(H0))M

KfC (P,X ).

Fix any K∗f -admissible complete cone decomposition Σ∗ for (P∗, h(X∗)). By pullback, thisΣ∗, together with the partial cone decomposition Σ considered in 12.10 induce a finiteK ′f -admissible partial cone decomposition Σ′0 for (P ′,X ′). Clearly,

MK′fC (P ′,X ′,Σ′0) ∼= M

K∗fC (P∗, h(X∗),Σ∗)×

MKTf

C (Gm,Q,h(H0))M

KfC (P,X ,Σ);

201

indeed, for any (P ′1,X ′1) as above we have a corresponding isomorphism

MK1′f

C (P ′1,X ′1,Σ′01) ∼= MK∗fC (P∗, h(X∗),Σ∗)×

MKTf

C (Gm,Q,h(H0))M

K1f

C (P1,X1).

By 12.8 and 12.10 these fibre products descend to the canonical models over E(P ′,X ′) =E(P,X ), so we may define

MK′f (P ′,X ′,Σ′0) := MK∗f (P∗, h(X∗),Σ∗)×MKTf (Gm,Q,h(H0))

MKf (P,X ,Σ).

This proves all of 12.4 for the given Σ′0.

More generally, let Σ′ be any finite partial refinement of Σ′0. Note that, since Σ′0 hasthe property that |Σ′0(X ′0, P ′1, p′f )| = C∗(X ′0, P ′1) for all p′f and all (P ′1,X ′1) as above, thisincludes any sufficiently fine decomposition “supported” only on these (P ′1,X ′1). Then, both

MK′fC (P ′,X ′,Σ′)→M

K′fC (P ′,X ′,Σ′0)

and the various

MK1′f

C (P ′1,X ′1,Σ′1)→MK1′f

C (P ′1,X ′1,Σ′01)

are allowable modifications in the sense of [KKMS] ch.II §2 p.87 def. 3. Since they areformally isomorphic, they correspond to the same “f.r.p.p. decomposition” (see [loc. cit.]p.86 def.2). By the equivalence between allowable modifications and f.r.p.p. decompositions[loc. cit.] p.90 thm. 6∗), one of them descends to E(P,X ) if and only the other does so. Butby 12.8 the second does descend, hence so does the first, and the compatibility is automatic.

12.12. Proof of 12.3 (a): Consider pure mixed Shimura data (P,X ) and neat Kf ⊂P (Af ). By 12.10, the canonical model extends to a model MKf (P,X )+ of M

KfC (P,X )+,

over E := E(P,X ). Define MKf (P,X )∗ as the closure of MKf (P,X )+ inside

PNE ∼= PΓ(MKf (P,X )+, ω[dlog]⊗n)

for any sufficiently large n. By 8.2, this is a model MKfC (P,X )∗, as desired.

The following lemma implies that every boundary component can be deformed, alonga given collection of tangent directions, into embedded mixed Shimura data of the specialtype considered in 12.11.

12.13. Embedding Lemma: Let (P1,X1) be a proper rational boundary componentof some mixed Shimura data (P,X ). Let U ⊂ U ′1 ⊂ U1 with dim(U ′1/U) = 1, and such that,for some (⇔ for all) connected components X 0 of X+ (see 4.11),

C(X 0, P1) ∩ U ′1(R)(−1) 6= ∅.

Then there exists irreducible mixed Shimura data (P ′,X ′), an embedding (P ′,X ′) →(P,X ), and a rational boundary component (P ′1,X ′1) of (P ′,X ′), such that (P1,X1) is therational boundary component of (P,X ) associated to (P ′1,X ′1) by functoriality (4.16), and

202

(a) P ′ contains U , and P ′/U is reductive,

(b) P ′1 ∩W1 = U ′1 and P ′1/U′1∼−−→ P1/W1, and

(c) (P ′/U)ad ∼= (P ′1/U′1)ad × PGL2,Q.

Proof. Without loss of generality we may assume U = 1. Since C(X 0, P1) is open,there exists an element u1 ∈ C(X 0, P1) ∩ U ′1(Q)(−1). Take a point x ∈ X 0 that maps to apoint x1 ∈ X1 whose imaginary part is u1. By definition (4.14), (int(u−1

1 ) hx1 : S→ P1,Ris defined over R. Fix a splitting P1 = W1 oG1, defined over Q. Then there exists a uniqueelement u′1 ∈ U1(R) so that int(u′1) int(u−1

1 ) hx1 factors through G1,R. After replacingx by u′1 · x, and x1 by u′1 · x1, which has the same imaginary part, int(u−1

1 ) hx1 factorsthrough G1,R.

Next, let ωx be as in 4.6. Since U = 1, is defined over R. Observe that, in the notationsof 4.3, there exists a unique nonzero element u0 ∈ U0(R)(−1) such that int(u−1

0 ) h∞ isdefined over R. The identity hx1 = ωx h∞ implies u1 = ωx(u0). Since dim(U ′1) = 1, itfollows that ωx(U0) = U ′1,R. Now, by 4.9 (b),

int(u1)(G1,C) · ωx(H0)C

is a subgroup of PC. Since u1 ∈ ωx(H0)(C), this is equal to G1,C · ωx(H0)C. As in the proofof 4.9 (a), this group has a parabolic subgroup

G1,C · U ′1,C · (ωx h0 w(Gm,C)),

which is contained in (G1 · U ′1 · Z(P ))C. Thus (G1 · U ′1 · Z(P ))C is a parabolic subgroup ofthe group

P ′C := G1,C · ωx(H0)C · Z(P )C.

In particular, some parabolic subgroup of P ′C is defined over Q. For the same reason as inthe proof of 4.9 (a), P ′C = P ′ ×Q C for a unique subgroup P ′ ⊂ P .

Now we are practically done. Defining X ′ to be the P ′(R)-orbit in X generated by x,we get an embedding (P ′,X ′) → (P,X ). There is an obvious rational boundary component(U ′1 o G1,X ′1) of (P ′,X ′). The assertions (a)–(c) hold by construction. If the (P ′,X ′) soconstructed is not irreducible, just replace it by an irreducible component. q.e.d.

Next we want to prove effectivity of descent in a certain general situation where thedescent data involves formal completions.

12.14. An analog of Cech-resolution: Let X be a locally noetherian scheme, andY a closed subscheme of X. Let X denote the formal completion of X along Y . Denotethe inclusion X := X r Y → X by j, the canonical morphism X → X by k. Since Xis locally noetherian, [B–AC] ch.III §4 thm. 3 implies that k∗OX is flat over OX . Since

j∐k : X

∐X → X is surjective, it is a faithfully flat morphism of formal schemes.

Claim. There is a canonical exact sequence

0 −→ OX −→ j∗OX ⊕ k∗OX −→ j∗OX ⊗OX k∗OX ,

with the morphisms given as s 7→ (s, s) and (s, t) 7→ s ⊗ 1 − 1 ⊗ t respectively. (If j∐k

were an open covering, this would be the Cech-resolution.)

203

Proof. The assertion is local in X, so we may assume X to be affine, say X = Spec(A).If I ⊂ A is the ideal of Y , then X is the formal spectrum of A, the I-adic completion of A.Fix an affine covering Spec(A)→ X, then by the sheaf property we have an exact sequence

0 −→ A0 := Γ(X,OX) −→ A −→ A⊗A A,

where the homomorphism on the right is given as a 7→ a ⊗ 1 − 1 ⊗ a. Since A is flat overA, A⊕ A is faithfully flat, and we get an exact sequence

0 −→ A −→ A⊕ A −→ (A⊕ A)⊗A (A⊕ A).

We shall prove the this sequence stays exact if the rightmost term is replaced by its quotientA⊗A (A⊕ A). Contemplating the commutative diagram

0 −→ A −→ A⊕ A −→ (A⊗A A) ⊕ (A⊗A A)

‖ ↑∪

↑∪

0 −→ A −→ A0 ⊕ A −→ 0 ⊕ A0 ⊗A A

we find that this will imply the exactness of the second line, as desired.

To prove the exactness of the first line K be the kernel of the map A → A. Sincesupp(K) ⊂ Y and A is noetherian, it is annihilated by some power of I, whenceK⊗AA ∼= K.Tensoring our sequence with the short exact sequence 0 → K → A → A/K → 0, a littlediagram chasing shows that it suffices to prove the assertion for A/K in place of A. In otherwords we may assume that the map A→ A is injective. In this case, by the flatness of A,the natural map A⊗A A→ A⊗A A⊗A A is also injective. Now let (a, a) ∈ A⊕ A so thata⊗ 1 = 1⊗ a in A⊗A A. In A⊗A A⊗A A we can calculate 1⊗ a⊗ 1 = a⊗ 1⊗ 1 = 1⊗ 1⊗ a,whence by injectivity a ⊗ 1 = 1 ⊗ a in A ⊗A A. This fact implies that our sequence staysexact when we drop the terms A⊗A (A⊕ A), as required. q.e.d.

12.15. Functoriality: Let us consider j : X → X ⊃ Y and k : X → X as in12.14, and another such situation with j′ : X ′ → X

′ ⊃ Y ′ and k′ : X ′ → X′. Any

morphism φ : X′ → X with φ

−1(Y ) = Y ′ induces compatible morphisms φ : X ′ → X and

φ : X ′ → X. In particular we have a commutative diagram

OX → j∗OX ⊕ k∗OX↓ ↓

φ∗OX′ → φ∗j′∗OX′ ⊕ φ∗k′∗OX′ ∼= j∗φ∗OX′ ⊕ k∗φ∗OX′

Claim. Ih φ is scheme theoretically dominant (cf. 5.6), this diagram is cartesian.

Proof. Contemplating the commutative diagram with exact rows

0 −→ OX −→ j∗OX ⊕ k∗OX −→ j∗OX ⊗OX k∗OX↓ ↓ ↓

0 −→ φ∗OX′ −→ φ∗j′∗OX′ ⊕ φ∗k′∗OX′ −→ φ∗(j

′∗OX′ ⊗OX k

′∗OX′)

we see it suffices to show that the vertical arrow on the right is injective. Again the assertionis local on X, so let A ⊃ I, A, A0, and A be as in 12.14. Let Spec(A′)→ X

′be an affine

204

covering of X′, and define A′, A′ likewise. Since A0 ⊗A A → A ⊗A A, it suffices to prove

the injectivity of the natural homomorphism

A⊗A A −→ A⊗A A′ ⊗A′ A′.

By the assumption on φ, we have A → A′. Thus, by tensoring with the A-flat A, weget A⊗A A → A⊗A A′ ∼= (A⊗AA′)⊗A′ A′. On the other hand, A⊗AA′ injects into A⊗A A′since

Spec(A⊗A A′) ∼= Spec(A)×X Spec(A′)

−→ Spec(A)×X X ′ ∼= Spec(A)×X X′ ∼= Spec(A⊗A A′)

is faithfully flat. Since A′ is flat over A′, the assertion follows. q.e.d.

12.16. Descent Lemma: Let T → S be a faithfully flat, quasicompact morphismof locally noetherian schemes. Let XT ⊂ XT ⊃ YT and XT be as in 12.14, this timeas schemes over T . Let LT be an invertible sheaf on XT , LT := LT |XT , and let LT

be the associated invertible sheaf on XT . Assume that we are given models (X,L) of(XT , LT ), and (X ,L) of (XT , LT ), over S. Consider another situation as in 12.14, this

time with schemes X ′ ⊂ X′ ⊃ Y ′, X ′ over S, and with compatible invertible sheaves

L′, L′, L′. Finally, suppose that we are given a morphism φ : X

′T → XT over T such

that φ−1

(YT ) = Y ′T , and an isomorphism ψ : φ∗LT

∼−−→ L′T . They induce isomorphisms

φ : X ′T → XT and φ : X ′T → XT∼= XT , and isomorphisms ψ : φ∗(LT ) ∼−−→ L′T and

ψ : φ∗(LT ) ∼= φ∗(LT ) ∼−−→ L′T . Let us assume that

(i) (φ, ψ) descends to S.

(ii) (φ, ψ) descends to S.

(iii) The morphism φ : X ′T → XT is scheme-theoretically dominant.

(iv) LT is relatively ample with respect to XT → T .

Claim. There exists a unique model (X,L) for (XT , LT ) over S, compatible with allthe morphisms above.

Proof. For any nonnegative integer n, we have a commutative diagram for L⊗nT , anal-

ogous to that in 12.15:

L⊗NT → j∗L

⊗nT ⊕ k∗L

⊗nT

↓ ↓φ∗L

′⊗nT → φ∗j

′∗L′⊗nT ⊕ φ∗k′∗L′⊗nT

∼= j∗L′⊗nT ⊕ k∗φ∗L′⊗nT

Since LT is locally free, 12.15 implies that this diagram is also cartesian. Denote thestructure morphisms by f : XT → T , f : XT → T , and f : XT → T respectively, andanalogously for X

′, etc. Applying f∗ to the above diagram, we obtain the diagram

f∗(L⊗nT ) → fT,∗(L

⊗nT )⊕ fT,∗(L⊗nT )

↓ ↓f′∗(L′⊗nT ) → f ′∗L

′⊗nT ⊕ f ′∗L′⊗nT

205

which is, by the left exactness of f∗, again cartesian. In particular, f∗(L⊗nT ) is uniquely

determined as a subsheaf of fT,∗(L⊗nT )⊕ fT,∗(L⊗nT ) by the bottom part and the right hand

part of this diagram. Now by our data, both these parts come from sheaves and morphismsover S. Invoking [SGA1] exp.VIII cor. 1.8, this shows that f∗(L

⊗nT ) comes from a canonical

quasi-coherent sheaf An on S.

Put A :=⊕

n≥0 AN , then its pullback to T is canonically isomorphic to the grated

algebra⊕

n≥0 f∗(L⊗nT ). By descent of homomorphisms of sheaves (see [SGA1] exp.VIII

cor. 1.2), the algebra structure descends to one on A. Put P := Proj(A), then by therelative ampleness of LT , there is a canonical open embedding XT → PT . The image ofthis embedding is the union of the images of XT and of YT , each of which is stable bythe descent data on PT . Thus, by [SGA1] exp.VIII cor. 1.9, the complement of XT in PTdescends to a closed subscheme of P , hence XT descends to an open subscheme of P , asdesired. The compatibility with the various morphisms holds by construction. q.e.d.

12.17. End of the proof of 12.4: It remains to prove 12.4 (a), and that in 12.4 (c)

the isomorphism of formal neighborhoods of Stab∆1([σ])Mπ[σ](K1f )(P1,[σ],X1,[σ]) descends to

E := E(P,X ). Let (P,X ), Kf , Σ be as in 12.4. It suffices to prove the assertion for every Σi,so we may assume Σ = Σi and Σ is finite. Then, by assumption and by 8.13, there is a line

bundle MK′fC (P ′,X ′,Σ′) → M

KfC (P,X ,Σ) so that, if M denotes the associated invertibele

sheaf,M⊗−1 is relatively ample with respect to MKfC (P,X ,Σ)→M

π(Kf )C ((P,X )/W )∗. By

7.13 we may further assume that the condition 7.12 (*) is satisfied for Σ.

We proceed by induction over the (finite) number of double classes in P (Q)\Σ/Kf .Fix pf ∈ P (Af ), and a rational boundary component (P1,X1) of (P,X ) . Let K1

f :=

P1(Af )∩pf ·Kf ·p−1f and Σ0

1 := (([ ·pf ]∗Σ)|(P1,X1))0, and fix a double coset [σ] ∈ P1(Q)\Σ0

1/K1f

such that σ0 ⊂ C(X 0, P1) for some X 0. By gluing , it suffices to prove the assertions in thecase that Σ is the smallest Kf -admissible partial cone decomposition that contains σ. LetΣ0 ⊂ Σ be the unique Kf -admissible partial cone decomposition that contains all properfaces of σ, but not σ itself. Let Σ′0 ⊂ Σ′ be analogous, then we may assume that both

MKf (P,X ,Σ0) and MK′f (P ′,X ′,Σ′0) exist. We have to prove the same for MKf (P,X ,Σ)

and MK′f (P ′,X ′,Σ′), and that the isomorphism 9.37 descends to E. This will follow byapplying 12.16. Without loss of generality, we may assume pf = 1. By 12.8, we may assumethat (P1,X1) is a proper rational boundary component of (P,X ).

In the setup of 12.16, put T := Spec(C) and S := Spec(E). As models over S we define

X := MKf (P,X ,Σ0), and let X be the formal completion of Stab∆1([σ])\MK1f (P1,X1,Σ

01)

along the stratum Y := Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X1,[σ]). We let XT := M

KfC (P,X ,Σ),

with the isomorphism XT∼= X ×S T given by 9.37. For the line bundle, fix any ample line

bundle N on Mπ(Kf )((P,X )/W )∗. Then, for some positive integer n, LT :=M⊗−1⊗N⊗nTis ample. By assumption, its restriction to XT has an obvious model over S, and the sameholds for its pullback to XT ; with MK′f (P ′,X ′,Σ′0)→MKf (P,X ,Σ0), respectively

Stab∆1([σ])\MK1′f (P ′1,X ′1,Σ′01 )→ Stab∆1([σ])\MK1

f (P1,X1,Σ01)

in place of MK′f (P ′,X ′,Σ′)→MKf (P,X ,Σ).

206

To define X′, we apply 12.13 to (P,X ) with all U ⊂ U ′1 ⊂ U1 such that σ0∩U ′1(R)(−1) 6=

∅. Denote by iα : (Pα,Xα) → (P,X ) all embedded mixed Shimura varieties thus obtained.

Denote by wβ1,f all elements of W1(Af ). For all α and β, let Kαβf be some open compact

subgroup inside Pα(Af )∩wβ1,f ·Kf ·(wβ1,f )−1, and let Σ∗αβ denote the smallest Kαβf -admissible

partial cone decomposition for (Pα,Xα) that contains Uα(R)(−1) ∩ σ · (wβ1,f )−1. By 12.11,

we can choose Kαβf and a refinement Σαβ of Σ∗αβ such that 12.4 holds for (Pα,Xα), Kαβ

f

and Σαβ. Observe that , using 12.1, the characterization in 12.13 implies E(Pα,Xα) =E(P1,X1) = E. The inverse image of XT under the map

[ ·wβ1,f ] [iα] : MKαβf

C (Pα,Xα,Σαβ)→MKfC (P,X ,Σ) = XT

is just MKαβf

C (Pα,Xα,Σαβ0) for some partial cone decompositionΣαβ0 ⊂ Σαβ. We let X′

be

the disjoint union of all MKαβf (Pα,Xα,Σαβ), with φ : X

′T → XT the disjoint union of the

above morphisms, and X ′ the disjoint union of all MKαβf (Pα,Xα,Σαβ0).

By 8.7, the pullback of the line bundle MK′fC (P ′,X ′,Σ′) → M

KfC (P,X ,Σ) is the line

bundle MKαβ′f

C (P ′α,X ′α,Σ′α) → MKαβf

C (Pα,Xα,Σαβ) for some (P ′α,X ′α), Σ′αβ and Kαβ′f , for

some α. Thus by 12.11 it has an obvious model over S. By the same argument as in 12.6,

the composite morphism [π]∗ [ ·wβ1,f ] [iα] descends to a morphism MKαβf (Pα,Xα,Σαβ)→

Mπ(Kf )((P,X )/W )∗. Using the pullback of N under this morphism, we get an obviousmodel of φ

∗(LT ) over S.

Now all objects needed in 12.16 have been defined. The conditions (i), (ii), and (iv)hold by construction. To get the desired assertion, it remains to show that the morphismφ : X ′T → XT is scheme-theoretically dominant. Recall how this morphism is defined: forall α and β, it comes by formal completion, and by taking the quotient by Stab∆1([σ]),from a morphism

[ ·wβ1,f ] [iα] : MKαβ1f

C (Pα1,Xα1,Σαβ1)→MK1f

C (P1,X1,Σ01),

where (Pα1,Xα1) is a rational boundary component of (Pα,Xα). It suffices to prove thesame assertion without taking the quotient by Stab∆1([σ]).

By 6.8–9, since (P1,X1) is a proper rational boundary component, MK1f

C (P1,X1,Σ01)

is a relative torus embedding over Mπ′(K1

f )

C ((P1,X1)/U1) with respect to the split torusGm,C ⊗ (U1(Q) ∩K1

f ). A neighborhood of our stratum is given by the affine relative torusembedding with respect to the cone λ(σ) ⊂ U1(R). Let us introduce new notation. Let Tbe the torus over C with cocharacter group R · λ(σ) ∩ U1(Q) ∩K1

f . Then

X := MK1f

C (P1,X1) −→ S := Mπ[σ](K

1f )

C (P1,[σ],X1,[σ])

is a T -torsor, and letting Xλ(σ) be as in 5.7, our formal scheme XT becomes a finite quotient

of Xλ(σ).

207

Next,

MKαβ1f

C (Pα1,Xα1,Σ0αβ1)→M

π(Kαβ1f )

C ((Pα1,Xα1)/Wα1)

is a relative torus embedding with respect to the torus with cocharcter group Uα1(Q)∩Kαβ1f .

We are only interessted in a neighborhood of those strata that map to the closed stratumof Xλ(σ); this is a torus embedding with respect to the subtorus Tαβ with cocharcter group

R · λ(σ) ∩ Uα1(Q) ∩Kαβ1f . The morphism

[ ·wβ1,f ] [iα] : MKαβ1f

C (Pα1,Xα1)→MK1f

C (P1,X1,Σ01)

is still defined if we replace Kαβ1f by Kαβ1

f · (R · λ(σ) ∩ Uα1(Q) ∩ K1f ), which corresponds

to dividing by a finite subgroup of Tαβ. Then Tαβ becomes canonically identified with asubtorus of T . Our neighborhood is the torus embedding with respect to some partial conedecompositions of R · λ(σ) ∩ Uα1(R) with support λ(σ) ∩ Uα1(R). Let τγαβ denote all those

cones in this decomposition for which (τγαβ)0 ⊂ λ(σ)0. Since W1 centralizes U1, we find

that neither Tαβ ⊂ T , nor the set of these τγαβ depends on wβ1,f . Therefore we may write

Tα and τγα . Fixing α and γ, we now let T γα ⊂ Tα be the subtorus with cocharacter groupR · τγα ∩ Uα1(Q) ∩K1

f . We let Xγα → Sγα be the disjoint union of the T γα -torsors

MKαβ1f

C (Pα1,Xα1)→Mπ

[τγα ]

(Kαβ1f )

C (Pα1,[τγα ],Xα1,[τγα ])

for all β. The maps [ ·wβ1,f ] [iα] induce a T γα -equivariant morphism (Ψγα, ψ

γα) : (Xγ

α, Sγα)→

(X,S). Since the induced morphism of the formal completions Ψγα : Xγ

α,τγα→ Xλ(σ) factors

through the morphism which we have to prove to be scheme-theoretically dominant, itsuffices to prove that the disjoint union of all Ψγ

α is scheme-theoretically dominant.

This will follow by applying 5.7. In fact, conditions (i) and (ii) of 5.7 hold by con-struction. To finish, it therefore remains to prove that every ψγα : Sγα → S is scheme-theoretically dominant. Since S is a normal scheme, it suffices to show that its imageis Zariski-dense. Recall that by 12.13 (b), Pα1 has the same reduced part as P1. ThusP1(Af ) = Pα1(Af ) ·W1(Af ), which by the definition of ψγα, implies that its image is thesame as the union of the images of the maps

[ ·p1,[σ],f ] [iα] : MKα1∗f

C (Pα1,[τγα ],Xα1,[τγα ]) −→Mπ[σ](K

1f )

C (P1,[σ],X1,[σ])

for all p1,[σ],f ∈ P1,[σ](Af ), and some Kα1∗f for which these morphisms are defined. By 11.7

this is Zariski-dense, as desired. This finishes the proof of 12.3 and of 12.4. q.e.d.

As an application of 12.4, we study q-expansions in our setting.

12.18. Completions and coherent sheaves: Consider irreducible mixed Shimuradata (P,X ), a neat open compact subgroup Kf ⊂ P (Af ), a Kf -admissible cone decomposi-tion Σ for (P,X ). Assume that Σ is concentrated in the unipotent fibre, and that there existsa morphism (P,X )→ (Gm,Q,H0). Then by 6.8–9 (and, of course, 12.4) MKf (P,X ,Σ) is a

208

relative torus embedding with respect to a split torus Gm⊗ΓU and a rational partial polyhe-dral cone decomposition ΣU of U(R). Fix a coset [σ] ∈ Σ/P (Af ), and assume that Σ is thesmallest Kf -admissible cone decomposition containing σ. Then ΣU consists of all faces ofthe single cone λ(σ), for λ associated to the connected component X 0 with σ ∈ Σ(P,X 0, pf ).Letting T := Gm ⊗ (ΓU ∩ R · σ), we have R · σ = Y∗(T )R, and MKf (P,X ,Σ) is also a torusembedding with respect to T . The closed stratum is just Mπ[σ](Kf )(P[σ],X[σ]) in the nota-

tion of 7.1. Let us denote the canonical projection MKf (P,X ,Σ) → Mπ[σ](Kf )(P[σ],X[σ])by πσ.

Every character χ ∈ X∗(T ) = Hom(ΓU ∩ R · λ(σ),Z) induces a morphism U(Q) ∩ R ·λ(σ)/Q · ker(χ) ∼= Q = U0(Q), and so defines the structure of a ((P0,X0) → (Gm,Q,H0))-torsor on (P,X )/〈ker(χ)〉 → (P[σ],X[σ]). This defines a Gm-torsor structure on

M (Kf mod〈ker(χ)〉)((P,X )/〈ker(χ)〉) −→Mπ[σ](Kf )(P[σ],X[σ]).

Let us denote the associated invertible sheaf by Lχ. By 5.13, the direct image under π0

of the structure sheaf of MKf (P,X ,Σ) is the direct sum of all those Lχ, for which χ isnonnegative on λ(σ). Denote by M the formal completion of MKf (P,X ,Σ) along theclosed stratum, and by πσ : M → Mπ[σ](Kf )(P[σ],X[σ]) the retraction induced by π0. By5.13, we have a canonical isomorphism

πσ,∗OM ∼=∏χ

Lχ,

the sum extended over the same χ ∈ X∗(T ).

More generally, let F be a locally free coherent sheaf on MKf (P,X ,Σ), and F the asso-ciated coherent sheaf on M . Assume that we are given a coherent sheaf G onMπ[σ](Kf )(P[σ],X[σ])

and an isomorphism F ∼= π∗σG, whence an isomorphism F ∼= πσ,∗G. Then we have canonicalisomorphisms

πσ,∗F ∼=⊕χ

Lχ ⊗ G and πσ,∗F ∼=∏χ

Lχ ⊗ G.

In particular, for any E = E(P,X )-algebra R the given data determines an isomorphism

Γ(M ×E Spec(R), F) ∼=∏χ

Γ(Mπ[σ](Kf )(P[σ],X[σ])×E Spec(R),Lχ ⊗ G).

12.19. Definition of q-expansions: Let MKf (P,X ,Σ) be a toroidal compactifica-

tion; for simplicity we assume thatKf is neat. Fix a stratum Stab∆1([σ])\Mπ[σ](K1f )(P1,[σ],X1,[σ])

of MKf (P,X ,Σ) rMKf (P,X ). Let M1 denote the formal completion of MK1f (P1,X1,Σ

01)

along Mπ[σ](K1f )(P1,[σ],X1,[σ]), and π1 : M1 → Mπ[σ](K

1f )(P1,[σ],X1,[σ]) the retraction, as in

12.18. Let F be a locally free coherent sheaf on MKf (P,X ,Σ), and F1 its pullback to M1

under the morphism 12.4. Assume that we are given an isomorphim F1∼= π∗1G1, for some

locally free sheaf G1 on Mπ[σ](K1f )(P1,[σ],X1,[σ]). Let E = E(P,X ) , and R an E-algebra. By

pullback of sections and by 12.18, this data determines a map

Γ(MKf (P,X ,Σ)×E Spec(R),F)

−→∏χ

Γ(Mπ[σ](K1f )(P1,[σ],X1,[σ])×E Spec(R),Lχ ⊗ G1),

209

the product being extended over all χ as in 12.18. This map can be interpreted as associatingto each section of F its “q-expansion coefficients”.

For certain F this can be made more explicit. Assume first that F is defined interms of the internal geometry of our toroidal compactifications: for instance in terms ofsheaves of differentials possibly of logarithmic poles (compare 8.1), for the given (P,X )or for other mixed Shimura varieties, and/or in terms of line bundles in 8.6. Then the

same definition gives rise to a coherent sheaf F1 on MK1f (P1,X1,Σ

01), and 12.4 (c) yields a

canonical isomorphism between F1 and the coherent sheaf on M1 associated to F1. Next,by 8.1 a canonical G1 exists in the case F = ω[dlog]. By functoriality, the same followsfor any other sheaf defined in terms of ω[dlog] for the given (P,X ) or for other mixedShimura varieties. In particular, for F = ω[dlog]⊗n and Σ complete, the above definitionof q-expanison applies to the space

Γ(MKf (P,X ,Σ)×E Spec(R), ω[dlog]⊗n)

of all R-valued automorphic forms f of a certain weight. The “q-expanison” of f is a (in

general infinite) collection of R-valued automorphic forms f1,α on Mπ[σ](K1f )(P1,[σ],X1,[σ])

of certain other weights. These automorphic forms are generalizations of Jacobi modularforms; the condition at infinity being played by extendability to some toroidal compactifi-cation.

12.20. A q-expansion principle for E(P,X )-rationality: In the situation of

12.19, let us now consider a finite number of strata Stab∆i([σi])\Mπ[σi]

(Kif )(Pi,[σi],Xi,[σi]),

and write Mi, Fi, πi accordingly. We assume that the map∐i

Stab∆i([σi])\Mπ[σi]

(Kif )(Pi,[σi],Xi,[σi])(C) −→MKf (P,X ,Σ)(C)

induces a surjection on the sets of connected components. Then, since MKf (P,X ,Σ) is anormal scheme, our assumption implies that pullback of sections induces injections

Γ(MKf (P,X ,Σ),F) →⊕

i Γ(Mi, Fi)∩↓

∩↓

Γ(MKfC (P,X ,Σ),F) →

⊕i Γ(Mi,C, Fi)

Clearly this diagram is cartesian, i.e. Γ(MKf (P,X ,Σ),F) consists of those sections in

Γ(MKfC (P,X ,Σ),F) whose pullback to every Mi,C is already defined over E. If for every

i we are given a product decomposition πi,∗(Fi) ∼=∏α(Gi ⊗ Li,α), then this means that

a section in Γ(MKfC (P,X ,Σ),F) is defined over E if and only if every “coefficient in the

q-expanion” is defined over E, i.e. lies in

Γ(Mπ[σi](Ki

f )(Pi,[σi],Xi,[σi]),G1 ⊗ CL1,α).

Sometimes the above surjectivity condition already holds for just one stratum. This is so forinstance if (P,X ) is irreducible, the semisimple part, (P/W )der, of P is simply connected,

210

and there exists an algebraic “ local cross-section” (P/W )/(P/W )der → P1. Indeed, underthese circumstances we have P (A) = P1(A) · P der(A), and the surjectivity follows from 3.9.

Remark. The same result has been obtained by M. Harris for arbitrary “arithmeticvector bundles” on pure Shimura varieties, in terms of the Baily-Borel compactification (see[Ha1],[Ha2]).

12.21. Example: Hilbert-modular varieties: Fix a totally real number field F .Let P := RF/QGL2,F , and X ∼= h(X ) be the P (R)-orbit generated by

h : S(R) ∼= C× −→ P (R) ∼= (GL2(R))[F/Q],

x+√−1y 7→ (

(x y−y x), . . . ,

(x y−y x)).

This orbit does not depend on the choice of√−1 (of course, using such a choice it can be

identified with (C r R)[F/Q]). The reflex field of (P,X ) is Q, not F ! There is precisely oneconjugacy class of proper rational boundary components of (P,X ), e.g. with representative

P1 :=

(Gm,Q

0

RF/QGa,F1

).

Here every hx1 for x1 ∈ X1 is of the form C× 3 z 7→(zz ∗0 1

), and (P1,X1) is isomorphic to

the [F/Q]-fold fibre product of (P0,X0) of 2.24 with itself over (Gm,Q,H0). In particular,(P1,X1)/W1

∼= (Gm,Q,H0).

It follows from 12.3 that the Baily-Borel compactification MKf (P,X )∗, the boundary

is a finite disjoint union of certain MK0f (Gm,Q,H0). In particular, the field of definition of

every geometric point of the boundary, i.e. of MKf (P,X )∗(C)rMKf (P,X )(C), is an abelianextension of Q. Of course, this field of definition may be larger than the field of definition ofthe associated connected component of MKf (P,X )∗(C). In the toroidal compactification, itis well-known how the arithmetic of F comes into play. This is described in M. Rapoport’sthesis [Rap], in a strictly more general setting (i.e. not restricted to characteristic zero).

12.22. Example: The group CU(n, 1): Let E ⊂ C be an imaginary quadraticfield (with a fixed embedding into C), V an E-vector space of dimesion n + 1 ≥ 3, andH : V × V → E a nondegenerate hermitian form that becomes of type (n, 1) over C. Welet P be the group of unitary similitudes of V with respect to H, i.e.

P := g ∈ RE/QGLE(V ) | H(gv, gv′) = t ·H(v, v′) for some t ∈ RE/QGm,E.

Every homomorphism h : S→ RC/RGLC(V ⊗E C) such that V ⊗E C becomes of Hodge type(−1, 0), (0,−1) induces another structure of C-vector space on V ⊗E C which commuteswith the old C-structure. Thus it defines a decomposition V ⊗E C = V + ⊕ V −, where V +

(resp. V −) is the subspace where the two C-structures coincide (resp. differ by complexconjugation). If we further require that V + and V − are orthogonal with respect to H andthat H|V − is negative definite (whence dim(V −) = 1, dim(V +) = n, and H|V + is positivedefinite), then h must factor through PR. The set X of all such h constitudes a P (R)-orbit,and (P,X ) is pure Shimura data. Since n ≥ 2, P (R) and hence X are connected. The reflexfield of (P,X ) is E.

211

Every maximal proper Q-parabolic subgroup Q ⊂ P is the normalizer of a nontrivialisotropic subspace V ′ ⊂ V . In the given case, we must have dim(V ′) = 1, and Q stabilizes(V ′)⊥ which is of codimension 1. If (P1,X1) is the rational boundary component of (P,X ),then for every x1 ∈ X1 and z ∈ S(R) = C×, hx1(z) acts on V ′⊗EC through multiplication byzz, on ((V ′)⊥/V ′)⊗EC through multiplication by z, and trivially on (V/(V ′)⊥)⊗EC. Sincethe semisimple part of Q is totally anisotropic over R, P1 is solvable, and, in fact, P1/W1

∼=RE/QGm,E . Since X is connected, we have |X1| = 1, and the associated homomorphism isthe isomorphism

S = RC/RGm,C ∼= (RE/QGm,E)×E C.

Moreover dim(U1) = 1, and dim(V1) = 2(n− 1).

By 12.3, the boundary MKf (P,X )∗ rMKf (P,X ) of the Baily-Borel compactification

is a finite disjoint union of certain Mπ(K1f )((P1,X1)/W1). In particular, the field of def-

inition of every geometric point of the boundary is an abelian extension of E. Sincedim(U1) = 1, there is a unique complete decomposition Σ for (P,X ). By 12.4, theboundary of the associated toroidal compactification is a finite disjoint union of certain

Mπ′(K1f )((P1,X1)/U1). The family Mπ′(K1

f )((P1,X1)/U1) → Mπ(K1f )((P1,X1)/W1) is a tor-

sor under a family of abelian varieties of dimension n − 1, isogenous to the (n − 1)-foldproduct with itself of an elliptic curve with complex multiplication by E. By 12.4 (c) the

normal bundle of Mπ′(K1f )((P1,X1)/U1) in MKf (P,X ,Σ) is isomorphic to the line bundle

MK1f (P1,X1,Σ1)→Mπ′(K1

f )((P1,X1)/U1). The coefficients of the q-expansion of a modularform

f ∈ Γ(MKf (P,X ,Σ), ω[dlog]⊗n)

are sections of all nonpositive powers of the invertible sheaf associated to this line bundle.(Since the stratum is blown down in the Baily-Borel compactification, the normal bundleis anti-ample: which explains in another way why we have the nonpositive powers of thisinvertible sheaf.)

Since V is of Hodge type (−1, 0), (0,−1), we can also consider the mixed Shimuradata V o (P,X ). The resulting family of abelian varieties inherits multiplication by E,which leads to a modular interpretation of MKf (P,X ). A suitable extension of this familyin terms of toroidal compactification can, as in 10.17–10.22, be described as degenerationof such abelian varieties. All these structures can, of course, be described in terms ofmorphisms in 12.4.

212

Bibliography

[AMRT] A. Ash, D. Mumford, M. Rapoport, Y. Tai, Smooth compactifiction of locallysymmetric varieties. Lie groups: History, Frontiers and Applications. Math. Sci.Press, Brookline, Mass. (1975)

[B] A. Borel, Indroduction aux groupes arithmetiques. Hermann, Paris (1969)

[Br] J.-L. Brylinski, “1-motifs” et formes automorphes (theorie arithmetique des do-maines de Siegel), prepublication du Centre de Math. de l’Ecole Polytechnique,(1979)

[BB] W. L. Baily, A. Borel, Compactification of arithmetic quotients of bounded sym-metric domains. Annals of Math. (2) 84 (1966), 442–528

[BPV] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Springer, Berlin(1984)

[BT] A. Borel, J Tits, Groupes reductifs. Publ. Math. IHES 27 (1965), 55–150

[B-AG] N. Bourbaki, Algebre commutative, Hermann, Paris (1961)

[B-Lie] N. Bourbaki, Groupes et algebres de Lie, IV–VI: Hermann, Paris (1968), VII–VIII:Paris (1975)

[C] C.-L. Chai, Compactification of Siegel moduli schemes, London Math. Soc. Lec-ture Notes Series 107, Cambridge Univ. Press (1985)

[CK] E. Cattani, A. Kaplan, On the SL2-orbits in Hodge theory, Preprints IHES/M/82/58(1982)

[D1] P. Deligne, Travaux de Shimura. Seminaire Bourbaki, Exp. n0 389, SLN 244(1971), 123–165

[D2] P. Deligne, Varieties de Shimura: Interpretation modulaire, et techniques de con-struction de modeles canoniques. Proc. Sympos. Pure Math. AMS, 33 part 2(1979), 247–290

[D3] P. Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58

[DR] P. Deligne, M. Rapoport. Les schemas de modules de courbes elliptiques. Modularfunctions of one variable, II, SLN 349 (1972), 143–316

[EGA-I] A. Grothendieck, J. A. Dieudonne, Elements de geometrie algebrique I, SpringerGrundlehren 166, Berlin (1971)

[EGA-II]A. Grothendieck, J. A. Dieudonne, Elements de geometrie algebrique II, Publ.Math. IHES 8, (1961)

[F1] G. Faltings, Arithmetic varieties and rigidity, in: Seminaire de theorie de nombresde Paris, 1982-83, Birkhauser, Boston (1984), 63–77

[F2] G. Faltings, Arithmetische Kompaktifizierung des Modulraumes der abelschen Va-rietaten, in Proceedings Arbeitstagung Bonn 1984, SLN 1111 (1985), 321–383

[G] R. Godement, Domaines fondamentaux des groupes arithmetiques, Sem. Bour-baki, exp. 257 (1963)

[GR] H. Grauert, R. Remmert, Coherent analytic sheaves, Springer Grundlehren 265,Berlin (1984)

213

[Ha1] M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varietiesI, Inventiones Math. 82 (1985) 151–189,

[Ha2] M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varietiesII, Compositio Math. 60 (1986) 323–378

[Ha3] M. Harris, Functorial properties of toroidal compactifications of locally symmetricvarieties, to appear in Proc. London Math. Soc.

[H1] R. Hartshorne, Ample subvarieties of algebraic varieties, SLN 156 (1970)

[H2] R. Hartshorne, Algebraic geometry, Springer GTM 52 (1977)

[KKMS] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings I, SLN339 (1973)

[K] D. Knutson, Algebraic spaces, SLN 203 (1971)

[Mi1] J. S. Milne, The action of an automorphism of C on a Shimura variety and itsspecial points. Arithmetic and Geometry, Pap. dedic. I. R. Shafarevich, Vol. I,Prog. Math, 35 (1983), 239–265

[Mi2] J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vectorbundles, to appear.

[M1] D. Mumford, Abelian varieties, Oxford University Press (1970)

[M2] D. Mumford, On the equations defining abelian varieties, I-III, Inventiones Math.1 (1966), 287–354; 3 (1967), 75–135 and 215–244

[M3] D. Mumford, J. Fogarty, Geometric invariant theory, Springer, Berlin (1982)

[M4] D. Mumford, An analytic construction of degenerating abelian varieties over com-plete rings, Compositio Math. 24 (1972), 239–272

[N] Y. Namikawa, Toroidal compactification of Siegel spaces, SLN 812 (1980)

[Rag] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer, Berlin (1972)

[Rap] M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal,Compositio Math. 36, no.3 (1978), 255–335

[SAG1] A. Grothendieck et al., Revetements etales et groupe fondamental. Seminaire degeometrie algebrique du Bois-Marie 1960–61, SLN 224 (1971)

[S] J.-P. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier 6(1955–56) 1–42

[Sh] G. Shimura, Indroduction to the arithmetic theory of automorphic functions,Princeton Univ. Press (1971)

[Sp] E. Spanier, Algebraic topology, McGraw-Hill (1966)

214

List of frequently used symbols

A, Af , AK , AK,f adele ring 0.2AdG, adG adjoint operation 0.3A[d] group of d-division points 10.1B(X 0), B(X 0, P1) set of commutators 8.11β(P1,X1, P

′1,X ′1, pf ), β(P1,X1, P

′1,X ′1, pf )

certain maps in U , U 6.11, 6.15C set of complex numbers 0.2C(P,X ) conical complex 4.24C(X 0, P1) open cone associated to boundary component 4.15C∗(X 0, P1) union of all boundary components of C(X 0, P1) 4.22(CSP2g,H2g) Shimura data for the symplectic group 2.7∆1 normalizer of a boundary stratum in the Baily-

Borel compactification 6.3, 6.18, 7.3eσ splitting associated to a cone σ 5.11, 8.5expL subgroup associated to sub-Lie algebra 0.3E(P,X ) refelx field 11.1e(X) canonical pairing H(X)×H(X)→ Gm 10.2E(X ′0, P ′1) set of splittings 9.2FPMC Hodge filtration 1.1G semisimple part P/W 2.1Gad, Gder adjoint, derived group 0.3(Gm,Q,H0), (Gm,Q, h(H0))

standard Shimura data 2.8h homomorphism SC → PC 1.4h equivariant map X → Hom(SC, PC) 2.1H0, h0, h∞ reference group and homomorphisms 4.3H(X) group of translations normalizing X 10.1HW conjugacy class of h : SC → PC 1.6im imaginary part 4.14intG interior automorphism 0.3int(p) interior automorphism of (P,X ) 3.5Kf (d), KP

f (d), KUf (d), KW

f (d)special open compact subgroups 10.7, 10.15

LP canonical invertible sheaf on MKf (P,X )∗(C) 8.1Lχ invertible sheaf occuring in q-expansion 12.18λ isomorphism Z→ Z(1) 2.8, 3.16, 8.5Mp,q Hodge decomposition 1.1M(n) Tate twist 1.11

MKf (P,X )(C), MKfC (P,X ), MKf (P,X )

(mixed) Shimura variety 3.1, 9.25, 11.5

MKf (P,X )∗(C), MKfC (P,X )∗, MKf (P,X )∗

Baily-Borel compactification 6.2, 9.25, 12.3

MKf (P,X )(C), MKfC (P,X ), MKf (P,X )

215

toroidal compactification 6.24, 9.25, 9.34, 12.4–5MKf (P,X )+(C) union of MKf (P,X )(C) with all boundary strata of

codimension 1 8.2M(d), MV (d), MW (d), M0(d), M0

U (d)special mixed Shimura varieties 10.7, 10.15

Md moduli scheme of abelian varieties 10.6, 11.16M0

d, X0d “moduli scheme” of roots of unity 10.15

µ canonical cocharacter of S 1.3ordT homomorphism T (C)→ Y∗(T )R 5.8ωTX/S sheaf of invariant differentials 5.26

ω[dlog], ωXΣ[dlog], ωXΣ

/S[dlog]sheaf of differentials with at most logarithmicpoles along the boundary 5.26, 8.1

(P,X ), (P,X , h) (mixed) Shimura data 2.1(P,X )/P0 quotient mixed Shimura data 2.9(P1,X1)×(P,X ) (P2,X2)

fibre product 2.20(P0,X0) unipotent extension of (Gm,Q,H0) 2.24(P2g,X2g) unipotent extension of (CSP2g,H2g) 2.25(P[σ],X[σ]) mixed Shimura data associated to [σ] 7.1π the positive real number 0.2π, π′ projections P → G = P/W, P → P/U 2.1π0(X) set of connected components 0.1πσ canonical projection Tσ → Tσ 5.2π[σ] projection (P,X )→ (P[σ],X[σ]) 7.1[π]∗ projection to Baily-Borel compactification 6.24, 9.25, 12.4Q admissible Q-parabolic subgroup 4.5RL/KX Weil restriction 0.2R set of real numbers 0.2S, S1 Deligne torus 1.3Σ, Σ(X 0, P1, pf ) admissible (partial) cone decomposition 6.4Σ0 subset of all cones along the unipotent fibre 6.5Σ|(P1,X1) restriction to a boundary component 6.5Σ[σ] induced cone decomposition 7.7Σ(f) cone decomposition associated to a piecewise linear

convex rational function 5.20Σ+

Σ , Σ′Σ, ΣΣ cone decompositions associated to Σ 9.12|Σ| support of a cone decomposition 5.1σ0 interior of a cone 5.1σ0 standard cone 5.10, 8.5, 9.1σ dual cone 5.1[σ] double coset of cones 7.1Σ locally polyhedral subset 9.8TΣ torus embedding 5.3Tσ affine torus embedding 5.2

216

Tσ orbit in a torus embedding 5.2U weight -2 subgroup of P 2.1U , U(P1,X1, pf ) covering of MKf (P,X )(C) 6.10U , U(P1,X1, pf ) covering of MKf (P,X ,Σ)(C) 6.13V weight -1 subquotient of P 2.1V [d], U [d] standard Z/dZ-modules 10.3, 10.7w “weight” of S 1.3W unipotent radical of P 2.1WnM weight filtration 1.1Xd → Ad →Md universal family of abelian varieties 10.10, 11.16XΣ relative torus embedding 5.5X∗(T ) charcter group of a torus T 5.2X+ open subset of X that maps to X1 4.11X ∗ union of hermitain symmetric domain with all its

rational boundary components 6.2Y∗(T ) cocharacter group of torus T 5.2Z(n) Tate Hodge structure 1.11

∼ equivalence relation on U or U 6.10, 6.16√−1 the complex number 0.2

[ , ] commutator 0.3[ ·pf ], [φ] morphisms of mixed Shimura varieties

3.4, 6.2, 6.25, 9.25, 12.3, 12.4[ ·pf ]∗Σ, φ∗Σ pullback of admissible cone decomposition 6.5

217

Index

abelian scheme 10.1admissible complete cone decomposition 6.4admissible partial cone decomposition 6.4admissible Q-parabolic subgroup 4.5almost (semi-)direct product 0.3arithmetic subgroup 0.5Baily-Borel compactification 6.2barycentric subdivision 5.24canonical model 11.5canonical projection 5.2canonical sheaf 5.26, 8.1class field isomorphism 11.3complete rational polyhedral decomposition 5.1concentrated in the unipotent fibre 6.5congruence subgroup 0.5conical complex 4.24convex rational polyhedral cone 5.1core 6.1cyclotomic character 11.3Deligne-torus 1.3embedding of mixed Shimura data 2.3face of a cone 5.1fibre product 2.20, 3.10finite admissible cone decomposition 6.4Gm-torsor 3.17, 5.10, 8.5, 10.1generalized elliptic curve 10.17Hilbert modular variety 12.21Hodge structure 1.1imaginary part 4.14improper rational boundary component 4.11irrducible mixed Shimura data 2.13locally polyhedral 9.8mixed Shimura data 2.1mixed Shimura variety 3.1moduli scheme 10.6, 10.15, 10.20morphism of mixed Shimura data 2.3neat subgroup 05.–6normalized Gm-torsor 10.4parabolic subgroup 4.1piecewise linear convex rational function 5.20polarization of Hodge structures 1.11proper rational boundary component 4.11properly discontinous 0.4pure Shimura data 2.1

218

pure Shimura variety 3.1q-expansion 12.19quotient of mixed Shimura data 2.9rational boundary component 4.11rational partial polyhedral decomposition 5.1reciprocity law 11.4reductive group 0.3refinement 5.1reflex field 11.1Satake topology 6.2scheme-theoretically dominant 5.6Shimura data 2.1Shimura variety 3.1Siegel modular variety 10.7smooth admissible cone decomposition 6.4smooth cone (decomposition) 5.2–3strictly convex 5.16symplectic d-structure 10.3Tai’s theorem 8.15Tate curve 10.17toroidal compactification 6.24torus embedding 5.2–5totally symmetric Gm-torsor 10.4unipotent extension 2.17variation of Hodge structures 1.9Weil restriction 0.2

219