A New Proof of the Global Torelli Theorem for K3 Surfaces

Annals of Mathematics

A New Proof of the Global Torelli Theorem for K3 SurfacesAuthor(s): Robert FriedmanSource: Annals of Mathematics, Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 237-269Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2006942 .

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Annals of Mathematics, 120 (1984), 237-269

A new proof of the global Torelli theorem for K3 surfaces

By ROBERT FRIEDMAN1

To my parents

Introduction

The global Torelli theorem for K3 surfaces seems to have been conjectured first in the late fifties by Andreotti and Weil. Together with Nirenberg, Kodaira, Grauert, and Tyurina, they showed that the periods of a K3 surface give local coordinates in moduli, work which was clarified and greatly generalized by Griffiths in his study of the period map. The global Torelli theorem was proved for algebraic K3 surfaces by Piatetskii-Shapiro and Shafarevitch in 1971 [22], and for Kahlerian (and hence all) K3 surfaces by Burns-Rapoport in 1975 [5], with additional clarifications and simplifications by Shioda [27] and Looijenga-Peters [17]. The theme common to these works is that of Kummer surfaces (or special Kummer surfaces), which are dense in any local universal family of K3 surfaces. There is a straightforward Hodge-theoretic criterion for a K3 surface to be a Kummer surface. Moreover any two Kummer surfaces with the same periods are isomorphic (in a strong sense); this is, after some work, reduced to the statement that a complex torus is determined by the periods of holomorphic one-forms. Combining the last two statements with the density result and a theorem of Bishop one is able to deduce global Torelli for all K~hlerian K3 surfaces.

The aim of this paper is to present an essentially different proof of the global Torelli theorem for algebraic K3 surfaces. Our starting point is the following recent classification of degenerations of K3 surfaces, due to Kulikov [16] and Persson-Pinkham [21]:

THEOREM (0.1). Let 7T: X -+ A be a semistable degeneration of K3 surfaces with all components of gV '(0) = U Vi algebraic, and let N = the logarithm of the monodromy on H2(Xt). Then, after birational modifications, it may be assumed

'Research partially supported by NSF Grants MCS-8114179 and MCS-8300860.



238 ROBERT FRIEDMAN

that Kx 0. In this case, 1'i(O) = UV? is one of the following: Type I: 1- l(o) = X0 is a smooth K3 surface and N = 0. Type II: T-1(O)= VO U V1 U ... UVr, V0, Vr are smooth rational,

V1, . . ., Vr-1 smooth elliptic ruled and Vi n V, $ 0 if and only if j = i ? 1. Vi n V, is then a smooth elliptic curve and a section of the ruling on Vi, if Vi is elliptic ruled. N $ 0, N2 = 0.

Type III: z-V1(0) = UVM, each Vi is smooth rational and all double curves are cycles of rational curves. The dual graph F is a triangulation of S2. N2 $ 0 and N3 = 0.

An easy consequence of the theorem is the suriectivity of the period map for algebraic K3 surfaces. With a little more effort, one can prove that the period map is in fact proper. The main technical point is the following result of N. I. Shepherd-Barron [26] (the Type 0 case, which is all that is needed for properness, is due independently to Dave Morrison):

THEOREM (0.2). Let 'n: X -- A be a semistable degeneration of K3 surfaces with Kx 0, and let Y' be a line bundle on X such that Y'IXt is numerically effective and (ylXt)2 > 0 for t # 0. After ?P has been twisted by Cx(- Z) where Z is effective and supported on XO, and various birational modifications performed on X which leave X semistable and Kx trivial, Y?Xt is numerically effective for all t.

(The necessary birational modifications are the elementary modifications, which are explained in further detail at the end of the introduction.)

Let 92k be the set of algebraic K3 surfaces with a primitive, numerically effective L E Pic(S) satisfying L2 = 2k. As an easy consequence of the properness, we show in Section 1 that the global Torelli theorem for all values of k follows from the assertion that the period map has degree one onto its image for one value of k, for example k = 1. Our strategy now is to enlarge the moduli space of polarized K3 surfaces, by allowing certain degenerations of K3 surfaces, and to enlarge period space by allowing certain "limiting" mixed Hodge structures (essentially one of Mumford's toroidal compactifications of Hermitian symmetric spaces). Next, we extend the period map and show that it is proper. To make a degree calculation, it suffices to locate a point in the enlarged moduli space lying in a singleton fiber and prove that the differential of the extended period map is injective at that point. The calculation of the fiber proceeds via an analysis of the limiting mixed Hodge structure and Carlson's work on the extensions of Hodge structures arising from surfaces with normal crossings [7].



GLOBAL TORELLI THEOREM 239

The injectivity of the differential is based on Clemens' formulas for the Picard- Lefschetz transformation.

It remains to discuss the partial compactifications of moduli space which we shall use. Now Shah [25], using geometric invariant theory, has worked out a compactification of 2, and it seems quite likely that one could prove a degree one statement for the period map on 2 by using his compactification as a point of departure. We have chosen not to do so, and indeed one of the main technical points of this paper is the construction of a partial compactification of 2k for all k > 1. Our compactification '2k is well-suited for Hodge-theoretic computations- k - '2k is a smooth divisor obtained, essentially, by adding on "stable" Type II degenerations-and its construction is hopefully of independent interest. The motivation for working with our 2k comes from the Kulikov-Persson- Pinkham theorem and the deformation theory for the central fibers> occurring in their classification (worked out in [11]), just as the definition of a stable curve is a natural consequence of the semistable reduction theorem for curves and the deformation theory of curves with ordinary double points. Thus, the relationship between our 2 and Shah's compactification is very much like that between 9N3, the moduli space of stable curves of genus 3, and the geometric invariant theory quotient of the space of semistable plane quartics.

Throughout this paper, we only work with a partial compactification of F2k

involving Type II degenerations. Correspondingly, we only compactify period space by adding those limiting mixed Hodge structures which might arise from Type II degenerations, i.e., those whose weight filtration is defined by a rational nilpotent matrix N with N2 = 0. The reason why it is sufficient to restrict ourselves to this case, or, equivalently, why the extended period map remains proper, is the monodromy statement in (0.1). The strata in Mumford's compactification corresponding to N2 = 0 are quite simple to describe, and do not depend on any careful choices involving the subtle combinatorics of toroidal compactifications. On the moduli side, Type III degenerations are likewise considerably more complicated to describe. Their combinatorics (worked out with the help of the global Torelli theorem) is the subject of a forthcoming paper with F. Scattone [13].

Having proved the degree one Torelli theorem for 2, one may easily adapt the Burns-Rapoport-Looijenga-Peters line of reasoning to prove the strong global Torelli theorem for Kilerian K3 surfaces. The only remaining necessary ingredi- ent is an easy density result due to Tyurina [29, Ch. IX, Theorem 5]; our result then splices neatly into Sections 7-9 of Looijenga-Peters.

Finally, we remark that the idea of exploiting Type II degenerations (in the form of rational surfaces with trivial dualizing sheaf and a simple elliptic



240 ROBERT FRIEDMAN

singularity of multiplicity 3) to study families of K3 surfaces dates back at least to Franchetta, and is described in Enriques' book [10, pp. 247-257].

Acknowledgements. This paper is a revised version of a portion of my Harvard Ph.D. thesis. I would like to take this opportunity to thank my teachers there, especially Heisuke Hironaka, Shigefumi Mori, and Wilfried Schmid. I learned much from the SAGS participants, especially Dave Morrison and Nick Shepherd-Barron, and Jim Carlson's ideas and results were helpful both for this work and for its predecessor [11]. I would especially like to thank my thesis adviser, Phil Griffiths, for his assistance, encouragement, and patience. Finally, I would like to thank Kate March for a skillful and patient job in typing this manuscript, and the referee for many constructive suggestions.

Notation and standard facts. (0.3) If 'T: X -a A is a semistable degeneration of K3 surfaces with Kx= 0

X will be called a Kulikov model. (0.4) Let X0 be a variety with normal crossings and D = (XO )sing' If

T~o = Ext'( x C's) is the Lichtenbaum-Schlessinger sheaf of local deformations of the singularities of X0, XO is d-semistable if Tx0 OD. If XO = VO u V1 is the union of two smooth surfaces glued along a smooth curve D,

X0 is d-semistable ND/v ? ND/v, = D

[11 (1.9), (1.13), (2.4)]. (0. 5) If ST: X -a A is a semistable degeneration of surfaces and C C X0 is a

smooth rational curve, then it is sometimes possible to perform certain birational modifications on X with center C, called elementary modifications (of types 0, I, II). These are described in [12, pp. 12-14]. We shall not be concerned with type II modifications here, and give a more detailed description of type 0 and I modifications:

type 0 modifications: Suppose that C n (XO)sing = 0, C2 = - 2 on the unique component of X0 on which C lies, and that C does not extend to a Cartier divisor on X. In this situation, C may be contracted on the threefold X, and the resulting singular threefold X is a one parameter smoothing of an ordinary double point. By Brieskorn's theory [4], X admits two different resolutions with exceptional set a smooth rational curve, X and X', say. The resulting rational map X ---+ X' is the type 0 modification along C (cf. [5, p. 260]). It may also be defined by a somewhat involved series of blowups and blowdowns on X.

type I modifications: Suppose that C meets (Xo)smg in one point and that C2 = _ 1 on the unique component of X0 on which it lies. Let X be the blowup of X along C. The exceptional divisor E P_ P x P' is ruled in two ways, and may be smoothly contracted along either ruling. One contraction yields X and




the other a smooth threefold X'; the resulting rational map X ---+ X' is the type I modification along C (cf. [20, (2.4.5)]). Pictorially, a type I modification looks as follows:

C \/ - 1\C

x x x

We say that the curve C has been "flipped" to C'. (0.6) We will call a line bundle L on a variety V a polarization or

quasi-ample if, for n >> 0, LIn has no base locus or fixed components and defines a birational morphism from V onto its image in some pN. If S is a smooth K3 surface, and L E Pic(S), then L is a polarization L is numerically effective and L2 > 0. In this case, LIn is birational onto its image for n ? 3 and its image is the normal model of S obtained by contracting all curves not met by L, which are rational double point configurations [19].

(0.7) 2k denotes the set of isomorphism classes of pairs (S, L), where S is a smooth K3 surface and L a primitive polarization on S (in the sense of (0.6)) with L2= 2k > 0.

(0.8) Let A be the unique even unimodular lattice of signature (3, 19); A -(- E8)2 @ H3, where H is the hyperbolic plane. If S is a smooth K3 surface, H2(S; Z) is isometric to A. If h e A is a primitive vector with h2 = 2k, then h is unique up to the action of the group of integral isometries of A. Define

AO = {h} 1 c A.

(0.9) Let t2k be the set of Hodge ifitrations on A ? C for which h is of type (1, 1), or equivalently the set of Hodge filtrations on A0 ? C. Z2k is a Hermitian symmetric space with compact dual Z2k. Let r = Autz(A) be the set of integral automorphisms of A and define

r2k= fy e F: y(h) = h).



242 ROBERT FRIEDMAN

It is easy to see that Z2k = the set of lines Cv C A0 ? C such that v2 = 0, v v5 > 0, and that ' 2k has two connected components, corresponding to the choice of a continuously varying orientation in the set of positive two planes in A0 ? R. An easy calculation shows that r2k acts transitively on the set of components of Z2k for all k 2 1; indeed, by (0.8), we may assume that h is in one H factor, and then Autz(H ED H) c r2k c Autz(A), where we use the remaining two copies of H in A, already acts transitively on the two components of i2k. Hence r2k \ Z 2k is connected. The period map is then the map

-i2 k _*r2 k \ in 2k;

we shall refer to r2k/sZ2k as period space. (0.10) The differential of the period map (at the level of the versal deforma-

tion, and without polarizations) may be read off from cup product

H'(Os) ? Ho( Q2) -) H'(0i1) [14].

It follows that the period map (0.9) is locally an isomorphism; we will refer to this fact as local Torelli for K3 surfaces.

1. Some consequences of properness

We begin by describing the reduction of the proof of global Torelli for K3 surfaces to the case k = 1. This will follow easily from the statement that the period map (0.9)

q: F2k r2k \ Z2k

is a proper morphism of algebraic spaces. To give '2k the structure of an algebraic space (even a V-manifold), let

U c Hilb(PN) be the open set of smooth K3 surfaces and K3 surfaces with rational double points embedded by the complete linear system L?", where L is a primitive polarization with L2 = 2k and n ? 3. If G = PGL(N + 1), the natural action of G on U is proper, by an easy adaptation of the result of Matsusaka-Mumford [18] (cf. Step III of the proof of (4.10)). Thus, a geometric quotient U/G exists by standard results in invariant theory (e.g. Popp [23]). The period map q is defined in the usual way for smooth K3 surfaces, and by passing (locally) to a finite cover and using Brieskorn's theory of simultaneous resolution ([4] or [1]) for K3 surfaces with rational double points. By Borel's extension theorem, the period map is then algebraic, and therefore its image is open and dense, by (0.10).

The following theorem is an immediate consequence of the monodromy statement in (0.1), the Type 0 case of (0.2), and the valuative criterion of properness. A slightly different proof, in a somewhat more general context, will be given in Section 4.




THEOREM (1.1). The period map 7 is proper. In particular, by the local Torelli theorem, 71 is surjective.

We now state the version of the global Torelli theorem which we shall prove in this paper.

THEOREM (1.2). For all k > 0, 1) '2k is irreducible; 2) 71 is an isomorphism; 3) If (Sj, Lj) and (S2, L2) are two polarized K3 surfaces and p*: H2(S2)

H2(S1) is an isomorphism of Hodge structures such that qp*(L2) = L1 in cohomnology, then, possibly after T* has been composed with Picard-Lefschetz transformations on H2(S1), p* is induced by an isomnorphism qp: S1 - S2 (which is unique, by [22] and [17, (7.5)]).

Remarks (1.3). 1) Let y E H2(S1; Z) be of type (1, 1) such that y2 = - 2. The Picard-Lefschetz transformation sY associated to y is by definition

s'(x) = x + (x *y)y;

it is an integral isometry of H2(S1) of order 2 preserving the Hodge structure. Since s,(y) = - y and, by Riemann-Roch, either y or - y is represented by an effective divisor on Si, s. can never be induced by an automorphism of S,.

2) The statement (1.2.1), which is hard to find explicitly stated in the modern literature, appears in the thesis of d'Orgeval. However, the proof given there does not appear to be complete.

The idea of the proof of (1.2) will be to use induction, with the crucial step, the case k = 1 of 2), occupying most of the remainder of the paper. We give some preliminary steps.

Step I. To prove 2), it suffices to prove that qj has degree one.

Proof Since q is finite (by local Torelli) and proper, and r2k \V 2k is normal, this is an immediate consequence of Zariski's main theorem.

Step II. For a given k, 3) follows from 2).

Proof Let R2 be the local ring of some point in ' 2k mapping onto the point of r2k \ Z2k corresponding to the polarized Hodge structure of (S,, h1), and G2 = the finite group of automorphisms of the polarized Hodge structure. Let R1 be the local ring of the versal deformation of (S1, h1), and G1 the finite group of automorphisms of (S,, hj). Let W be the abstract Weyl group generated by the reflections in - 2 curves orthogonal to h1 on S,. Then it is known that the local ring of 2k at the image of (S,, h1) is of the form R'G1 . By 2),



244 ROBERT FRIEDMAN

Rw7G = R 2, and we have a diagram

1 2

Rwil R2c

1 ~~2'

where the top horizontal isomorphism follows from the local Torelli theorem. Hence, the action of W * G1 on R2 is identical to that of G2. It remains to deal with the case where G2 does not act faithfully on Z2k, which is covered by the following elementary lemma:

LEMMA (1.4). Let 9 be an integral automrmphism of A fixing h with h2 = 2k. Suppose that q9 is norntrivial but that p is trivial on 2k. Then pIA0 = - Id and k = 1. In this case, the corresponding K3 surface is hyperelliptic and 9p is induced from the hyperelliptic involution.

Proof Since Z 2k C PH2, pIHo2 is a scalar. As qp is orthogonal, 9H2 = + Id. Suppose lIH02 = - Id, and assume (as we may, by (0.8)) that

h = a1 + ka2, a = a2 = a1*2=1

Then 4p(h) = h = al + ka2. But h * (a, - ka2) = 0, so that

9(al - ka2) = - a1 + ka2.

In this case, 2k9)(a2) = 2al, or k9)(a2) = a,. Hence k = 1, Q.E.D.

Step III. If (1.2) is true for 2k' then (1.2.1) is true for 2k+2*

Proof By the properness of the map 2k+2 -+ r2k+2\ 2k+2' we can locate, in every component of 2k+2' a polarized surface (S, L) with a given Hodge structure and such that

Pic(S) = Z L ED Z * y

where y2 = - 2 and L * y = 0. Thus, (S, L) is a K3 surface whose projective model has a rational double point. Set

L' = L + y; (L')2 = 2k

(so that (S, L') is "the projection of S from the double point").

LEMMA (1.5). For a unique choice of sign, L' is ample.

Proof By Riemann-Roch, either y or - y is effective. Assuming that y is effective, take L' = L - y, so that L' y = 2. Now let C be an arbitrary




irreducible curve on S. It suffices to show that L' * C > 0. Write

C= aL + by, a,b e Z.

We may assume that C2 = - 2; otherwise, as C is irreducible, ICl has no fixed components, so that C * L' is automatically > 0. Hence

2(k + 1)a2 - 2b2 = - 2, or

(k + 1)a2 = b2 - 1.

But

C * L' = 2(k + 1)a + 2b; this is positive (k + 1)a > - b. As

(k + 1)2a2 = ((k + 1)a2)(k + 1) = (b2 - 1)(k + 1), and a > O as C L > O, it suffices that (b2 -1)(k + 1) > b2. But (b2 - 1)(k + 1)-b2 = k(b2 - 1)- 1, which is > O unless b = ?1, in which case C = y, Q.E.D.

Returning to the proof of Step III, let (S, L) be as described, and suppose rq(S, L) = q(S1, L1) for some (S,, L1) e -2k+2. Repeating the above construction for (Sp, L1), we obtain two points (S, L') and (S1, L") E F2k with

(* ) ql(S, L') = (S1, L).

Using (1.2) for 2k we have an isomorphism g: S -S 1 with g*L" = L', where g induces the isomorphism on Hodge structures (*). Hence g*L1 = L, and (S,, L1) _ (S, L). In particular, (S, L) occurs in every component of 2k+2. As Zk+2 is locally analytically irreducible, it is irreducible.

Step IV. If (1.2) is true for '2k, (1.2.2) is true for '2k+2*

Proof The argument of Step III yields an (S, L) e '2k+2 with -1q(S, L) - {(S, L)}. By Step I and the properness of q, it suffices to prove that the differential of q is injective at (S, L).

Now the integral automorphism s. defined in (1.3.1) is an elliptic element of order two acting on the primitive cohomology of S. By choosing q(S, L) sufficiently general subject to Pic(S) = Z L @ Z y, we may assume that sY is the unique elliptic element acting on H2.m(S; Z), by standard arguments in the theory of arithmetic groups.

Let S be the image of S under the complete linear system L; S is a singular surface with a smooth rational curve C contracted to an ordinary double point, and the class of C in Pic(S) is y. Let L be the induced Cartier divisor on S and 7r: S --+S the natural map. Finally, let T be the germ of a smooth analytic space



246 ROBERT FRIEDMAN

representing the universal deformation space of the pair (S, L) and T the corresponding germ for (S, L).

LEMMA (1.6). The simultaneous resolution map (Res -+ Def, in the terminology of [1]) realizes T as a two sheeted branched cover of T with branch locus B C T those K3 surfaces for which C remains a Cartier divisor.

Proof. Without the polarizations L and L, (1.6) follows immediately from the fact that Res -- Def, for an Al double point, is a two-sheeted branched cover [1], [4], and the fact that the deformations of S are versal for the double point (cf. Burns-Wahl [6], especially (1.10) and (2.14)). The statement for polarized deformations follows from the fact that the locus in the universal deformation space for S where L is Cartier is a smooth divisor meeting the corresponding divisor where C is Cartier transversally, and an easy local calculation.

Returning to the proof of Step IV, let U be a small neighborhood of ij(S, L) in r2k+2 \ Z2k+2 and U 5 2k+2 a connected open set such that U/{ sy } = U. Then we have the following diagram:

B c T-+ UC 2k+2

2 -11

-y= U C r2k+2 \ Z2k+2'

It suffices to prove that the differential of ij is infective, which follows from the usual cohomological calculation of the local Torelli theorem (0.10).

Step V. (1.2.1) and (1.2.2) are true for k = 1.

Proof. It is well-known that the general K3 surface with a polarization of degree two is a double cover of p2 branched along a smooth sextic. Since the space of all such is irreducible, 1) follows. The proof of 2) will be given in Section 5.

2. Polarized Type II degenerations of K3 surfaces

This section is concerned with technical results on arranging useful models of polarized Type II degenerations, refining Shepherd-Barron's theorem (0.2) in the Type II case. We begin with some motivation for the definition of a stable Type II K3 surface. In the Type II case of the Kulikov-Persson-Pinkham theorem (0.1), with 7r-r(0) = X0 = Vo U ... U Vr, V0 Vr rational and V1,, . . , V,-1 elliptic ruled, the elliptic rule components are, in a certain sense, redundant. Indeed, via type I modifications (0.5), we can arrange that Vl,...,Vr_- are minimal ruled surfaces by successively flipping exceptional curves in fibers to the




end surfaces V0, V,. By standard results on contractions, there is a (singular) threefold X obtained by contracting the divisor V1 U ... U V>_ on X to a curve, and a degeneration ST: X -- A with central fiber X0 = VO U Vr, where VO is glued to VX along the obvious isomorphism of the double curves defined by the rulings of the Vi, 1 < i < r - 1. That XO is again d-semistable follows immediately from

LEMMA (2.1). Let S be a minimal elliptic ruled surface, E1 and E2 two disjoint sections on S and ap: E1 -- E2 the natural isomtorphism. Then

(* ((E2) e dEI(El) =EI.

Proof Since Pic(S) Pic(El) ED Z[El], we may write, in Pic(S),

[E2] = [E1] +a .f, where a E Pic(E1) and af stands for p'- (a), where p: S -) E1 is the ruling. But then

*9 E2C(E2) = a = E1(- E1),

proving (2.1), Q.E.D.

The following theorems, the main results of this section, are concerned with arranging a projective contraction and controlling the embedded deformation theory of the image.

THEOREM (2.2). Let En: X -) A be a Type II degeneration of K3 surfaces and ? a line bundle on X such that Y?X, is a polarization for t # 0. After suitable elementary modifications on X have been performned and ?? twisted by (x(Y2iaiVi), for suitable integers ap,

1) Y?f is numerically effective and there exist sections of H?(X, ?) which are nonvanishing on Vi for every i;

2) V1, ... ., V- 1 are minimal ruled; 3) YJVi is a sum of fibers, 1 O for all i.

THEOREM (2.3). If rr: X -) A and ? are as in (2.2), then, with Y? replaced by ?)n , n ? 40, and with certain modifications as in (2.2), 2 satisfies the conditions (2.2.1)-(2.2.4), and in addition,

1) ? defines a morphism X PN for an appropriate N with image X which is birational to X.

2) If X0 is the image of X0, X0 is obtained by contracting all curves C in VO U V, such that Y. C= 0 and gluing the resulting surfaces V0, VM along



248 ROBERT FRIEDMAN

3) The curves C contracted in 2) form rational double point configurations disjoint from Do U D, -1, and

ND/ ? ND/ ND. I V. D,-1/IV, =(DO

under the natural identification Do-=D 1.

Remark (2.4). (2.3.3) exactly means that, aside from some mild double point singularities, X0 is a d-semistable variety. We will use (2.3) to construct moduli spaces in Section 4. On the other hand, to make the computations of Section 5, it is easier to use (2.2) before passing to a large tensor power.

Proof of (2.2) and (2.3). The proof will be in several steps. We begin by recalling the algorithm in [26] for proving (0.2) in the Type II case. Let 's: X -l A be a Type II degeneration and ?' as in (0.2). By an easy application of the semicontinuity theorem, 2= (x(E) for some effective divisor E on X. After twisting by (x(- Z), Z supported on X0, we may assume that E does not contain any component of X0, so that g2'1 Vi is effective for all i. It is easy to check that if C C Vi is some irreducible curve with ?f C < 0, then C is smooth rational and either (C2), =- 2 and C does not meet any double curves (call these (?) curves) or (C2)v =- 1 and C meets exactly one double curve (call these (t) curves). The algorithm in [26] for proving (0.2) is then:

1) Make elementary modifications along all (?) curves C such that ?- C < 0; this process terminates.

2) Make an elementary modification along a (f) curve C such that ?- C < 0 and then repeat step 1).

It is shown in [26] that this procedure cannot be repeated indefinitely, so that eventually 2' becomes numerically effective.

Proof of (2.2). By (0.2) we may assume from the beginning that ?' is numerically effective and that there are sections of ?' which do not vanish on any component of X0. Now, if V1 is not minimal, there are (f) curves C meeting D1 which may be flipped to V2; we repeat this until V1 is minimal. Similarly, if V2 is elliptic ruled and not minimal, we may flip all (t) curves to V3, and so on. In the end, we obtained a birational model X' of X with V,', ... ,X'>, minimal ruled. Let ?'1 denote the strict transform of ? on X'. Note that V0 is unaffected and that Y V0' = ?' I V0.

We may write ?1 Vi = ai Di + a combination of fiber components,

1 0. Hence

' l Vi' = aiDi + a sum of fibers, where ai > 0.




Now modify Yj as follows. Twist Yj by Qx(- al(V2' + V3' + ... +Vr- =

ox(a1(Vo' + V1')). This has the effect of subtracting a1D1 from S2lV,', adding a1Dj to ?fIV2', and leaving Yj unchanged elsewhere. In particular, ? 1tVy' - a D1 is a sum of fibers, and 2'1! V2' + a1D1 = a'D2 + fibers if V2' is elliptic ruled, where a' is again ? 0. After iterating this procedure for V2', V3' ..., 5Vr 15 we obtain a line bundle Y'2 satisfying:

(i) Y21V ' is a sum of fibers, 1 < i < r - 1; (ii) '2t1 Vr = YN IV,' + aD_ - where a ? 0.

Note that Y21 V/ is effective, since it is the proper transform of I V', which is effective, plus a * Dr, - a ? 0.

To prove (2.2.1)-(2.2.3), apply the following modification of the algorithm of (0.2) to 2?:

(1) Make elementary modifications along all (?) curves C lying on V0' or V7' such that Y'2 * C < 0.

(2) Given a (t) curve C' on V0' or V7' such that Y'2 * C < 0, make a sequence of type I modifications to flip C to the opposite end, i.e. to either Vr' or V0'; then repeat (1).

Although Y'2 need not have sections which are nonvanishing on every component, an easy modification of the proof of (0.2) in [26] shows that the algorithm above terminates. Indeed, let E be the divisor of a section of H?(X, ?) which does not vanish on any component Vi of X0, and E' be its proper transform on X', so that

Y?1 = Ox (E)

Then Y21 Vo = a v,(E') and $'21 Vri = (v,(E + aDr- 1) The proof of Theorem 1, Case I in [26, p. 138] applies verbatim to show that

(1) of the modified algorithm above terminates. As for (2), note that E' remains effective after birational modifications of types 0 and I. Hence E' Do ? 0 after each stage of the algorithm. Moreover, since Y21 Vj' is a sum of fibers, 1 < i < r - 1, and (2) leaves the ruled components minimal, for all i,

92 * DI = E' * Do 2 0.

However, as in the proof of Case II of [26], the quantity ?2 * Di strictly decreases after a sequence of type I modifications along a (t) curve C with Z2 * C < 0 as in (2); so the modified algorithm terminates as well. If the resulting line bundle and degeneration are relabeled Y and X, note that 2 V0 and Y V, are effective and numerically effective and ?'j Vi is a sum of fibers, 1 0 or



250 ROBERT FRIEDMAN

(Y Vr)2 > 0. Suppose that (1V0)2 > 0. By the Hodge index theorem, (Do)2v < 0, and an application of (2.1) yields

(Do)27 + (Dr_1)27 = 0?

so that (D 1)2v > 0 and 1 Vr is trivial. By Riemann-Roch, [21 V0I defines at least a pencil on V0, and so contains

enough sections to sweep out V0. As Ah' Do = 0, there must be a (connected) element of 2t VJ of the form Do + F. where F is effective. Since (Do)2v < 0, Fe Do > 0.

Replace 2' by 1, where

1 =S? Ye OX(Vo) ? OX (Vo + V1) C) * x(Vo + + V.-j. Since Ox(V0 + * Vr) = Ox(X0) = Ox, we see that

Y Vo = (1 Vo} ? C'1f- D) = Cvo(F).

YlVi is a sum of fibers, 1 0.

By [26, (2.17)], there is a section E of H0(X, Y') whose restriction to VO is Do + F. Moreover, on Vr, the map

H0((v,(Dr-i)) HO((D _1(Dr- )) is surjective. Since (Dr 1)2v > 0, IDri is at least a pencil on V, and contains smooth irreducible members. Perform the following algorithm on X:

(1) Make all type 0 modifications along (?) curves C c VO such that 1 * C < O.

(2) If C is a (f) curve on VO with Y2 * C < 0, perform a sequence of type I modifications to flip C to V, and repeat (1).

By using the section E of H0(X, 2) above, (1) terminates. After performing a sequence of type I modifications as in (2), the line bundle Y, I V, is replaced by Cv,(D,'Q1). Moreover (Dr,_1)2vt = (F' Do") vo 0, where F' is the transform of F' on V0'. Since (D, 1)2, < (D- 1)2,, this procedure terminates; let the result be relabeled as ., X, etc. Then Y2- Do = Fee Do = (Dr- )V. Hence, if .,? Do-0 =0, (F) is trivial and 21 V, = Cv)(D-1). As (2X0)= (1V42 > 0, this last possibility is absurd, so that (2.2.4) holds. Clearly ?'' VO and ?1 V, are effective and numerically effective. By [26, (2.17)] and Lemma 4 of the appendix to [26], Y satisfies the second part of (2.2.1), completing the proof of Theorem (2.2).




Before giving the proof of (2.3), we record the following result from the appendix to [26]; it will also be used in Section 5.

PROPOSITION (2.5). Let S be a smooth rational surface and D a reduced member of I - KI. Let L be a line bundle on S which is effective and numerically effective, and such that no component of D is a fixed component of L. Write L = Lf + Lm, where Lf is the fixed part of L and ILmI has no fixed components. Then:

(2.5.1) Lf # 0 only if L2 > 0. If L2 > 0, either Lf = 0 or (2.5.1.1) Lf = C is smooth rational, C2 = - 2, and Lm = kE where E is smooth elliptic, E2 = E * D = 0, E * C = 1, and k ? 2. (2.5.1.2) Lf= ElCi, where C2 = -2, 1 1. All Ci's are smooth rational, and Lm either contains smooth irreducible members or is of the form kE as in (2.5.1.1).

(2.5.2) If L2 = 0, then either (2.5.2.1) L = kC, C smooth rational, C2 = 0, C D = 2, and the morphism q)c: S -+ P1 exhibits S as a (not necessarily minimal) ruled surface, or (2.5.2.2) L = kE where E is as in (2.5.1.1).

(2.5.3) If L2 > 0, then nL has no fixed components or base locus for n > 2. If n ? 3, TnL is a birational morphism to its image and cpnL(S) is the normal model of S obtained by contracting all curves C such that L - C = 0; these are smooth rational curves C with C2 = - 2 or - 1.

(2.5.4) If Lf = 0, L has a base locus only if L * D = 1, and L contains smooth members.

Proof of (2.3). We assume that Y satisfies the conclusions (2.2.1)-(2.2.4). Since (1X0)2 = (r1 V0)2 + (21V)2, we may assume, after perhaps relabeling, that (?'V)2 > 0. Moreover, if C is a (f) curve with ?- C = 0, we may assume C c VO. Indeed, if C c V4, performing a sequence of type I modifications flips C to V0, without violating any of the assumptions (2.2.1)-(2.2.4). Define

.0 = ,yo sn (2 7 -V 2 CX( o V 2 {3 ...

0(- Vo VR

Then

rI V0 = ( V0) O(Do);

Yl Vi is a sum of fibers, 1 0 for i =0 ,r, O-Do > 0, and O- C>0 for all (f) curves C.

Proof Since V0, Vr are rational surfaces, (Ki)2 ? 9 for i = 0, r. From

(Do)2 + (Dr_ 1)2 = 0

DI E|- KVI, we obtain - 9 < (DO)2 < 9 and similarly for (D_1)2v. Set Li = " Vi and let D denote either Do C VO or Dr -1 C Vo. Since Li D > 0 by (2.2.3),

(nLi ? D) - D = n(Li - D) ? D2>2n-9 > 0 if n ? 10,

and

(nL0 + D)2 = n2(Lo)2 + 2n(Lo - D) + D2 > 2n -9 > 0.

By (2.5.4), there exist smooth irreducible sections of Lr on V1, so that

- 2 < 2pa(Lr) - 2 = (L )2 (Lr D)

(nLr -D)2 = n2(L)2 - 2n(L, * D) + D2

? n2(Lr)2 - 2n((Lr)2 + 2) + 9

= (n2 2n)(Lr)2 - 4n - 9.

Since, by hypothesis, (Lr)2 > 0 and n2 - 2n > 0,

(n2 - 2n)(L)2-4n-9 2 n2-6n-9 > 0 if n ? 8.

By Riemann-Roch, then, (nL0 + D) and (nLr - D) are effective with positive self-intersection. To check that they are numerically effective, it suffices to check that they meet all (?) and (f) curves non-negatively. If C is a (?) curve, C * D = 0, so that

(nLi ? D) - C = nLi - C > 0.

If C is a (f) curve, either C C V0, C Lo 2 0 and C - D > 0 or C c V, C -L> 0 and C -(nLL - D) ? n - 1 > 0, so that JP- C> 0 as desired.

To complete the proof of (2.3), we let JP be as above. Then by [26, Theorem 2(i) and Theorem 2W], ?1'? has the properties (2.3.1) and (2.3.2) for m ? 4. The remaining property (2.3.3) follows since JP meets all (f) curves positively and by (2.1), Q.E.D.

Remark (2.7). In case X0 = VO U V1 consists of only two components, there is a simple geometric interpretation of passing from ? to ?9. Essentially, the morphism defined by ?9m, m >> 0, contracts some (f) curves to threefold




double points. In the resulting threefold, V0 and V1 are no longer Cartier divisors, and the morphism defined by ? ' is precisely the blowup along V0.

3. The mixed Hodge structure on H2(X 0)

The purpose of this section is to collect various facts, mostly well-known, about the mixed Hodge structure on H2(Xo), where X0 is the central fiber of a Type II degeneration of K3 surfaces, and to discuss the relationship between H2(X0) and the limiting mixed Hodge structure LH2(X0). The following definition is motivated by (2.2) and (2.3).

Definition (3.1). A stable K3 surface of Type II, X0 = V0 U V1, is a d-semistable surface (0.4) with normal crossings such that V0, V1 are smooth rational and D = (XO)sing E I - Kv I for i = 0, 1, where D is a smooth elliptic curve.

THEOREM (3.2) [11, (5.10)]. If X0 is a stable K3 surface of Type II, there exist semistable degenerations n: X -- A with central fiber X0. More precisely, the versal deformation space of X0 contains a unique smoothing component of dimension 20 whose discriminant locus is a smooth 19-dimensional divisor consisting of the locally trivial deformations of X0 which remain d-semistable.

(3.3) If 7T: X --a is a semistable smoothing of X0, the condition of d-semistability may be rephrased by saying that X0 has a canonical Cartier divisor

(3.3.1) = XNo)Ixo or equivalently

(3.3.2) Jv 0 = Ov(- D),

lVi= ev1(D).

Changing the ordering of V0, V1 simply replaces ( by -

The next lemma follows easily from the Mayer-Vietoris exact sequence for X0 = Vo U V1 and the fact that, if S is a rational surface, h2(S) = 10 - K'S.

LEMMA (3.4). Let X0 be a stable K3 surface of Type II. 1) dim H2(X0) = 21. 2) W2H2(Xo)/WlH2(Xo) has dimension 19 and is of pure type (1, 1). 3) WJf2(X0) _ H'(D) as Hodge structures.

Let mr: X -, A be a semistable smoothing of X0 as in (3.3) and LH2(Xo) the limiting mixed Hodge structure. It is not canonically defined, but since N2 = 0(0.1), W2LH2(Xo) is intrinsically defined (cf. also [11, ?3]).



254 ROBERT FRIEDMAN

LEMMA (3.5). 1) The Clemens-Schmid exact sequence [14, (10.16)],

H4( X0 ) -H2(XO) LH2(XO) 2(XO),

is exact over Z. 2) W1H2(X0) W1LH2(X0). 3) W2LH2(Xo)/W1LH2(Xo) I{} '/Z (, if ( is viewed as a class in

H2(fX) = H2(V0) e H2(V1).

4) The signature of the intersection pairing on W2LH2(Xo)/W1LH2(Xo) is (1, 17).

Proof 1) follows from the fact that the Clemens-Schmid exact sequence is obtained by splicing together the Wang sequence (X * = X - XO):

0 = H'(Xt; Z) -* H2(X*; Z) H2(Xt; Z) NH2(Xt; Z)

and the exact sequence of the pair (X, X*):

Hl(X*;Z) -3 H4(X0;Z) - H2(Xo;Z) - H2(X*;Z) -H3(XO;Z)

together with the straightforward calculation that H3(XO; Z) = O. Since the Clemens-Schmid sequence is an exact sequence of mixed Hodge structures, 2) is clear, and 3) follows similarly from the observation that the image of H4(XO) in H2(X0) is (. Finally, 4) is a consequence of the Hodge index theorem.

In Section 4, we shall use the following generalization of (3.5.1).

LEMMA (3.6). Let T: X -, A be a semistable degeneration of K3 surfaces of Type II. Then the Clemens-Schmid sequence

H4(XO) - H2(X0 ) -*) LH2(X LH

is exact over Z.

Proof By the Wang sequence and the exact sequence of (X, X *) above, and since the Clemens-Schmid sequence is exact over Q. it suffices to check that H3(Xo; Z) is torsion-free. Let X0 = U'.O?V with V0, Vr rational and the remaining surfaces elliptic ruled. From the Mayer-Vietoris sequence of XO, we obtain (notation as in ?2):

? = 3Or lH3(Di;Z) (130rH3(Vj;Z) H3(Xo;Z) (13Or lH2(Di;Z) = zr.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3Vj Z 3XO Z 0 2(i;Z

Since H3(Vj;Z) = 0 if V, is rational and Z2 if V, is ruled, H3(Xo;Z) is torsion-free, Q.E.D.




The usefulness of (3.5.2) is that it reduces computations on LH2(XO) to computations on H2(Xo). To exploit this information, we shall need the following result of Carlson:

THEOREM (3.7) [7]. 1) The mixed Hodge structure on H2(Xo) determines a homomorphism

4: W2H2(Xo; Z)/WlH2(Xo; Z) -* D.

2) If L = (Lo L2) e PicVo e PicV1 is a class satisfying

Lo .D= L1' D

and [L] e W2H2(X0)/W1H2(X0) is the corresponding cohomology class, then

{([L]) = CD(Lo) ? (D(- L1) E Jac(D) = D.

Hence, in particular, 4([L]) = 0 for any Cartier divisor L on XO, and therefore A(( = O.

PROPOSITION (3.8). Let 7T: X A-* be a semistable degeneration of K3 sutrfaces with central fiber X0 a stable Type II K3 surface and N the logarithm of the monodromy on H2(Xt; Z), so that N = I - T. Then N is a primitive integral matrix.

Proof Let c,: X, -* X0 be the Clemens collapsing map [8, (4.1)], [20, (2.3)], and let UL = c7-'(VX)' Then D = U0 n U1 is a homotopy S1 bundle over D. Let { a, /P1 be a standard homology basis for H1(D; Z) and &, /3 the S1 bundles over a,.3 in D; we may view &,/3 as elements of H2(D), H2(Ui) or H2(X,). By [8, (5.6)], we have the following formula for N acting on H2:

N(y) = (y * 3) &- (y - &)/3. In particular, { &, 131' = W2LH2(Xo) c H2(Xj), so that { a, 13} generates a two dimensional subspace of H2(X,). Moreover, N is primitive if the subspace generated by { a, /3) over Z is a primitive subspace of H2(Xt; Z).

Up to homotopy, Uj = Vi - D, so by the Gysin sequence,

H3(Vi; Z) = 0 -* H1(D; Z) -* H2(Ui; Z) -* H2(Vi; Z)

is exact. As H2(Vi; Z) is torsion-free, the image of H1(D; Z) = span{&,,1} is primitively embedded in H2(LUi; Z).

From the Mayer-Vietoris sequence,

H2(Uo; Z) e H2(U1; Z) -* H2(Xt; Z) -* H1(D; Z)

is exact. It will therefore suffice to prove that the image of H2(Uo; Z) E H2(U1; Z) in H2(Xt;Z) is primitive. Since the statement of (3.8) is independent of the choice of birational model of X, we may assume (after type I modifications) that



256 ROBERT FRIEDMAN

(D2)Vo = (D2)V1 = 0. In this case the homotopy S' bundle D is a trivial bundle over D, so that

HI(D; Z) = Z3 is torsion-free, Q.E.D

Remark (3.9). In general, if X0 = VO U V1 U ... UVr-j U Vr, with r -1 elliptic ruled components, it is easy to see that the logarithm of monodromy is equal to rN', where N' is the primitive integral.

Proposition (3.8) describes the behavior of the period map near the singular variety X0, in directions normal to the discriminant divisor. To describe the differential of the period map in directions tangent to the discriminant locus, we state without proof the following result from [13] (Remark (3.8)).

PROPOSITION(3.10). If X0 is a stable Type II K3 surface, the differential of the variation of limiting mixed Hodge structure on the discriminant locus of the smoothing component of the versal deformation space of X0 is an isomor- phisrn.

We will use (3.10) in the next section in the construction of partial compactifications of moduli space. For the reader who is only interested in global Torelli theory, a way to avoid (3.10) will be sketched in Section 5.

Finally, we discuss polarizations on X0 briefly.

Definition (3.11). A polarization L on stable K3 surface X0 of Type II is a Cartier divisor L which is numerically effective and satisfies L2 > 0. Two polarizations L1 and L2 are equivalent if

1 L2mod(;

i.e., L1 and L2 define the same classes in H2(X0; Z)/Z (. Finally, L is primitive if its image in H2(X0; Z)/Z - is primitive.

4. Construction of partial compactifications

The purpose of this section is to construct partial compactifications of ~2k and r2k \ X 2k and to extend the period map. The compactification of r2k \ Z 2k

turns out to be a piece of Mumford's toroidal compactifications [2], and we shall use it to construct the corresponding compactifications of 2kI following a suggestion of Mori. The arguments involved are not quite standard, as the group actions that arise are not proper.

(4.1) By well-known arguments in the theory of Hermitian symmetric spaces, the Baily-Borel boundary components of r2k \ Z 2k are either points or quotients of the upper half plane, corresponding to maximal real parabolic subgroups which are stabilizers of rational isotropic subspaces of dimension one




or two, respectively. Define (r2k \ Z2k)o to be the Baily-Borel compactification minus its point boundary components and set

r2k \ Z2k = inverse image, under the natural map, of (r2k \ Z2k)o in any of Mumford's toroidal compactifications [2].

(It is easy to see that the space r2k \ 2k is independent of the choice made.) The following theorem is a result of the constructions of [2].

THEOREM (4.2). 1) r2k \ Z 2k is a quasiprojective V-manifold containing r2k \ Z2k as a dense open subset.

2) The complement r2k \ Z 2k - r2k \ t)2k is, modulo finite group actions, a smooth divisor in r2k \ Z 2k'

3) Let f: Aln X A* -+ r2k\ Z 2k be a map, locally liftable to Z 2k with unipotent monodromy T such that, if N = log T,- then N2 = 0. Then f extends to a map

f: A' r2k \ Z2k

4) In the situation of 3), the differential of f in directions normwal to the divisor An X {0) C An ' is injective 4 N is a primitive integral matrix.

Here, (4.2.3) and (4.2.4) follow easily from the statement and proof of the theorem on page 279 of [2]. (One should interpret (4.2.4) as only holding modulo elliptic fixed points, or at points f(x, 0) which are general points of r2k \ V)2k- r2k \ V 2k)

We must make precise the connection between boundary points of r2k \ X 2k

and limiting mixed Hodge structures. For each r2k conjugacy class of primitive integral nilpotent N in the Lie algebra of 0(2, 19)R, define

U(N)z = {exp(kN): k eZ,

U(N)c = {exp(XN): X e C).

It is not hard to show that the set U(N)c * Z2k is just the set of filtrations F0 D F' D F2 of A0 ? C which, together with the monodromy weight filtration W1 C W2 C W3, yield a mixed Hodge structure on AO ? C. Therefore,

U(N)C \ U(N)C - )Z2k

is the set of nilpotent orbits of such limiting mixed Hodge structures, and Mumford's construction of r2k \ Z 2k iS obtained by gluing onto r2 k/ 2k

(U(N)c \ U(N)c * Z 2k) rN

for each conjugacy class N, where TN is the set of y E r2k which preserve the monodromy weight filtration associated to N.



258 ROBERT FRIEDMAN

Via the proof of [2, theorem on p. 279], in the situation (4.2.3) of a map f: A'n X >K* + r2k \ Z2kI the extended value f(x, 0) is the limiting mixed Hodge structure of Schmid (cf. [24, (6.16)]) and hence of Steenbrink, by [28, (5.10)].

Cattani and Kaplan (unpublished) have also considered Mumford's toroidal compactifications from the point of view of mixed Hodge structures.

We turn now to moduli. Let X0 be a stable K3 surface of Type II (3.1) and h a polarization on X0 (3.11). Let (, ) denote the Yoneda pairing

Ext'(0, ex0) ? H'(X1, --i H2(Xo, Xo0) = C.

By Serre duality, Yoneda pairing is dual to cup product

H0(XO X wxo) ? H1( X0, x) XX

Since XO - Yoneda pairing is perfect.

PROPOSITION (4.3). The polarized versal deformation space of X0 has a unique smoothing component of dimension 19 with discriminant locus a smooth divisor of dimension 18.

Proof Without the statement on the polarization, this is (3.2). The Zariski tangent space to polarized deformations is given by the subspace of TX1, Ext'(i2k1, e'o) defined by

{a EETxo: (a, h) = 0.

Let W c Tx' be the hyperplane corresponding to the smoothing component of the versal deformation of X0.

Claim (4.4): W= (a E Tx1: (a,f) = 0).

Proof of the claim. Note that ( E H'(Xo, ao) is non-trivial, as its image in H'(V0 o1) e3 H'(V 1, 1) is not zero. By counting dimensions, it suffices to prove that W is contained in the right hand side of (4.4), or, equivalently, that ( survives as a Cartier divisor on any one-parameter family X -- A with central

fiber X0 and Kodaira-Spencer class ( a ) E W. But this is obvious if

X -a A is a semistable smoothing, by (3.3.1), and obvious if X -a A is a locally trivial family of d-semistable varieties by (3.3.2).

Now we return to the proof of (4.3). Since Yoneda pairing is perfect, it suffices to show that ( and h are independent classes in H'(Xo0 ao) But, in H1(V0, S4~) e H1(V1 Q1V1), 42 = 0 and h2 > 0, which proves (4.3).

LEMMA (4.5). Let X0 be a stable Type II K3 surface and h a polarization on XO so that Th(XO) C PN is the normal model of X0 obtained by contracting




rational double point configurations on X0 disjoint from D. If X0 is the image of Wh, the embedded deformation space of X0 C pN is formally smooth over the polarized versal deformation space of X0.

Proof Consider the normal bundle sequence (Tk0 = Ext'(Q40, cP'), 0 - TX-* H0 (X0, e pNJX0) -) HO(XONX/P TX'- H'(X0, eNjxO)(,

and the Euler sequence

0 __ _D 1X(1) )pNIX 0.

One sees easily that Hl((9xo(1)) = H2(CpX?(1)) = 0 and that H2(0X0) = H?(Xo) = C. Hence, the image of H0(XO NXO,/pN) in TX' is a hyperplane and is contained in the set of polarized deformations of X0, hence is. equal to these. Finally, TXO = 0 since TXO = 0, by an easy calculation along the lines of [11, ?5]. Hence the action of PGL(N + 1) on the Hilbert point corresponding to X0 is formally free, and this proves (4.5).

Remark (4.6). It is not hard to show, by the methods of Burns-Wahl [5], that the deformations of X0 are versal for the double points of X1.

(4.7) Let k > 0 be given and let n be any integer ? 40. Set

N= 2 *n2 2k+ 1 = kn2 + 1.

Let U C Hilb(PN) be the open set consisting of smooth K3 surfaces S and K3 surfaces S with rational double points with minimal resolution p: S -* S, such that p*Cg(l) = nL, L is primitive and L2 = 2k; thus U is the analogue of the set U introduced in Section 1. Let U C Hilb(PN) consist of U, together with X% a stable Type II K3 surface, possibly with rational double point configurations disjoint from D, such that, if

p: X0 _+X0

is the minimal resolution of X0 at the rational double points, p*C)x(l) = H, where H is equivalent to nL, L is a primitive polarization with L2 = 2k (3.11). It follows from (4.3) that U is open in its closure in Hilb(PN), and PGL(N + 1) acts on U.

LEMMA (4.8). The period map p: U -k r2k \ Z 2k extends to a morphism fr U-'r2k V2k.

Proof Let (XO, h) E U - U and let V be a small neighborhood of (XO, h). After a finite base extension by a certain Weyl group W, say V, the family over V can be simultaneously resolved over the set of rational double points in XO, by Brieskorn's theory. The resulting family is pulled back from the polarized versal



260 ROBERT FRIEDMAN

deformation space of (X0, p*h), where p: X0 -k X% is the minimal resolution of the double points. By (4.3) and (4.2.3), there is an induced extension of the period map P: V -_ r2k \ Z 2k* Since W is naturally represented in F2k by Picard-Lefschetz transformations and P is easily seen to be W-invariant, P descends to a map p, as desired. The situation is summarized in the following diagram:

v pr~~k \ 22k

V

The remaining point is the construction of F2k* Set G = PGL(N + 1).

Definition (4.9). A useful quotient of U by G is a normal quasi-projective variety Y~k and a morphism r: U -+2k such that

1) r is surjective. 2) 7r(p) = rr(q) - every G-saturated open set containing p meets every

G-saturated open set containing q. 3) '7(U) = '2k is a coarse moduli space for K3 surfaces (possibly with

double points) with a primitive polarization of degree 2k. 4) There is a commutative diagram:

-F2k r2k \ ) 2k

where 7q is proper and finite.

THEOREM (4.10). A useful quotient of U exists.

Proof Following a suggestion of Mori, consider U, r2k \V 2k as schemes and take all sheaves in the Zariski topology. Then set

'F2k = SpeC e]ak\Zak

where - denotes the integral closure of r2k\Z2k in 0u. (Note that p is dominant.) Thus, "2k is normal and separated and there exists a diagram

qr ff r2k \ q) 2k

2k

where 7q is a finite morphism. Hence g2k is quasi-projective.




The rest of (4.9) will be broken up into several steps.

Step I. ST is surjective.

Proof Clearly, the image of S is open and dense. Let p E i2k. Then there exists a map f: A 2k with f(O) = p and f(A*) C iT(U); after finite base change, there exists a lifting h of f to Hilb(PN) with h(A*) C U. Let T: X -* A be the flat family corresponding to h. After base change and birational modification, we may assume that X is semistable and Kx = , and such that ?2x (1) = nLt for Lt E Pic(Xt), t # 0. Since h is a lifting of a map from A to F2k\ Z2k, the monodromy of X -- A must satisfy N2 = 0, and hence X is of Type I or Type II. We assume that X is of Type II (the Type I case is much simpler).

LEMMA (4.11). There exists Ye Pic(X) with 2'J XXt- = 9x(1), uwhere (x(l) is the natural line bundle induced from X C A x PN.

Proof Let 1 e H2(Xt; Z) be the class of Lt. Then 1 is invariant under monodromy, since nl is. By (3.6), there exists a class 1 E H2(X0; Z) H2(X; Z) whose restriction to Xt is 1. Applying the exponential sheaf sequence on X, we obtain

H'( ) - H2(X; Z) -* H2(X; C9X) = H0(A, R2'T*Cx).

Viewing the class 1 as a section of H0(A; R27T*Z), one sees that its image in H2(X, (9X) is 0, and hence that there exists Y e H 1(C9*) inducing 1, which proves (4.11).

Returning to the proof of Step I, let ? be as in the lemma. After further modifying X and ?'?(n as in (2.3), we may assume that ?1'?n satisfies the conclusions of (2.3). Thus, choosing a basis for 7T *(Y9?) gives a map g: A U such that

I7TO gA* = iT ?hlA* = fIA*.

In particular, 7T o g(0) = f(0) = p, so that p e Imr.

Step II. (4.9.2) holds.

Proof Let p e '2k Then iV-1(p) is a union of G-orbits. On the other hand, by (4.5) and (3.9), the dimension of 7V-1(p) = dimG, and since Aut(Xo) is discrete for X0 a K3 surface or a stable Type II K3 surface, i - 1(p) is a union of a finite number of G-orbits 01 5 ?m Write Oi = G * pi for pi (e . Then there exist locally closed algebraic subsets Pi of U containing pi such that the Kodaira-Spencer map is an isomorphism, at all points of Pi of the tangent space at x onto the hyperplane in Tx1 consisting of smoothing directions, if Xx is



262 ROBERT FRIEDMAN

singular and corresponds to x E Pi, and is an isomorphism onto H'(X.; ex ) if Xx is smooth. Again by (3.9), jPP is quasi-finite. We may assume further that PinP1= 0,ifi~j.

Let P = UP, c U. Then G equivalence defines a relation R0 c P x P. R0 is an open equivalence relation with discrete fibers. Define R to be the closure of R0 in P X P.

LEMMA (4.12). 1) R is discrete, i. e. R [ p] is discrete for all p. 2) A categorical quotient P/R exists as an analytic space. 3) The natural map P/iR -* 2k is an open immersion.

Proof 1) Clearly R[p] is an algebraic set. Moreover, -p factors through R0, hence through R. Therefore, dim R[p] = 0 because of (3.10).

2) This is in [15, ?4]. 3) Since P/R is categorical, - induces a map P/R -k

This map is quasi-finite, since the map to r2k \ Z2k is. By the analytic case of Grothendieck's formulation of Zariski's Main Theorem [9, (8.12)], P/R -* 2k iS an open immersion, Q.E.D.

To prove (4.9.2), we may assume that q E P. in the above notation, and that r7(p) = ir(q). Then (4.9.2) is clear from the definition of R.

Step III: 'r(U) = 2k as coarse moduli spaces.

Proof By monodromy considerations, 7T(U) = q -1(r2k \ 2k) is open, and 7T(U) n 'T(U - U) = 0. It suffices to check that the action of G on U is proper, and hence that it is proper in the algebraic sense. This essentially follows by the theorem of Matsusaka-Mumford. Briefly, the argument is as follows: by the valuative criterion for properness and an easy reduction, it suffices to show that if (X, Y') and (X', ?'") are two polarized families of K3 surfaces over A and ff: X X' is a birational map which is an isomorphism except over 0 such that f * = Y', then T,(X0) - , ) as polarized varieties. But let S be the ring

S = e * n?0

and similarly for S'. Since f is an isomorphism in codimension one, f *S' = S. Let X = ProjjS = ProjjS'. Then, for T: X A- , w(X0) = sT-V(O) = ,(X'), which proves Step III and (4.10).

Remark (4.13). If (X, 2') and (X', 2"') are polarized degenerations of K3 surfaces of Type II and f: X ---b X' is a birational map which is an isomorphism away from 0, then by [26, (3.1)], f is a sequence of elementary modifications of types 0 and I. Thus, (X, 2) and (X', 2") differ exactly by making such modifications and twisting by components of the central fiber. This result (which we shall not need) explains the nonproperness of the action of G on U.




5. Torelli at the boundary in degree two

In this section, we specialize the preceding results to k = 1; the general K3 surface with a polarization h with h2 = 2 is a double cover of P2, branched along a smooth sextic. We identify four distinct arithmetic possibilities for the Baily- Borel boundary components of Type II, and prove a Torelli theorem for one of these.

We use the following notation, for (XO, h) a polarized stable K3 surface of Type II with h2 = 2.

(5.1): 1) W2/W1 = W2LH2(XO)/W1LH2(Xo). 2) (W2/Wi)prm = {h}' in W2/W1, of dimension 17. 3) Y?(W2/W1)o = the integral vectors in (W2/Wi)prnm. 4) ( c- ?(W2JW1)0, defined by

oF = {y e y(W2/JW1)0: y2 =-2).

It will turn out that (1 is always a root system in (W2/Wl)o. Next, we describe various ways of constructing (XO, h). The situation will always be as follows: start with YO, Y1, two copies Of p2, with Di C Yj a smooth cubic and p: Do-* D1 an isomorphism. Blow up 18 points on Do U D1 and let the resulting surfaces be V0, V1. Glue these back together along w. There are 20 moduli involved: one for the j-invariant of Di, one for the choice of q9, and 18 for the choice of the points {Pk } on Do U D1. However, we will also be given a polarization h = (h0, h,) E H2(V) ED H2(V1). There will be one condition to make V0 U ,Vl d-semistable and one condition to make h the class of a Cartier divisor on X0 = V0 U V1; these conditions may always be arranged, and we shall always assume this has been done in what follows.

(5.2): Let 10 and 11 be the class of lines in Y0, Y1 pulled back to Vi and let el, ... ., el8 be the classes of the exceptional divisors in H2(V) ED H2(V1). We proceed to list the possibilities for h.

(5.2.1) h = (l1, 11): The linear system on X0 maps each Vi to Yj, then onto p2, contracting all exceptional curves. The general member of h consists of two smooth rational curves meeting at 3 points. Here

(1 = {e - e: < ?i ,j< 18,i j)

with simple roots { ej - e2+ 1 }. 1 is of type A17, with all systems of simple roots obtained by permuting the e2 's.

(5.2.2) h = (310- - - * - e8, 311 - -e - e16): We consider cubics on Y0 passing through the points p1,... , P8 with the ninth point of intersection of the general cubic in this series with Do = q, say, and do the same on Y1, arranging that q(q) = the unique base point in the series. (Compare



264 ROBERT FRIEDMAN

(2.5.4).) The general member of h is thus two smooth elliptic curves meeting transversally at one point. The simple roots are given by

el - e2, ... e7- es, lo - el - e2- e3;

e9 - e10,.. ., el5- e16, 11 - e9 - e10 - el,;

e17- el.;

so = E8 + E8 + Al. (5.2.3) h = (310 - el - e7, 11 - e8): We consider cubics on YO

passing through 7 points of Do and meeting Do residually in 2 points q1 and q2, and the pencil of lines in Y1 passing through one point of D1. The general member of h consists of a smooth elliptic curve and a smooth rational curve meeting transversally at two points. The simple roots in 1' are given by

el- e2, ..,e6- e7,10- el - e2-e3;

eq - e 10 11- e8 - e9 - eloelo-eel, ...,e17-el;

so '> = E7 + D10. (5.2.4) h = (2(10 - el), 21 - e2 - e3): We consider conics in Y1 passing

through 2 points of D1 and meeting D1 residually in 4 points, along with twice the fibers of a ruling in YO. The general member consists of three smooth rational curves CO, C1, C2 with C1 n C2 = 0 and C1 CO= C2 CO= 2 with transverse intersections. The simple roots are given by

e2 - e3;

e- e5

10- el - e4- e5 e5 -e6,...,e17 e18 -(11 -e2 -e3-e8);

so ? = D16 + A1.

Remark (5.3). Namikawa and Kitaoka (unpublished) have shown that the possibilities for 1s are exhausted by (5.2.1)-(5.2.4). This also follows from:

THEOREM (5.4). If (XO, h) is a polarized stable K3 surface (3.11) with h2 = 2, then (XO, h) is a specialization of a surface described in (5.7.1)-(5.7.4), possibly after modifications as in (2.2).

Proof Via (2.2), we may assume hi = hIVi is effective and numerically effective, with h2 > 0 and hi * D > 0. Consider first the case where the general member of hi is connected for i = 0, 1. Then

2pa(hi) - 2 > - 2.




As 2pa(hi) - 2 = h-2 - hi - D, we obtain

2pa(ho) + 2Pa(hi) = ho + h2 -ho h D - * D + 4

= 6 - 2ho* D.

Thus, Pa(ho) + Pa(hl) + ho D = 3. Moreover, we cannot have pa(ho) = 2, or else h, - D = 1, pa(hl) = 0 so that hl = - 1. Hence Pa(hi) = 0 or 1. We list the possibilities:

(5.4.1) Pa(hO) = Pa(hi) = 0, hi D = 3,

(5.4.2) Pa(ho) = pa(hl) = 1, hi D = 1,

(5.4.3) pa(ho) = 1, Pa(hi) = 0, hi D = 2.

In case (5.4.1), hi has no fixed components or base locus, by (2.5), and Ihi contains smooth members Ci. From Cj2 = 1 and

0 H-(* - H0(C0,(Ci) -- H0((Pc(Cj)) -?

we obtain dim Ihi = 2 and Thi: Vi P2 is a birational morphism; as hiD =3, D is the proper transform of a smooth cubic, and (XO, h) is as described in (5.2.1).

In case (5.4.2), h2 = 1. Refer to (2.5.1.2); if hi has fixed components they must be as described there. In particular, we may contract successively Cr, Cri, ..,Ci until we attain h2 = 1, hi - D = 1 and IhiI is a linear series on Vi without fixed components and with the unique base point hi D. Let Vi' be the result of contracting any possible Ci's and then blowing up hi D, and let h' be the strict transform of hi on Vi'. Thus, the general member of I 'I is a smooth elliptic curve and I 'I is an elliptic pencil with h' - D' = 0. where D' = the strict transform of D. Since the sections of I h sweep out Vi' and h' - D' = 0, aD' + YDE E h1' for some a > 0, Di effective and irreducible. As Ih1' has a section (the blowup of hi * D), a = 1. Moreover, all Di's are smooth rational with Dj2 < 0, and each connected component of ED, is an exceptional curve.

Let Vi" be the result of contracting Cr, ... ., C1 on Vi. Then (Kvi)2< 1 and I-v,, I contains a smooth elliptic member, D". An elementary argument shows that Vi" dominates P2 in such a way that D" is the strict transform of a cubic. As D" + ED. E jh"j, we are clearly in case (5.2.2) where the possible Ci's arise if we make (possibly infinitely near) blowups at the base point of I V.

In case (5.4.3), an argument similar to that above shows that VO dominates p2 with ho as in (5.2.3). On V1, jh1I defines a pencil of irreducible rational curves, which exhibits V1 as a (nonminimal) rational ruled surface, by (2.5.2.1). Since V1 has a smooth effective anti-canonical divisor, an elementary argument shows that V1 actually dominates Fo F1, or F2, and will fail to dominate F1 only



266 ROBERT FRIEDMAN

if it is minimal. However, after making a type I modification along one of the exceptional divisors in VO not meeting h0, we may always assume V1 is nonminimal. Thus, V1 and hi are again as in (5.2.3).

The remaining case is where one of h0, hi is not connected. If, say, hi fails to have connected members, then h2 = 0 and h1 = k C, where C is smooth rational and C D = 2. (Recall that h is connected, so that hi * D # 0.) Then ho = 2 and the members of ho are smooth and connected. Hence, as pa(h) = 2, necessarily k = 2 and ho * D = 4. Hence, pa(ho) = 0. Let Z be a smooth member of I ho. From

0 -O H0(Cv9) -* H0(CvO(Z)) -- H0(C9(Z)) -,

h0(Cv(Z)) = 4. In particular, Ihol defines a morphism from V0 to a quadric in P3. An easy argument, as in the previous cases, shows that we are in case (5.2.4),

Q.E.D.

COROLLARY (5.5). 1) The root systems described in (5.2) are the only ones which occur.

2) The root system determines the geometric configuration (XO, h); i.e., the cases (5.2) and their specializations are mutually exclusive.

3) If the root system is A17, the polarization h has no fixed components or base locus and realizes X0 as two copies of p2 glued along a smooth cubic and blown up at 18 points.

Remark (5.6). Shah [25] has classified semi-stable degenerations of double covers of p2 branched along a smooth sextic in the sense of geometric invariant theory. E. Looijenga (unpublished) has determined the corresponding root systems. The correspondence is as follows: In [25], Theorem (2.4), group II means a Type II degeneration. Then we have the following table:

root system semi-stable sextic

A17 a smooth cubic with multiplicity 2

E8 + E8 + A1 three conics passing through 2 fixed points and meeting doubly at each

E7 + D1o four confocal lines, plus a fifth line counted with multiplicity 2 which does not pass through the point of intersection of the other four

D16 + A1 two conics, one of them double, which meet in 4 distinct points.




The following is the main result of this section.

THEOREM (5.7). Let (X0, h) be a polarized stable K3 surface of Type II with h2 = 2 and M'(W2/W1)0 a root system of type A17. Then the limiting mixed Hodge structure and the class of the polarization in LH2(Xo) determine (XO, h) up to type I modifications.

Proof. We know that (XO, h) is as described in (5.5.3). We must recover the j-invariant of D and the 18 marked points, up to projective equivalence. The j-invariant of D is immediate from the Hodge structure on W1.

Next, by (3.7), the extension of mixed Hodge structures ? -

W1 -*(WA) (Wf2/Wl)O ?-

defines a map A: 2(W2/W1)0 -- J1W1 = D. Choosing a system of simple roots Yi in Y(W2/W1)0, we obtain 4(yi) E D. In the notation of (5.2.1) we may assume D is blown up at {Pl '...P181 and that yi = ei - ei+1 for some ordering of the set { p1, , P18 1. By (3.7),

+(yi) = pi - Pi+, Ee D. We have thus the system of equations on D

P1iP2 = A(YJ

(5.7.1) 4 P17 P18 = OY17)

lPl + +P18 =0,

where we assume 1iJD = 3(0) and the last equation reflects d-semistability:

0 = Ov(D)ID X Pv1(D)ID- PD(9 (() + 9 0() P1 P2 P18).

Unfortunately, (5.7.1) has 182 solutions. Indeed, if 18 - q = 0 on D, then for (Pi ... P1) a solution, (P + q... p18 + q) is a solution as well (and all such solutions are of this form). On the other hand, addition of a 3-division point is a projective automorphism of D, so that the configurations (pl1. P18) and (p1 + qI.. , P18 + q) are projectively equivalent if 3 - q = 0 on D.

We have not, however, used the full information of LH2(XO).

LEMMA (5.8). There is a unique 8 E (W2/Wl)z satisfying h 8= 1, r1 8 = 3, Yi * 8 = 0,i > 1.

Proof As { h, y1, ... 17 } is a Q-basis for W2/W1, the uniqueness is clear. As for the existence, we may assume (Xo, h) is realized as in (5.2.1), and that



268 ROBERT FRIEDMAN

yi = ei - ei+1, Set

8 = l0 - 3e1 (mode).

Then (8 D) v = (d * D)v,, so that 8 is indeed a class in (W2/W1)z, and the remaining statement is clear, Q.E.D.

To complete the proof of (5.7), note that A(8) is an additional invariant of the mixed Hodge structure. Replacing pi by pi + q replaces A(8) by A(8) - 3q, and these will be equal if and only if 3 - q = 0, hence only if the configurations (D, pi,.. , PJ) and (D, P1 + q... , p18 + q) are projectively equivalent. We have proved (5.7), Q.E.D.

Completion of the proof of Theorem (1.2). It suffices to show that the map 2

- F2 \ '2 has degree one. Let (X0, h) E- 2 be a stable Type II K3 surface with periods of type A17. By (5.7),

71 1(Xos h) = {(XO, h)}.

Letting S prorepresent the germ of the smoothing component of the polarized versal deformation of (X0, h), we have a diagram

S

I2 \ \2-

2

By (3.10), (4.24), and (3.7), the differential of v is infective at (X0, h), and hence the same must be true for i by (4.12.3), Q.E.D.

Remark (5.9). We may eliminate the appeal to (3.10), i.e. the calculation of the differential of q at (X0, h) E 2k - F2k. This was used to construct jk

above. Without actually constructing gk' we may prove (5.7) directly. Then, we may enlarge 2 by adding in only those polarized stable K3 surfaces whose periods have the requisite A17 root system and similarly for F2 \ Z 2. The proof of (4.10) works equally well in this case, where e.g. we obtain (4.12) since (X0, h) is determined uniquely by its limiting mixed Hodge structure. Finally, we may appeal to Sard's theorem: if qj is a bijection from a component of 'F2 - 'F2 to one of F2 \ Z2 - F2 \ Z22 then, on a dense open set of F - F2, its differential restricted to the tangent directions of 9 - A2 must be infective.

COLUMBIA UmVERsrrY, NEW YoRK, N.Y.




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(Received March 7, 1983)

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A New Proof of the Global Torelli Theorem for K3 Surfaces