ON THE COHOMOLOGY OF REDUCTIVE BOREL-SERRE

ON THE COHOMOLOGY OF REDUCTIVE BOREL-SERRECOMPACTIFICATIONS

ARVIND N. NAIR

Abstract. We prove new results on the cohomology of the reductive Borel-Serrecompactification of a noncompact locally symmetric variety by a mixture of analyticand geometric methods. We show that the Hecke-invariant part of the cohomologyis a summand given by the cohomology of the compact dual. We show that thissummand is given by Chern classes of automorphic vector bundles. These resultsare used to prove Lefschetz properties for the cohomology of reductive Borel-Serrecompactifications with respect to restriction maps.

The reductive Borel-Serre (RBS) compactification M of a noncompact locally sym-metric space M = Γ\G(R)/KR was introduced by Zucker in [34] by modifying theoriginal construction of Borel and Serre [7], and has found extensive use since. Itscohomology ring H∗(M) (taken with coefficients Q or C, according to the context)enjoys a number of good properties: It carries a fundamental class in top degree. Thesum of hypercohomology groups of any complex of sheaves on M (e.g., the weightedcomplexes of [12]) is a module over H∗(M). It has an action of the Hecke algebraat level Γ. It has a de Rham description in terms of a complex of C∞ differentialforms on M or in terms of relative Lie algebra cohomology ([23]). The Poincare dualgroups W dH∗(M) ∼= HdimM−∗(M) can be computed using automorphic forms for Γ([23, 10]). When D = G(R)/KR is a Hermitian symmetric space, so that M has analgebraic structure (it is a connected component of a Shimura variety at finite level),H∗(M) carries a natural mixed Hodge-de Rham structure with weights like those ofa complete variety ([36, 25] and §2), and is even motivic ([2]).

In this note we prove some new results on the cohomology of reductive Borel-Serre compactifications by a mixture of analytic methods (automorphic forms) andgeometric methods (Hodge theory).

In §1 we study H∗(M) using analytic methods. The C∞ de Rham model for thecohomology groups H∗(M) and for the dual cohomology groups W dH∗(M) given by[23] is the following: There are (g, KR)-modules B ⊂ R of functions on Γ\G(R) suchthat

H∗(M) = H∗(g, KR, B) (0.0.1)

W dH∗(M) = H∗(g, KR, R). (0.0.2)

The space B consists of functions which are bounded up to certain logarithmic terms,cf. 1.2. (The relative Lie algebra complexes for B and R are identified with certaincomplexes of differential forms on M .) There are inclusions C ⊂ B ⊂ R where C is

1

2 ARVIND NAIR

the space of constant functions, so that if D is the compact symmetric space dual toD there are maps

H∗(D)→ H∗(M)→ W dH∗(M).

The fundamental result of Franke [10] allows us to replace R by a certain space ofautomorphic forms FinΘR in (0.0.2). Franke’s filtration on FinΘR allows us to showthat H∗(D) is a direct summand (as a Hecke module) of W dH∗(M) and there areno other Hecke-trivial constituents. By duality we conclude that H∗(M) containsH∗(D) as a direct summand and has no other Hecke-trivial constituents (Theorem1.4.1). These results apply to arbitrary locally symmetric spaces.

From §2 onwards we assume that M is a locally symmetric variety, i.e D has aHermitian structure. In §2 we recall geometric results from [25, 26] that we use in§§3,4. Let M∗ be the minimal compactification (of Satake and Baily-Borel [4]) andlet MΣ be a smooth projective toroidal compactification of M as in [1]. We havemappings

Mp−→M∗ π←−MΣ.

In [26] it is shown that the direct image Rp∗QM can be realized via a truncation (inthe sense of Morel [20]) of the object Rπ∗QH

MΣin Saito’s derived category of mixed

Hodge modules on M∗. This puts a mixed Hodge structure on H∗(M) (cf. also[36]). It also shows that the pullback π∗ : H∗(M∗) → H∗(MΣ) factors through amap γ∗ : H∗(M)→ H∗(MΣ), which was earlier constructed by Goresky and Tai [14].These statements also hold in the setting of Hodge-de Rham structures by takingadvantage of the fact that M∗,MΣ and π : MΣ → M∗ can be defined over a numberfield k. We can thus define the algebraic de Rham cohomology H∗dR(M/k) over thefield of definition, with a comparison isomorphismH∗dR(M/k)⊗kC = H∗(M,C). Sincethe action of Hecke correspondences is by morphisms of HdR structures we concludefrom §1 that the Hodge-de Rham structure on H∗(M) contains a direct summandwith underlying C-vector space H∗(D), namely the space of Hecke-invariants. (1)

In §3 we show that the space of invariants H∗(D) in H∗(M) is spanned by Chernclasses of automorphic vector bundles. The Chern forms of an invariant connection inan automorphic vector bundle are bounded with respect to an invariant metric; suchbounded differential forms belong to the standard complex computing H∗(g, KR, B)and so we have classes in H∗(M) which we will call Chern classes. An alternatedefinition of these classes is given by pulling back the Chern classes in H∗(M∗) definedby Goresky and Pardon in [13]. We prove several properties of these classes onM : They are uniquely characterized by invariance under Hecke operators and theirpullback to a smooth toroidal compactification MΣ. They are rational for the Bettirational structure (given by H∗(M,Q)) and (up to some powers of 2πi) for the deRham rational structure (given by H∗dR(M/k)). They generate the direct summandH∗(D) of invariants in H∗(M) from §1. (These properties are summarized in Theorem

1Some of the results in §2 can also be deduced from [2], which became available after the firstversion of this paper was written in 2009. We use the results of [26, 25] as they are (in comparisonwith [2]) elementary.

ON THE COHOMOLOGY OF RBS COMPACTIFICATIONS 3

3.2.1. They answer, on M , questions raised by Goresky and Pardon in [13] for theirChern classes on M∗.) The main point is that the results of §1 show that the pullbackmap γ∗ : H∗(M) → H∗(MΣ) to a suitable toroidal compactification is injective onthe summand of invariants. We then conclude properties of these classes on M fromtheir properties on MΣ (which are known by work of Mumford [21] and Harris [17]).

In §4 we apply the results of §2 and §3 to the situation of a semisimple subgroupH ⊂ G and the induced map f : MH →M . The RBS construction is not functorial,i.e. f does not usually extend continuously to MH → M . Nevertheless, there is anatural homomorphism

f ∗ : H∗(M)→ H∗(MH)

with the properties one would expect of a pullback. This is easily seen in the C∞

de Rham model (0.0.1) and does not require Hermitian structure for M or MH . Wegive another construction in the Hermitian case, i.e. f : MH → M is a morphismof locally symmetric varieties, using the results recalled in §2. This shows that f ∗

is a morphism of mixed Hodge-de Rham structures and has other good properties(Theorem 4.1.1). There is a dual Gysin map f! : W dH∗(MH) → W dH∗+2c(M)(c)(here c = dimCM − dimCMH) and we can define a cycle class

ξ := f!(1) ∈ W dH2c(M)(c).

We show that a suitable linear combination of Hecke translates of ξ gives the class ofthe compact dual DH in H2c(D)(c) under the embedding H∗(D) ⊂ H∗(M) (Theorem4.2.1). (This is the analogue of a result of Venkataramana [31, Theorem 1] in thecompact case and given the results of §1-§3 the same proof works.) In particular,ξ 6= 0. The results of §§1–3 and §§4.1, 4.2 allow us to adapt the methods of [31] fromthe compact case in a straightforward way to the top weight quotient ⊕iGrWi H i(M)(which is isomorphic to the image of γ∗ : H∗(M) → H∗(MΣ)). There is a Lefschetzproperty (i.e. an injectivity property for f ∗,) for this space in the cases where f(MH)has codimension one (Theorem 4.3.2) and for a certain subspace of the cohomologyH∗(M) (Corollary 4.3.3). There is a nonvanishing result for cup products in certainunitary and orthogonal cases (Theorem 4.4.1).

I thank N. Fakhruddin, M. Goresky, V. Vaish, and T. N. Venkataramana for helpfuldiscussions, and S. Zucker for a conversation about [36] at Banff in May 2008, whichgot me thinking about M again. I thank the Institute for Advanced Study, Princetonfor its support and hospitality in the Fall of 2008, when much of this work was done.This work was partially supported by Swarnajayanti Fellowship DST/SF/05/2006(2008-2013) from the Department of Science and Technology of the Government ofIndia.

1. (Co)Invariants in cohomology

Cohomology groups in this section have coefficients C.

1.1. Notation. Fix an isotropic connected semisimple Q-group G. Let KR be amaximal compact subgroup of G(R)0. The Lie algebra of G(R) will be denoted by gand its complexification by gC.

4 ARVIND NAIR

Fix a maximal Q-split torus A0 in G such that A0(R) is stable under the Cartaninvolution of G(R)0 given by KR. Let M0 be the centralizer of A0 in G; this is aminimal Levi subgroup. For any standard Levi subgroup M ⊃ M0 the split centreis a torus AM ⊂ A0 and this gives the dual vector spaces aM = LieAM(R) andaM = X∗(M) ⊗ R = X∗(AM) ⊗ R. Restriction of characters by M0 ⊂ M gives anembedding aM ⊂ a0. Restriction by AM ⊂ A0 gives a projection a0 aM inverse toaM ⊂ a0. The roots of A0 in G define a root system Φ0 in a0.

Fix a minimal parabolic subgroup P0 ⊃ M0. This fixes positive roots Φ+0 ⊂ Φ0,

a system of simple roots ∆0 ⊂ Φ+0 , a Weyl chamber a+

0 and a positive cone +a0,and a+

0 ⊂+a0. The half-sum of roots in Φ0 is denoted ρ0; it belongs to a+0 . For any

standard P , the half-sum of roots of AM appearing in the nilradical of LieP (R) is

denoted ρP ∈ aM . An element ν ∈ a+0 determines a standard parabolic P (ν) and

standard Levi M(ν): The root group of α ∈ Φ0 is contained in P (ν) if and only if(α, ν) ≥ 0. Then ν ∈ aM(ν).

Fix a good maximal compact subgroup K = KR×∏

p Kp of G(A), so that G(A) =

KP0(A).For a standard Levi M let HM : AM(R)0 → aM be the logarithm map. Since

M(A) = M(A)1 × AM(R)0 (where M(A)1 is the subgroup of g with |χ(g)|A = 1 forall χ ∈ X∗(M)) we get a map HM : M(A)→ aM by composing with the projection.(When M = M0 we write H0.) Let KM

R = KR ∩M(R)0. Then λ ∈ (aM)C defines aone-dimensional (m, KM

R )×M(Af )-module CMλ by taking the multiples of the function

m 7→ e〈λ,HM (m)〉 with the action of M(A) by right translation.We will use an induction functor IndGP from (p, KM

R )×P (Af )-modules to (g, KR)×G(Af )-modules. This is the functor used in [10, §4] or [32, 3.3]; in particular it isnormalized. (Our main concern will be its effect at a finite place, where it is the usualnormalized induction of smooth representations.)

1.2. Functions on G(Q)\G(A). For t > 0 let

A0(t) := a ∈ A0(R)0 : 〈α,H0(a)〉 > t for all α ∈ ∆0.

For a compact subset ω ⊂ P0(A) and t > 0 define the adelic Siegel set

S = S(t, ω) = pak : p ∈ ω, a ∈ A0(t), k ∈ K.

Reduction theory says that for ω large enough and t small enough we have (1) G(A) =G(Q) S and (2) γ ∈ G(Q) : γS ∩S 6= ∅ is finite. Thus S is a coarse fundamentaldomain for G(Q) in G(A). We use this to define certain (g, KR)×G(Af )-modules offunctions on G(Q)\G(A) via growth conditions. Let ‖ · ‖ be a norm on a0.

Let S(G) denote the space of smooth KR-finite functions of uniform moderategrowth on G(Q)\G(A). Recall that this means there exists N ∈ N such that for allX ∈ U(g), there exists C such that for g = pak ∈ S,

|(Xf)(pak)| ≤ CeN‖H0(a)‖.

(This is denoted S∞(G(Q)\G(A)) in [10].)


Let R(G) be the subspace of S(G) of functions satisfying the following condition:For all X ∈ U(g), there exists C such that for any N ∈ Z and pak ∈ S,

|(Xf)(pak)| ≤ Ce〈2ρ0,H0(a)〉(1 + ‖H0(a)‖)N .(This is the space Sρ−τ−log(G(Q)\G(A)) in [10] for the choice τ = −ρ0.)

Let Slog(G) be the space of functions in S(G) satisfying the following: There existsan N ∈ N such that for all X ∈ U(g), there exists C such that for pak ∈ S,

|(Xf)(pak)| ≤ Ce〈ρ0,H0(a)〉(1 + ‖H0(a)‖)N .(This is denoted Slog(G(Q)\G(A)) in [10].)

Finally define B(G) to be the space of functions satisfying: ∃N ∈ N such that∀X ∈ U(g) ∃C such that for pak ∈ S,

|(Xf)(pak)| ≤ C(1 + ‖H0(a)‖)N .(These are the functions which are bounded up to logarithmic factors, and havesimilarly bounded U(g)-derivatives.) (This is the space Sρ−τ+log(G(Q)\G(A)) in [10],for τ = −ρ0.)

The inclusionsB(G) ⊂ Slog(G) ⊂ R(G) ⊂ S(G)

are (g, KR)×G(Af )-module homomorphisms.

1.3. Automorphic forms. Fix a Cartan subalgebra h of gC containing a0 (hencecontained in (m0)C). This gives a root system Φ = Φ(h, gC) and Weyl group W =W (h, gC). Fix a system of positive roots Φ+ ⊂ Φ = Φ(h, gC) compatible with Φ+

0 (i.e.if β ∈ Φ+ then β|a0 ∈ Φ+

0 ∪ 0). The half-sum of roots in Φ+ will be denoted ρh; soρh|a0 = ρ0.

Recall that the Harish-Chandra isomorphism Z(g) ∼= S(h)W identifies infinitesimalcharacters with the W -orbits in h and ideals of finite codimension in Z(g) with finiteW -invariant sets in h. For a finite W -invariant set Θ ⊂ h with corresponding idealIΘ and a (g, KR)-module V let

FinΘV = v ∈ V : ∃n such that InΘ v = 0.Thus FinΘS(G) and FinΘR(G) are spaces of automorphic forms for G. The directsum of FinΘS(G) as Θ runs over W -orbits in h is the space of all automorphic forms.

For a set of finite primes S containing all but finitely many finite primes let KS =∏p∈S Kp. In the spherical Hecke algebra HS = ⊗p∈SHp where Hp = H(G(Qp)//Kp)

we have the maximal ideal IS annihilating the trivial representation. Consider thespace of KS-spherical vectors in FinΘR(G) which are killed by some power of IS:

FinΘR(G)KSIS = f ∈ FinΘR(G)KS : ∃n s.t. InSf = 0.

The direct limit over all such S is denoted

FinΘR(G)I := lim−→SFinΘR(G)KS

IS .

This is a direct summand of FinΘR(G). (Briefly, for any KS, FinΘR(G)KSKSis finite-

dimensional and so has a decomposition according to maximal ideals of HS. Taking

6 ARVIND NAIR

a direct limit over KS and then over S gives a decomposition of FinΘR(G) in whichFinΘR(G)I is one summand.) Henceforth let

Θ := W · ρh.

This orbit corresponds to the trivial character of Z(g), so that FinΘR(G)I containsthe constant functions. The following is a simple consequence of results of Franke:

1.3.1. Theorem. FinΘR(G)I is the space of constant functions.

Proof. We will use one of Franke’s filtrations on the space FinΘS(G), which we recallfollowing [10, §6] or [32, 4.7,6.4].

First we need an elementary construction. For a finite set Θ ⊂ h define another

finite set Θ+ ⊂ a+0 as follows: For θ ∈ Θ and a standard Levi M let θM be the

restriction to aM . Considering Re(θM) ∈ aM as an element of a0, let Re(θM)+ ∈ a+0

be the closest point to Re(θM) in the closure of the Weyl chamber. Taking the unionover M and θ ∈ Θ gives Θ+. There is a natural way to filter Θ+. Let Θ0

+ be the set

of maximal elements in the standard ordering (viz. λ ≤ µ ⇐⇒ µ − λ ∈ +a0). Forp > 0 define Θp

+ inductively to be the set of maximal elements of Θ+−Θ6p−1+ and set

Θ6p+ = Θ6p−1

+ ∪Θp+.

Let f ∈ FinΘS(G). For each standard parabolic P the constant term of f along Padmits a Fourier expansion in terms of characters of aM (cf. [10, §6]); the charactersappearing form a finite set ExpP (f) in (aM)C (⊂ (a0)C), the P -exponents of f . Theseare related to the infinitesimal character by:

f ∈ FinΘS(G) and λ ∈ ExpP (f) =⇒ Re(λ)+ ∈ Θ+. (1.3.1)

Define a finite decreasing filtration

FinΘS(G) = F 0S ) F 1

S ) F 2S ) · · ·

by the condition

f ∈ F pS ⇐⇒ for all P and λ ∈ ExpP (f) we have Re(λ)+ ∈ Θ6p

+ .

A fundamental result of Franke [10, Theorem 14] is a description of the graded piecesof this filtration. The graded piece F p

S/Fp+1S is the sum of the induced modules

IndGP (ν)(CM(ν)ν ⊗ π) (1.3.2)

where ν ∈ Θp+ and π is an automorphic representation on M(ν) with unitary central

character. (The notation P (ν),M(ν) is as in 1.1. Franke gives a more detaileddescription of π which appear, but we will not need it here.)

Now define a filtration (F p)p∈N of R(G) by:

F p := F pS ∩ FinΘR(G).

Recall that in the notation of [10], R(G) is the space Sρ−τ−log(G(Q)\G(A)) for τ = ρ0.

Since ρ0 ∈ +a0∩a+0 , we have a version of Theorem 15 of [10] available (cf. the remarks


on p. 242 after the proof of Theorem 15). This says that f ∈ FinΘS(G) belongs toFinΘR(G) if and only if, for each standard P ,

λ ∈ ExpP (f) =⇒ Re(λ) ∈ ρ0 −+a0. (1.3.3)

This condition is compatible with the way the filtrations are defined, so that thegraded quotient F p/F p+1 is a sum of induced modules like (1.3.2) over those elementsν ∈ Θp

+ which satisfyν ∈ ρ0 − +a0 (1.3.4)

(cf. p. 242 of loc. cit.). Let FN be the last nonzero step of the filtration (correspond-ing to ΘN

+ = 0). To prove the theorem it suffices to prove:

(1) (F p/F p+1)I = F pI/F

p+1I = 0 for p < N

(2) FNI is the space of constant functions.

We will use the following lemma, which will be proved later:

1.3.2. Lemma. Let P be a parabolic subgroup with Levi factor M , τ an irreducible

unitary (m, KMR ) ×M(Af )-module and ν ∈ (aM)C with Re(ν) ∈ a+

M . If the inducedrepresentation Π = IndGP (CM

ν ⊗ τ) has a constituent which is trivial at all but finitelymany finite primes, then Re(ν) = ρP .

To prove (1), note that each summand of F p/F p+1 is an induced module as in thelemma for some ν ∈ Θp

+. By the lemma it makes no contribution to F pI/F

p+1I unless

ν = ρP for some P . But this possibility is excluded by (1.3.4) since ρ0−ρP ∈ +a0−+a0.To prove (2) we use the description of FN = FinΘSlog(G) given in [10, Theorem

13]. The precise result does not matter for us, it is enough to note that this space isa direct sum of modules of the form

Π =(

IndG(A)P (A) DF ⊗ π

)W (M)

with notation as follows: P is a parabolic subgroup with Levi M , W (M) is a finitegroup (acting via intertwining operators), π is an automorphic representation of Mwith unitary central character appearing in the L2 discrete spectrum, F ⊂ iaM is afinite set (depending on Θ), and DF = ⊕λ∈FDλ is the space of distributions supportedon F . For λ ∈ (aM)C the space Dλ of distributions supported on λ is a (m, KM

R ) ×M(Af )-module in such a way that it has a filtration (not of finite length) with gradedquotients the modules CM

λ defined above. If P 6= G the previous lemma implies

that ΠKSIS = 0 for any S. If P = G then Π = π is a discrete L2 automorphic

representation for G and hence ΠKSIS = 0 unless Π is the space of constant functions

(by weak approximation). This completes the proof of (2) and of the theorem.

Proof of Lemma 1.3.2. The constituents of Π = IndGP (CMν ⊗ τ) are of the form

π = ⊗pπp where for each p, πp is a constituent of IndG(Qp)

P (QP )ν ⊗ τp, and for almost all p

where τp is spherical, πp is the unique Kp-spherical constituent of IndG(Qp)

P (QP )ν⊗τp. (The

component at p of CMν is the unramified character of AM(Qp) given by ν ∈ (aM)C,

which we continue to denote by ν.) Let p be such that τp is spherical. Then τp is the

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spherical subquotient of IndM(Qp)

M0,p(Qp)χ where M0,p ⊂ M0 is a minimal Levi of G/Qp

and χ is an unramified character of M0,p, unitary on AM(Qp), and we may assumethat χ is dominant ([8]). Thus πp is a constituent of

IndG(Qp)

P (Qp)IndM(Qp)

M0,p(Qp)νχ = IndG(Qp)

M0(Qp)νχ.

We use the following fact: The unramified principal series representation IndG(Qp)

M0(Qp)λ

has no trivial constituent unless λ ∈ W0ρ0,p, in particular for dominant λ we musthave λ = ρ0,p. (Here W0 is the relative Weyl group. This fact follows easily from thestructure of the Jacquet module of the unramified principal series [8].) Thus νχ = ρ0,p

and hence νχ|AM = ρP . Since χ is unitary on AM(Qp) we have Re(ν) = ρP .

1.4. Cohomology. For simplicity we assume thatG is semisimple, simply-connected,and isotropic. Then G(R) is a connected Lie group and D = G(R)/KR is a symmet-ric space. For a compact open subgroup K ⊂ G(Af ) and the congruence subgroupΓ = K ∩ G(Q), the strong approximation theorem gives an adelic description of thelocally symmetric space:

MΓ := Γ\G(R)/KR = G(Q)\G(A)/KRK. (1.4.1)

Let MΓ be the reductive Borel-Serre compactification of MΓ (cf. [34, 12]). The directlimit

H∗(M) := lim−→H∗(MΓ)

has an action of G(Af ) by ring automorphisms: Right translation by g ∈ G(Af ) givesan isomorphism M gKg−1∩G(Q) →MK∩G(Q) and pullback defines the action of g.

We will need the (C∞) de Rham description of H∗(MΓ) and H∗(M) and its dualtheory, in terms of the spaces B(G) and R(G) defined earlier, given by [23]. There isa natural isomorphism

H∗(M) = H∗(g, KR, B(G)). (1.4.2)

At a finite level Γ = K ∩G(Af ) one recovers

H∗(MΓ) = H∗(M)K = H∗(g, KR, B(G)K).

(Take λ = 0 in Theorem 3.8 of [23] and note that the weighted complex W 0C(C) of[12] is quasiisomorphic to the constant sheaf CM , cf. §19 of loc. cit.) The Poincaredual groups are the weighted cohomology groups W dH∗(MΓ) defined in [12]. Thedirect limit W dH∗(M) = lim−→Γ

W dH∗(MΓ) has a G(Af )-action defined in the same

way as for H∗(M) above. It has the de Rham description:

W dH∗(M) = H∗(g, KR, R(G)).

For Γ = K ∩G(Af ) one has:

W dH∗(MΓ) = W dH∗(M)K = H∗(g, KR, R(G)K). (1.4.3)

(Take λ = 2ρ0 in [23, 3.8]; the corresponding weighted complex W dC(C) is thedualizing complex, cf. [12, §19].)


In these de Rham descriptions the actions of G(Af ) are induced by the right trans-lation actions on B(G) and R(G). The action of the Hecke algebra at level K onH∗(M)K = H∗(MΓ) is the familiar geometric action via finite correspondences onMΓ. The integration pairing B(G)×R(G)→ C induces the Hecke-invariant Poincareduality pairing H i(MΓ) × W dHdimMΓ−i(MΓ) → C and a G(Af )-invariant pairingH i(M)×W dHdimD−i(M)→ C in the limit.

The compact dual symmetric space of D is

D = Gc/KR

where Gc is the compact real form of G(R), i.e. the Lie subgroup of G(C) with Liealgebra LieKR + ip (where g = LieKR + p is the Cartan decomposition of g). ThenH∗(g, KR,C) = H∗(D). The inclusion of the constants in B(G) and R(G) induce

H∗(D)→ H∗(M) (1.4.4)

andH∗(D)→ W dH∗(M). (1.4.5)

(In each case the image is the subspace represented by G(R)-invariant differentialforms on D.) We apply the analytic result Theorem 1.3.1 to conclude:

1.4.1. Theorem. The homomorphism H∗(D)→ H∗(M) is a split inclusion onto theG(Af )-invariants of H∗(M) and induces isomorphisms

H∗(D) ∼= H∗(M)G(Af ) ∼= H∗(M)G(Af ).

The same statements hold for H∗(D)→ W dH∗(M).

Proof. The main result of [10] tells us that the inclusion of FinΘR(G) in R(G) inducesan isomorphism in (g, KR)-cohomology. (Apply Theorem 16 of loc. cit., noting that

R(G) is the space Sρ−τ−log for τ = −ρ0 and ρ0 ∈ +a0 ∩ a+0 .) For a set S of finite

primes containing almost all primes we have

W dH∗(M)KSIS = H∗(g, KR,FinΘR(G)KS

IS )

= H∗(g, KR,C)

= H∗(D).

Thus W dH∗(M)I = H∗(D) is a direct summand of W dH∗(M) and the complementcontains no almost-everywhere-trivial constituents, in particular neither invariantsnor coinvariants. This proves the assertions about W dH∗(M); the statements aboutH∗(M) follow by duality.

1.4.2. Remarks. (i) The injectivity of (1.4.4) and (1.4.5) can be proved without usingautomorphic forms, at least in the case where D has a Hermitian structure. Sup-pose we know that (1.4.5) is nonzero in top degree, i.e. one can fix a generator forHdimD(MΓ) using D. A straightforward argument using Poincare duality for H∗(D)then shows that (1.4.5) is injective in all degrees. The injectivity in top degree re-duces to showing that there is a G(R)-invariant form of top degree which has nonzero

10 ARVIND NAIR

integral over M . In the Hermitian case this is easily checked either using a powerof the Kahler form (which is given in terms of the Killing form) or else using theBorel embedding D ⊂ D. Thus in the Hermitian situation one can conclude thatH∗(D) is a direct summand of H∗(M) and W dH∗(M) without using Theorem 1.4.1.However, one cannot rule out other invariant classes (or trivial constituents) withoutusing analytic methods, and we will need this stronger result in §4.

(ii) At a finite level Γ we conclude that the invariants for the Hecke algebra at levelΓ in H∗(MΓ) and in W dH∗(MΓ) are given by H∗(D).

(iii) In [11] Franke computes the space FinΘS(G)I and the spaces of invariants andcoinvariants in the cohomology H∗(M) = lim−→Γ

H∗(MΓ), a more delicate result thanours.

(iv) For an analogue of the theorem for toroidal compactifications see (3.3) below.

2. Hodge-de Rham theory of M

In this section M is a locally symmetric variety. We summarize some results from[25, 26] about the mixed Hodge(-de Rham) structure in the cohomology of M . Someof these results could also be deduced from stronger motivic versions due to Ayoub andZucker [2], but we prefer to use the cited references to keep the discussion relativelyelementary.

2.1. Mixed Hodge structures. For an algebraic variety X, a stratification of X(by which we mean a partition X =

⊔ri=0 Xi into locally closed subvarieties with each

Xd open in X −⊔i<dXj), and any function from the set of strata to Z ∪ ±∞,

Morel [20, §3] defines a t-structure (and hence associated truncation functors) onSaito’s derived category DbMHM(X) of mixed Hodge modules [28, 29] on X. We willbe interested in the particular function dim defined (for any stratification) by:

dim(S) := dimC S.

The associated truncation functors are w6dim and w>−dim. Recall that there is aforgetful functor rat : DbMHM(X)→ Db

c(QX) to the derived category of constructibleQ-sheaves compatible with all the standard operations.

Let j : M → M∗ be the minimal (or Satake-Baily-Borel) compactification of M([4]). This has a canonical stratification in which each stratum is a (connected) locallysymmetric variety. Choose a smooth projective toroidal compactification MΣ of M[1]. We have morphisms

Mp−→M∗ π←−MΣ.

Here π (resp. p) is a proper morphism (resp. proper continuous map) extending theidentity on M . We have (cf. [25, 26]):

2.1.1. Proposition. There are natural isomorphisms

Rp∗QM = rat(w6dimRπ∗QHMΣ

) = rat(w6dimRj∗QHM).


The proposition puts a rational mixed Hodge structure on the cohomology of M .(This was done earlier in [36] by a different method and also follows from [2] usingrealisations.) This mixed Hodge structure is like that of a complete but possiblysingular variety, i.e. the weights in H i(M) are ≤ i (cf. [25, 26]). The ring structureon H∗(M) is by morphisms of mixed Hodge structure.

The Poincare dual groups W dH∗(M) are computed by the complex W dC(QM)Verdier dual to W 0C(QM) = QM . They acquire a mixed Hodge structure by thenatural isomorphism dual to Prop. 2.1.1:

W dC(QM) = rat(w>−dim(Rπ∗QHMΣ

[2n](n))).

There is an action of H∗(M) on W dH∗(M) coming from the fact that the complexof sheaves W dC(QM) from [12] computing W dH∗(M) is a sheaf of W 0C(QM) =QM -modules. (Under the identification W dH∗(M) = H2 dimC M−∗(M) this is the capproduct action of cohomology on homology.) The action is by morphisms of mixedHodge structure.

2.1.2. Remark. If j : M →M∗ is the inclusion then we also have

Rp∗QM = rat(w6dimRj∗QHM) = rat(w>dim′j!QH

M)

where dim′(S) = dimC S+1 if S is a boundary stratum and dim′(M) = dimCM . Moregenerally, in [25] we show that the weighted complexes of [12] are quasiisomorphic torat(w6aRj∗QH

M) for suitable a.

2.2. Prop. 2.1.1 implies that the pullback π∗ : H∗(M∗) → H∗(MΣ) factors as π∗ =γ∗ p∗ where the homomorphism of mixed Hodge structures

γ∗ : H∗(M)→ H∗(MΣ)

is induced by the morphism w6dimRπ∗QHMΣ→ Rπ∗QH

MΣin DbMHM(M∗). This pull-

back homomorphism was constructed earlier by Goresky and Tai [14]. There is thefollowing analogue of a well-known fact ([9, 8.2.5]) about mixed Hodge structures ofcomplete singular varieties (cf. [25]):

2.2.1. Lemma. The map γ∗ : H∗(M)→ H∗(MΣ) induces a natural isomorphism

GrWi Hi(M) = im[H i(M)

γ∗→ H i(MΣ)]. (2.2.1)

It factors through any inclusion IH∗(M∗) → H∗(MΣ) coming from the decompositiontheorem [5, 28] and induces a natural isomorphism

GrWi Hi(M) = im[H i(M)→ IH i(M∗)]. (2.2.2)

We will also use:

2.2.2. Lemma. The restriction map GrWi Hi(M) → H i(M) is injective for i ≤

codim (M∗ −M).

Proof. By the previous lemma the map factors as GrWi Hi(M) → IH i(M∗) →

H i(M), and IH i(M∗)→ H i(M) is injective for i ≤ codim (M∗ −M).

12 ARVIND NAIR

2.3. Mixed Hodge-de Rham structures. Recall the notion of mixed Hodge-deRham (HdR) structure over a subfield k of C. (We ignore integral structures here.)This is a triple ((VB,W•), (VdR, F

•), I) where VB is a Q-vector space with increasingfiltration W•, VdR is a k-vector space with decreasing filtration F •, I : VB ⊗Q C 'VdR ⊗k C is an isomorphism, and (VB,W•, VB ⊗Q C, F •C) is assumed to be a rationalmixed Hodge structure. The basic example is the following: For an algebraic varietyX over k, we have the singular cohomology H i

B(X) := H i(X(C),Q) with its weightfiltration W•, the algebraic de Rham cohomology H i

dR(X/k) = Hi(X,Ω•X/k) with its

Hodge filtration F •, and the comparison isomorphism I : H iB(X)⊗QC ∼= H i

dR(X/k)⊗kC. The triple ((H i

B(X),W•), (HidR(X/k), F •), I) is a (mixed) HdR structure over k.

More generally, a mixed motive over k with coefficients in Q should give rise to sucha structure. Mixed Hodge-de Rham structures form an abelian category and there isa notion of Tate twist, namely tensor with the inverse (i.e. Hom to the trivial HdRstructure) of the structure appearing in H2(P1).

It is well-known that M , being a connected component of a Shimura variety at fi-nite level, has a model over a number field k contained in C. (The theory of canonicalmodels shows that this is, up to an abelian extension, independent of the congruencesubgroup Γ, but we do not need this here.) The canonical nature of M∗ implies thatit, too, has a model over k. For suitable Σ, MΣ and the morphism π : MΣ →M∗ canalso be defined over k ([17, 2.8]). We can keep track of this structure by working inthe theory of mixed sheaves M (·) for varieties over k defined by Saito in [30, 1.8(ii)].In essence, for a variety X/k, M (X) is the category of mixed Hodge modules onX(C) with a k-rational structure on the underlying bifiltered D-module. There isa forgetful functor to perverse sheaves on X(C) factoring through MHM(X(C)) andthese functors extend to derived categories. For a variety X/k there is an objectQMX ∈ DbM (X) realizing to QH

X and further to QX , and the k-structure on its coho-mology gives the de Rham rational structure given by the comparison isomorphismH∗dR(X/k) ⊗k C = H∗(X(C),Q) ⊗ C with algebraic de Rham cohomology. Morel’struncation functors are defined in the categories DbM (X) for a choice of stratificationof X/k and function to Z∪±∞. (The arguments of [20] carry over to the setting of[30] mutatis mutandis.) In our case M∗/k can be stratified in a way that coarsens thecanonical stratification, and we have the function dim for this stratification. (Thisstratification over k may not have geometrically connected strata, but the strata areof pure dimension.) Thus we have an object

w6dimRπ∗QMMΣ∈ DbM (M∗)

mapping under the forgetful functors to w6dimRπ∗QHMΣ∈ DbMHM(M∗) and (by

Prop. 2.1.1) to Rp∗QM ∈ Dbc(QM∗). If a : M∗ → Spec(k) is the map to a point

then H i(Ra∗w6dimRπ∗QMMΣ

) ∈M (Spec(k)) is a mixed Hodge-de Rham structure, i.e.

a pair consisting of a filtered k-vector space, which we denote H idR(M/k), and the

rational mixed Hodge structure of 2.1 on H i(M,Q), with a comparison isomorphismH∗dR(M/k) ⊗k C ∼= H∗(M,Q) ⊗Q C which identifies the decreasing (i.e. Hodge)filtrations. We shall refer to H i

dR(M/k) as the algebraic de Rham cohomology of M


and to the k-structure it defines on H∗(M,C) as the de Rham rational structure. Allthe results in 2.1 and 2.2 hold in the category of mixed Hodge-de Rham structures,i.e. with QH

M and QHMΣ

replaced by QMM and QM

MΣetc. (the proofs in [26, 25] carry

over without change).

3. Chern classes of automorphic vector bundles

We use the results of §1 and §2 to make some remarks about Chern classes of auto-morphic vector bundles on M . In this section G is an isotropic connected semisimpleand simply-connected Q-group, and we assume that D = G(R)/KR is a Hermit-ian symmetric domain. All cohomology groups have C coefficients unless indicatedotherwise.

3.1. Automorphic vector bundles. The complexified tangent space to D at eKRadmits a decomposition pC = p+ ⊕ p− into holomorphic and antiholomorphic parts.The subgroup P− = KC

R exp(p−) is a maximal parabolic subgroup of G(C) withP− ∩G(R) = KR. (Here KC

R is the complexification of the compact group KR.) Thisgives an alternate description of the compact dual as a flag variety:

D = G(C)/P−.

Moreover G(R) ⊂ G(C) induces a G(R)-equivariant holomorphic open embeddingD → D, the Borel embedding. A finite-dimensional algebraic representation of P−on a complex vector space V gives a homogeneous vector bundle

V = G(C)×P− V = G(C)× V/(g, v) ∼ (gp, p−1 v)

on D. Restricting V by the Borel embedding and dividing by Γ produces an auto-morphic vector bundle V on M = Γ\D. (If V is a representation of G(C) then V hasa flat connection (i.e. is the vector bundle associated with a local system).) Amongthese bundles are those, usually called fully decomposed, which come from representa-tions of KR which are complexified and inflated to P− by letting exp(p−) act trivially.Since every automorphic vector bundle is C∞-isomorphic to a fully decomposed one,it will suffice for our purposes to work with these.

3.2. Chern classes. Let π : MΣ → M∗ be a smooth projective toroidal desingular-ization of M∗ as in [1]. We will assume that MΣ is defined over k and MΣ −M isa simple normal crossings divisor (cf. [17]). Mumford [21] (for irreducible V ) andHarris [17] (for general V ) showed that there is a functorial way of extending an au-tomorphic vector bundle V to a vector bundle VΣ on MΣ, the canonical extension.(When V is flat this is Deligne’s canonical extension (cf. [17, §4]).) Thus there are(topological) Chern classes

ck(VΣ) ∈ H2k(MΣ,Z)

and they vanish if V has a flat connection. Mumford’s generalization of Hirzebruch’sproportionality theorem shows that they generate a copy of H∗(D,Z) in H∗(MΣ,Z)(cf. Lemma 3.2.3 below).

Automorphic vector bundles do not, in general, extend to vector bundles on M∗.Nevertheless, Goresky and Pardon [13] showed that the Chern classes of automorphic

14 ARVIND NAIR

vector bundles come from H∗(M∗,C), i.e. for an automorphic vector bundle V on Mthere are classes

c∗k(V ) ∈ H2k(M∗,C)

such that for any toroidal desingularization π : MΣ →M∗,

π∗(c∗k(V )) = ck(VΣ).

(Note that c∗k(V ) is merely (suggestive) notation, i.e. this is not the Chern class of abundle.) The construction in [13] proceeds via explicit differential forms on M withprescribed behaviour near M∗ −M and a number of questions present themselves(cf. [13, 1.6]) about these Chern classes: To what extent are they unique? Are theyrational for the Betti rational structure? Do they vanish for flat V ? Do they generatea copy of H∗(D) in H∗(M∗)? What are their Hodge types? Are they rational for thede Rham rational structure? (2) These seem difficult to answer.

We show here that all these questions can be answered on M , i.e. for the classes

ck(V ) := p∗(c∗k(V )) ∈ H2k(M,C).

(An alternate definition of ck(V ) is given by Lemma 3.2.2 below.) One way to ap-proach this would be to show that automorphic vector bundles extend over M andto use a suitable complex of differential forms computing this (e.g. in terms of log-bounded forms as earlier). However this would leave unanswered some questions (e.g.de Rham rationality). Instead we use the analytic input of Theorem 1.4.1. and com-bine it with the fact that these statements are known to hold for Chern classes on atoroidal compactification. We have:

3.2.1. Theorem. For an automorphic vector bundle V on M the classes ck(V ) ∈H2k(M,C) are uniquely determined by two properties:

(i) They are invariant under Hecke operators.(ii) They lift to the Chern classes of canonical extensions, i.e. for any π : MΣ →

M∗,γ∗(ck(V )) = ck(VΣ).

Moreover, they have the following properties:

(iii) They are rational for the Betti rational structure H∗(M,Q).(iv) The classes (2πi)k ck(V ) are rational for the de Rham rational structure on

H∗(M,C) given by the comparison isomorphism H∗(M,C) = H∗dR(M/k)⊗kC.(v) They generate the direct summand H∗(D) of invariants in H∗(M). This direct

summand of H∗(M) is (in each degree) a pure Hodge-de Rham structure overk of Tate type.

(vi) They vanish if V is flat, i.e. comes from a representation of G.

The proof follows the next two lemmas.

2An automorphic vector bundle admits a model over the field of definition of M ([17, 1.4]),so if V on M/k were to extend to a vector bundle V ∗ on M∗/k one would have Chern classescdRi (V ∗) ∈ H2i

dR(M∗/k)(i) related to 1(2πi)i ci(V ∗) ∈ H2i(M∗, C) by the comparison isomorphism.

(A theory of Chern classes in algebraic de Rham cohomology exists by [15] and [16].)


3.2.2. Lemma. The class ck(V ) ∈ H2k(M) is represented by the kth Chern form ofthe invariant connection on V and is therefore Hecke-invariant.

Proof. The class c∗k(V ) of [13] is defined via the kth Chern form σk(∇pM) of a certain

connection ∇pM on V . This is a controlled differential form on M , hence bounded,

hence belongs to the (C∞) de Rham model we have for M . Thus ck(V ) = p∗(c∗k(V ))is represented by the bounded differential form σk(∇p

M). On the other hand we havethe kth Chern form σk(∇inv

M ) of the invariant connection ∇invM (cf. e.g. [13, Ex. 5.5]),

which is an invariant differential form. The difference σk(∇pM) − σk(∇inv

M ) = dη isexact and the usual proof of this fact (e.g. Lemma 5 of Chp. XII of [19]) gives abounded η. Thus they represent the same class in our de Rham model for M .

3.2.3. Lemma. The subalgebra of H∗(MΣ,Z) generated by Chern classes of canonicalextensions of automorphic vector bundles is isomorphic to H∗(D,Z).

Proof. Following a suggestion of N. Fakhruddin we use K-theory to prove this. LetK(·) denote the topological K-theory of a space. The integral cohomology of D istorsion-free, so the Atiyah-Hirzebruch spectral sequence degenerates integrally. AsD = G(C)/P has a cell decomposition, H∗(D,Z) is nonzero only in even degrees so

the Chern character gives an isomorphism of Z-algebras ch : K(D)'→ H∗(D,Z) (cf.

p.19 of [3]). Let Rep(·) denote the representation ring of a compact group. SinceD = Gc/KR, it is known (cf. [3, Thm. 5.2]) that the natural map Rep(KR)→ K(D)is surjective, and moreover, if Z is a Rep(Gc)-module via the homomorphism dim :Rep(Gc) → Z, then Rep(KR) ⊗Rep(Gc) Z → K(D) is an isomorphism. Note thatRep(KR)⊗Rep(Gc) Z = Rep(KR)/I · Rep(KR) where I = ker(dim) ⊂ Rep(Gc). DefineRep(KR)→ H∗(MΣ,Z) by V 7→ ch(VΣ) for a representation V of KR and extendinglinearly. It is a ring homomorphism because canonical extension commutes withtensor product (cf. [17, 4.2]) and ch is a ring homomorphism. If V ∈ Rep(Gc) then theChern classes of VΣ vanish and ch(VΣ) = 0. Thus we have a map K(D)→ H∗(MΣ,Z).Define

κ : H∗(D,Z)→ H∗(MΣ,Z)

by first inverting K(D) ∼= H∗(D,Z) and then applying K(D)→ H∗(MΣ,Z). It is analgebra homomorphism and takes ck(V ) to (−1)kck(VΣ) (cf. [21, p. 263]). It remainsto show that κ is injective, which we can do with coefficients Q or C.

The proportionality theorem [21, 3.2] implies the following: for any integers k1, . . . , krwith

∑i ki = n and representations V1, . . . , Vr we have

ck1(V1,Σ) ck2(V2,Σ) · · · ckr(Vr,Σ) ∩ [MΣ] ∼ ck1(V1) ck2(V2) · · · ckr(Vr) ∩ [D]

where ∼ means up to some fixed nonzero constant and [ · ] denotes the fundamentalclass. This fact plus Poincare duality for D implies the injectivity of κ: For nonzeroα ∈ H i(D) choose β ∈ H2n−i(D) such that α · β 6= 0. Since K(D) ∼= H∗(D) bythe Chern character, both α and β are linear combinations of monomials in Chernclasses, so α · β is a linear combination of monomials of top degree in Chern classes.Then κ(α) · κ(β) ∩ [MΣ] = κ(α · β) ∩ [MΣ] ∼ α · β ∩ [D] 6= 0. So κ(α) 6= 0.

16 ARVIND NAIR

Proof of the theorem. Recall that the pullback by π : MΣ →M∗ factors as:

H∗(M∗)p∗→ H∗(M)

γ∗→ H∗(MΣ)

LetH∗Chern(M) be the subalgebra ofH∗(M) generated by the classes ck(V ) = p∗(c∗k(V )).By Lemma 3.2.2 and Theorem 1.4.1:

H∗Chern(M) ⊂ H∗(M)inv ∼= H∗(D).

On the other hand, the classes γ∗(ck(V )) = π∗(c∗k(V )) = ck(VΣ) generate a copy ofH∗(D) in H∗(MΣ) by Lemma 3.2.3. Thus γ∗ : H∗Chern(M) → H∗Chern(MΣ) ∼= H∗(D)is surjective, hence injective, and moreover we have

H∗Chern(M) = H∗(M)inv.

By Theorem 1.4.1 this is a direct summand, rational for either Betti or de Rhamrational structures because Hecke operators act rationally for either. The assertionsof the theorem follow from this. (The de Rham rationality of the Chern classescdRk (VΣ) = (2πi)kck(VΣ) follows from the fact that D has a natural k-rational structureand the functor V 7→ VΣ from homogeneous bundles on D to vector bundles on MΣ

is k-rational, cf. [17, 4.2].)

3.2.4. Remarks. (i) In [35] Zucker proves the Betti rationality of the classes ck(V )by identifying them with the Chern classes of the topological vector bundle on Mextending V constructed in [14, 9.2].

(ii) The method of proof suggests an approach to proving a similar theorem for theclasses c∗k(V ) on M∗: Give a de Rham description of H∗(M∗) in terms of a suitable(g, KR)-module of functions, reduce to the space of automorphic forms therein, andthen compute the space of Hecke-invariants in cohomology. If, as one might hope, thisis reduced to H∗(D), then all questions are answered as above, i.e. by reduction tothe corresponding statements on MΣ. This would also give information on H∗(M∗),which is scarce (see [13, 1.6, §16]).

3.3. There is an analogue of Theorem 1.4.1 for toroidal compactifications, which westate without proof. Let

H∗(Mtor) := lim−→Γlim−→Σ

H∗(MΓ,Σ).

Here the first (i.e. inner) limit is over all Γ-admissible rational partial polyhedraldecompositions Σ and the second is over all congruence subgroups Γ. In the adeliccontext this is a smooth, but not usually admissible, module for G(Af ). Properties ofcanonical extensions verified in [17] and Lemma 3.2.3 give an injective homomorphismH∗(D)→ H∗(Mtor).

3.3.1. Theorem. The G(Af )-invariants and G(Af )-coinvariants in H∗(Mtor) areidentified with the algebra of Chern classes H∗(D), which is canonically a direct sum-mand of H∗(Mtor).

The proof, which is purely geometric (i.e requires no analytic input), uses thedecomposition theorem.


4. Locally symmetric subvarieties

All cohomology groups have complex coefficients in this section.

4.1. Restriction maps. Let f : H → G be a homomorphism of connected semisim-ple groups over Q and assume that f has finite kernel. Choose a maximal compactsubgroup K of G(R); f−1(K) is a maximal compact subgroup in H(R). The inducedembedding DH ⊂ D of symmetric spaces induces a map

f : MH →M

which is an immersion in the differential geometric sense. Simple examples show thatthere is (in general) no continuous extension f : MH → M ; nevertheless, there is anatural homomorphism in cohomology

f ∗ : H∗(M)→ H∗(MH) (4.1.1)

satisfying the properties one would expect of a pullback or restriction map. The easi-est way to see that such a map exists, at least with C-coefficients, is using the de Rhammodel of [23] recalled in 1.4. Restriction of functions gives a map B(G)→ B(H) be-tween logarithmically-bounded functions, which gives, in Lie algebra cohomology, therequired map.

Rather than use this definition, we restrict ourselves henceforth to the case whenboth DH and D are Hermitian and we choose the complex structures so that DH → Dis holomorphic. We will refer to f : MH → M as a locally symmetric subvariety bya slight abuse of terminology. We give another construction of the pullback map,showing in particular that it is a morphism of mixed Hodge(-de Rham) structures(over a field of definition for the morphism f).

4.1.1. Theorem. Let f : MH → M be a locally symmetric subvariety given by ahomomorphism H → G of connected semisimple groups. Then there is a pullbackring homomorphism of mixed Hodge-de Rham structures

f ∗ : H∗(M)→ H∗(MH)

compatible with f ∗ : H∗(M) → H∗(MH). There is a Gysin homomorphism of mixedHodge-de Rham structures

f! : W dH∗(MH) −→ W dH∗+2(n−m)(M)(n−m)

(where n = dimCD,m = dimCDH , and (·) is the Tate twist) compatible with f! :H∗c (MH)→ H∗c (M). These are related by

f!(β · f ∗(α)) = f!(β) · α (for α ∈ H∗(M), β ∈ W dH∗(MH)). (4.1.2)

For the embedding f : DH → D of compact duals induced by f , there is a commutativediagram

H∗(D)f∗−−−→ H∗(DH)

κ

y yκHH∗(M)

f∗−−−→ H∗(MH)

(4.1.3)

18 ARVIND NAIR

where κ, κH are the inclusions given by Theorem 3.2.1.

Proof. The existence of f ∗ is proved in a general setting in [26], but we outline the

argument here. It is well-known that f extends to an algebraic map f : M∗H → M∗

which is the composite of a finite morphism and a closed immersion. By [17] we maychoose smooth projective toroidal compactifications to get a commutative diagram:

MH,ΣHτ−−−→ MΣ

πH

y yπM∗

H

f−−−→ M∗

(4.1.4)

Here τ is the composite of a finite morphism and a closed immersion. Applying Rπ∗to the homomorphism QH

MΣ→ Rτ∗QH

MH,ΣHgives

Rπ∗QHMΣ→ Rf∗RπH∗QH

MH,ΣH(4.1.5)

in DbMHM(M∗). Since f is finite we have Rf∗ = f∗. Let us refine the canonicalstratification of M∗ so that the images of strata of the canonical stratification ofM∗

H are strata of M∗, and let w′6dim be the truncation functor on DbMHM(M∗)corresponding to the function S 7→ dimC S for this new stratification. Then w′6dimhas the same effect as w6dim on Rπ∗QH

MΣ(see e.g. [26, 3.1]). Thus applying w′6dim

we get

w6dimRπ∗QHMΣ−→ w′6dimRf∗RπH∗QH

MH,ΣH

and using the identity w′6dimRf∗ = Rf∗w6dimH (where dimH is the function S 7→dimC S for the canonical stratification of M∗

H) we get a map

w6dimRπ∗QHMΣ−→ Rf∗w6dimHRπH∗Q

HMH,ΣH

. (4.1.6)

f ∗ is the homomorphism induced by (4.1.6) in hypercohomology. It is easy to checkthat it is a ring homomorphism (see [26, 4.1.5]).

Let D be the Verdier duality functor. The canonical orientations of MΣ and MH,ΣH

fix isomorphisms D(QHMΣ

[n]) = QHMΣ

[n](n) and D(QHMH,ΣH

[m]) = QHMH,ΣH

[m](m) which

are compatible under the Gysin isomorphism τ !QHMΣ

= QHMH,ΣH

[−2c](−c) (for c =

n−m). Applying D to (4.1.6) then gives a homomorphism

Rf∗w>−dimH (RπH∗QHMH,ΣH

[2m](m))→ w>−dim(Rπ∗QHMΣ

[2n](n)). (4.1.7)

(Alternately, dualize (4.1.5) and apply w>−dim.) f! is the map induced in hypercoho-mology after the identification Rp∗W

dC(QM) = rat(w>−dim(Rπ∗QHMΣ

[2n](n))) dualto Prop. 2.1.1.

(4.1.2) is proved in the standard way: By duality it is enough to show that forα ∈ H i(M), β ∈ W dHj(M) and any γ ∈ H2m−i−j(M), 〈γ, f!(β ·f ∗(α))〉 = 〈γ, f!(β)·α〉.


We have:

〈γ, f!(β · f ∗(α))〉 = 〈f ∗(γ), β · f ∗(α)〉 = 〈f ∗(α) · f ∗(γ), β〉= 〈f ∗(α · γ), β〉 = 〈α · γ, f!(β)〉 = 〈γ, f!(β) · α〉.

These arguments in the setting of mixed Hodge modules work verbatim in anytheory of mixed sheaves in the sense of Saito [30], in particular the theory M (·)mentioned in 2.3, where we assume f : MH →M to be defined over k. (The diagram(4.1.4) can be defined over k by [17].) Thus all statements hold in the context ofmixed HdR structures.

For the commutativity of (4.1.3) we argue as follows: The diagram

H∗(D)f∗−−−→ H∗(DH)

κ

y yκHH∗(MΣ)

τ∗−−−→ H∗(MH,ΣH )

commutes by the functorial property of canonical extensions: τ ∗VΣ∼= VΣH ([17,

4.3]). Moreover, one knows that κ and κH factor through κ : H∗(D) → H∗(M) andκH : H∗(DH)→ H∗(MH). To show that (4.1.3) commutes it is enough to note that

H∗(M)f∗−−−→ H∗(MH)

γ∗y yγ∗H

H∗(MΣ)τ∗−−−→ H∗(MH,ΣH )

commutes (by construction of f ∗), that f ∗ carries invariant classes to invariant classes,and that γ∗H is injective on the invariant classes (cf. the proof of Theorem 3.2.1).

4.1.2. Remark. The map defined here agrees with the one given by restriction offunctions B(G)→ B(H). The proof of this fact, which we do not need to use below,is a little involved, so we skip it here. (The commutativity of (4.1.3) is immediate inthis description.)

4.2. Cycle classes. Let us assume we have a morphism f : MH → M of locallysymmetric varieties coming from a homomorphism H → G with finite kernel. Forsimplicity we make the assumption that G is simply-connected. Let n = dimCM andm = dimCMH . From now on we work with C coefficients, so we will ignore Tatetwists. For a congruence subgroup Γ define the cycle class

ξΓ := f!(1) ∈ W dH2c(MΓ).

The following is the analogue of [31, Theorem 1] and the proof there works with theobvious changes, given what has gone before:

4.2.1. Theorem. The G(Af )-submodule of W dH2c(M) generated by ξΓ contains [DH ] ∈H2c(D).

20 ARVIND NAIR

Proof. Let V be the submodule generated by ξΓ. By Theorem 1.4.1 we can writeV = V 0 ⊕ V 1 where V 0 ⊂ H2c(D) and V 1 has neither invariants nor coinvariants(in fact, no trivial constituents). Write ξΓ = ξ0

Γ + ξ1Γ with ξiΓ ∈ V i. Since V 1 has no

coinvariants,

α · ξΓ = α · ξ0Γ for α ∈ Hm(D).

The commutativity of (4.1.3) implies that

α · ξΓ = α · [DH ] for α ∈ Hm(D).

Thus α · ξ0Γ = α · [DH ] for all α ∈ Hm(D). By duality for D, ξ0

Γ = [DH ].

4.2.2. Remarks. (i) The theorem shows in particular that ξΓ 6= 0. This also followsfrom the fact that the image of ξΓ in homology, i.e. under W dH2c(M) = H2m(M)→H2m(M∗), is nonzero. Note that the pullback map H∗c (M)→ H∗c (MH) on compactlysupported cohomology defines a cycle class in H2c(M) (to which ξΓ maps underW dH∗(M)→ H∗(M)). However, this class is not always nonzero.

(ii) The cycle class ξΓ ∈ W dH2c(M) defined here lifts into the intersection coho-mology IH2c(M∗) = W>−2ρ0H2c(M): Indeed, if γ! : H∗(MΣ) → W dH∗(M) is themap dual to γ∗ then ξΓ = f!(1) = γ!(τ!(1)) is pure of weight 2c. The dual assertion toLemma 2.2.2 gives that IH i(M∗) Wi(W

dH i(M)) for all i, so that ξΓ can be liftedto IH2c(M∗). But there does not seem to be a natural or canonical lift in general.Any lift has the property that a sum of G(Af )-translates of it is equal to [DH ]. Ido not know whether there is a canonical lift when the image of M∗

H is allowable (inthe sense of Goresky and MacPherson for middle perversity) cycle in M∗. (Thereare, of course, numerous situations in which one will have a canonical lift, e.g. ifW dH2c(M) = IH2c(M∗). For example, if M is a Hilbert modular variety of anydimension, then H∗(M) = IH∗(M∗) (cf. the computations in [34, §6]) and so duallyW dH∗(M) = IH∗(M∗), so that all cycles have lifts.)

(iii) The group W dH2c(M) has a direct sum decomposition according to associateclasses of parabolic subgroups by [10, Theorem 6]. It is interesting to ask when ξΓ

has a nontrivial projection to the cuspidal summand. This would require knowingsomething about periods of cusp forms along the subgroup H.

(iv) In the non-Hermitian case, the inclusion H∗(D) ⊂ H∗(MΓ) of §1 allows us tofix generators for the top cohomology of all MΓ at once, i.e. for Hn(M), and similarlyfor MH . This also defines the dual map f! and Theorem 4.2.1 holds with the sameproof.

4.2.3. Lemma. If Γ′ < Γ is a normal and of finite index then 1|Γ/Γ′|

∑γ∈Γ/Γ′ γ(ξΓ′) =

ξΓ.

Proof. The Γ/Γ′-invariants ofW dH∗(MΓ′) are identified withW dH∗(MΓ) using (1.4.3).It is easy to check that the averaged expression satisfies the defining property of ξΓ.


4.3. Lefschetz properties. Taking direct limits we get a map

f ∗ : H∗(M)→ H∗(MH).

For g ∈ G(Q) the action of g−1 on D induces an isomorphism MΓ → M g−1Γg and

hence defines an algebra automorphism of H∗(M). (This is the same as the actiondefined in 1.4 under the diagonal embedding G(Q) ⊂ G(Af ).) Following Oda [27](and many subsequent authors using a variety of methods, e.g. [22, 18, 31, 6]; wefollow Venkataramana’s formulation [31] here) we can consider the restriction map

Res :=∏

g∈G(Q)

f ∗ g : H∗(M) −→∏

g∈G(Q)

H∗(MH).

Since the restriction maps are morphisms of mixed Hodge structure we also have amap

Res : GrWi Hi(M) −→

∏g∈G(Q)

GrWi Hi(MH).

We say that a Lefschetz property holds in degree i if Res is injective in degree i. Wehave the following criterion analogous to [31, Theorem 2]; the proof is the same as in[31] given the results of earlier sections:

4.3.1. Proposition. If α · [DH ] 6= 0 then Res(α) 6= 0.

Proof. Suppose that Res(α) = 0 for α ∈ H i(M). Let g ∈ G(Q). Let Γ be such thatα ∈ H i(MΓ). Since Res(α) = 0 we have that f ∗(g(α)) = 0. Now g(α) belongs toH i(MΓ′) where Γ′ may be assumed to be a normal subgroup of Γ. Applying f! at thelevel Γ′ we deduce that g(α) · ξΓ′ = 0. Now any γ ∈ Γ preserves H i(MΓ′) (since Γ′ isnormal), so we deduce that 0 = γ−1g(α) · ξΓ′ = g(α) · γ(ξΓ′). Using Lemma 4.2.3 wededuce that g(α) · ξΓ = 0 for all g ∈ G(Q), and hence (by density) for all g ∈ G(Af ).Since g(α) · ξΓ = α · g−1(ξΓ) we have that α · g(ξΓ) = 0 for all g ∈ G(Af ). UsingTheorem 4.2.1 we conclude that α · [DH ] = 0.

Now suppose D is irreducible of complex dimension n. In this case H2(D) = C anda generator is given by the first Chern class of the ample line bundle O(1)BB in theBaily-Borel projective embedding of M∗. (The automorphic line bundle associatedwith the representation of KR on Λnp+ extends to M∗ [21, Prop. 3.4(b)], and O(1)BB

is a power of the extension.) Thus if DH has codimension one, [DH ] ∈ H2(D) isa nonzero multiple of c1(O(1)BB), and so [DH ] has the hard Lefschetz property onIH∗(M∗) (by [5]). So cup product with [DH ] is injective on IH i(M∗) for i < n andhence on GrWi H

i(M) for i < n (using Lemma 2.2.2). This gives:

4.3.2. Theorem. If D is irreducible and DH has codimension one then Res is injectiveon GrWi H

i(M) for i ≤ n− 1.

This applies to the following situations:

(1) G(R)nc = SU(n, 1) and H(R)nc = SU(n− 1, 1)(2) G(R)nc = Spin(n, 2) and H(R)nc = Spin(n− 1, 2)(3) G(R)nc = Sp(4,R) and H(R)nc = Sp(2,R)× Sp(2,R).

22 ARVIND NAIR

(Here G(R)nc is the product of the noncompact factors of G(R). In each case theembedding is the standard one.) Thus we have a Lefschetz property in degrees ≤ nin cases (1) and (2) and degrees i ≤ 2 in case (3). If G and H are anisotropic overQ, so that M,MH are compact this is Theorem 4 of [31]. If G is the split Sp(4) overQ and H is SL(2)× SL(2) over Q one gets a result of Weissauer [33].

If H(R)nc = SU(1,m) embedded in G(R)nc = SU(n, 1) in the standard way thenthe same argument shows that Res is injective on GrWi H

i(M) for i ≤ m. (If G isanisotropic this is Theorem 5 of [31].)

There is also a restriction map on open varieties

Res :=∏g

f ∗ g : H i(M) −→∏g

H i(MH) (4.3.1)

where f ∗ : H∗(M)→ H∗(MH). Lemma 2.2.2 gives:

4.3.3. Corollary. Res in (4.3.1) is injective on the image of H i(M) → H i(M) fori ≤ n− 1 in the unitary case (1) and for i ≤ n− 2 in the orthogonal case (2).

4.3.4. Remarks. (i) I do not know how big this subspace of H i(M) is, for example,whether it is all of H i(M) in these degrees, or indeed, whether it is all of WiH

i(M).(This would be the case were M a smooth algebraic variety.) I do not know inprecisely which degrees one expects the Lefschetz property for H i(M) to hold.

(ii) In the unitary case above WiHi(M) = H i(M) for i ≤ n − 1, thus all the

cohomology in these degrees comes from a smooth compactification. In this case onecan prove the Lefschetz property on all of H i(M) for i ≤ n − 3 by making use ofthe canonical smooth algebraic compactification available in this case [24]. (I do notknow what happens in degrees i = n− 1, n− 2.)

4.4. Cup products. The criterion of Proposition 4.3.1 can be applied to the diagonalembedding G ⊂ G×G to prove results about cup products in the top weight quotientGrW∗ H

∗(M) := ⊕iGrWi H i(M) of H∗(M). The arguments of [31, 8.2, 8.3] yield thefollowing analogues of Theorems 8 and 9 of loc. cit.:

4.4.1. Theorem. (1) Suppose that G(R)nc = SU(n, 1). If α, α′ ∈ GrW∗ H∗(M) are

nonzero classes of degree k and k′ with k+k′ ≤ n, then there is a g ∈ G(Q) such thatα · g(α′) 6= 0.

(2) Suppose that G(R)nc = Spin(n, 2). If α, α′ ∈ GrW∗ H∗(M) are nonzero classesof degrees ≤ bn/2c, then there is a g ∈ G(Q) such that α · g(α′) 6= 0.

Lemma 2.2.2 gives results on cup products in the image of H∗(M) in H∗(M).

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School of Mathematics, Tata Institute of Fundamental Research, 1 Homi BhabhaRoad, Mumbai 400005, India

E-mail address: [email protected]

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