Global Smoothings of Varieties with Normal Crossings

Annals of Mathematics

Global Smoothings of Varieties with Normal CrossingsAuthor(s): Robert FriedmanSource: Annals of Mathematics, Second Series, Vol. 118, No. 1 (Jul., 1983), pp. 75-114Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2006955 .

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Annals of Mathematics, 118 (1983), 75-114

Global smoothings of varieties with normal crossings

By ROBERT FRIEDMAN'

Introduction

The goal of this paper is to develop a formalism to study the deformation theory of compact analytic spaces with normal crossings, and in particular to study special cases in which these spaces are smoothable. One ingredient in the theory is the calculation of the cotangent complex of Lichtenbaum-Schlessinger [LS], along. with some simple local algebra related to normal crossings singularities. The other ingredient is, perhaps surprisingly, the theory of limiting mixed Hodge structures from the point of view of Steenbrink [S]. Our motivation is the following. By Mumford's Semi-Stable Reduction Theorem, any degeneration 7T: -> \ A of algebraic varieties may, after base change and birational modification, be brought into a standard form

X c 9C

0 E A ,

9 smooth, X = V- '(0) a reduced divisor with normal crossings. Now a general variety with normal crossings may not be smoothable at all,

or it may be smoothable in many "different" ways. Intuitively, however, an X as above has a natural smoothing direction (given by A). In this case, using the one variable degeneration, Steenbrink has constructed a limiting mixed Hodge structure on X.

From our point of view, the condition that X be a fiber in a semi-stable degeneration implies intrinsic conditions on X, both topological ("the triple point formula") and, more subtly, analytic (d-semi-stability (1.13)). Our guiding principle has been that any construction possible with a central fiber in a semi-stable degeneration is possible with a d-semi-stable variety as well. The main example of this principle is in Section 3, where we construct an intrinsic "limiting" mixed

'This material is based upon work partially supported by the National Science Foundation under Grant No. MCS-8114179.



76 ROBERT FRIEDMAN

Hodge structure on any d-semi-stable variety X (in a suitably modified sense), which agrees with that of Schmid and Steenbrink in case X is a central fiber in a semi-stable degeneration. In particular, the Betti numbers and Hodge numbers of any smoothing of X are a formal consequence of the geometry of X itself and do not depend on the choice of a smoothing component.

In Section 5, we present the major application of our techniques. Using the recent results of Miranda-Morrison [SAGS] we complete the classification of degenerations of K3 surfaces due to Kulikov and Persson-Pinkham [K], [PP], by showing that all those which, combinatorially, can arise in Kulikov's list do in fact occur. We are also able to describe explicitly the local structure of the versal deformation space. The analysis of Section 5 has at least two further applications, to the study of smoothings of cusp singularities (joint work with R. Miranda) and to the study of partial compactifications of the moduli space of K3 surfaces which arise in the study of the global Torelli problem for polarized K3 surfaces (to appear).

For notational simplicity, we have only considered the case of global normal crossings in this paper. However, all of our results go through with only minor modifications in the case of local normal crossings. The sheaves in question must simply be modified by certain locally constant sheaves (the transition functions are + 1), which are trivialized in case of global normal crossings by choice of an ordering of the components. The necessary formalism is explained in [DI, p. 75].

This paper is a revised version of a portion of my Harvard Ph.D. thesis. It is a great pleasure to thank my many teachers there, including Heisuke Hironaka, Shigefumi Mori, and my thesis adviser, Phillip Griffiths. I had helpful conversa- tions with Francesco Scattone. Finally I would like to thank the NSF for its generous financial support.

1. Kihler differentials

Throughout this paper, X = U N1X will be a compact connected variety with normal crossings of pure dimension n. Thus, each Xi is a compact connected complex manifold, dimcXi = n, and X is normalized by

N

a: X = H ~ Xi X. i=1

Moreover, locally at each point p of X, X is isomorphic to the closed subset of an open polydisk in Cn l defined by

{(Z1 ...* Zn+) z1 ... Zk = 0).



SMOOTHINGS OF VARIETIES 77

Define

ap : =IP I (xo n ..n Xi) X> io< ... Hip ?P

note that X?01 = X, XI1] = (X sng), and in general X[PI is a smooth variety of dimension n - p.

Recall that, if X is any analytic space locally embedded in a smooth variety V, the Kdhler differentials S1 can be defined locally by the conormal sequence

lx/lx -*x ?- 0,

where lx is the ideal sheaf of X in V. If moreover X is reduced and a local complete intersection (as will be the case when X has normal crossings), the conormal sequence becomes

0 -I Ixj - _ > ?, 0,

where IX/I2 is locally free as an Qx module. If 5 - A q14 there is a natural derivation d: Q- -* Q+1.

LEMMA (1.1). If X has normal crossings, the complex (Q- 5 d) is a resolution of the constant sheaf C. In particular

Hk(X; C) = H k(X, U.

Proof This is a special case of a more general theorem of Reiffen [R]. By functoriality, there is a natural map

Uk a-*- 2 a k = (ao)*, Qk [0]

whose kernel rX consists of those differentials in Qk supported on Xsing (since a is an isomorphism away from Xsing and (a 0)* Q2k[0] is torsion free).

Definition (1.2). Let Tr be the subcomplex of Q2x consisting of sections supported on Xsing' By abuse of language, call Tx the subcomplex of torsion differentials of X.

We will usually write TX = TX

Definition (1.3). Let Fp q = (ap)* * [pi and set

j.. =ED cp q. p, q

6F is a double complex with differentials d, 6 as follows: d is the de Rham operator X X[P}. 6 is the Cech boundary operator, an alternating sum of restriction maps

based on a choice of an ordering for the index set (1,..., N).



78 ROBERT FRIEDMAN

Let D = d + 8 be the total differential, 6jyk- = p+q=kjp q, and (N.Y; D) the associated single complex.

LEMMA (1.4). (N.; D) is a resolution of the constant sheaf C.

Proof [GS]. Let 5k{(6Y ) be the ith cohomology sheaf of (N.Y, D). Then there is a spectral sequence abutting to 5Ck(' () whose E2 term is

E2 8 Ca~d ( 6f f

By the holomorphic Poincare lemma, (6P,, d) is a resolution of the constant sheaf C on X[P], and a straightforward argument (e.g. [GS], p. 70) shows that

(a0)*C -> (al)*C -* ...

is a resolution of the constant sheaf C on X. Hence 'Ep q = 'Ep- q = C in dimension 0, and the same must be true for 'X('(6).

Splicing together the map Uk- a*Qk with the 8 complex in Definition (3.1), we obtain

(K.D.) 0 -Tk k -* (aO )*2k[oI _* (al)*E2kw

PROPOSITION (1.5). 1) The sequence (K.D.) is exact. 2) The complex (0i /jx, d) is a resolution of the constant sheaf C. 3) If each Xi is a Kdhler manifold, the associated spectral sequence

converging to Hk(X, U. /X) = Hk(X; C) whose E1 term is

ElPq = Hq(X, Sx/rP)

degenerates at the E1 term. 4) With assumptions as in 3), the corresponding filtration (FP) on Hk(X; C)

is the Hodge filtration arising from the functorial mixed Hodge structure on Hk(X; C) defined in [DII] and [GS, ? 4].

Proof 1) The exactness of

0 -> -> -* (ao )*X[?o]

was remarked prior to Definition (1.2). At the next stage, we may assume, as the question is local, that X c V, where V is an open neighborhood of 0 in Cn I', and X is defined by

(1.5.1) X = {(Z *, Zn~l) Zl *) Z = 0) C V.

The conormal sequence implies that

-kVX

k \~ Xk a X5




so it suffices to check that

Im(i2vIx ' a *,2 ) - ker (a *)*2[ ).

Clearly, the image is contained in the kernel. For the opposite inclusion, use induction on dim X and on 1. Thus, we may write, for Xi = {zi = 0) c V,

X = X' U Xl, Y= X'I ) Xi

where X' = U i=,Xi and Y C Xi is a divisor with normal crossings. Given q9 = (q( ..., (PI) E ker(a*2k -* (a,)* U2k[,1), by induction there ex-

ists X E 2 k with woXi = qi, i * 1. Moreover (PI - wIX, vanishes on Y, so that, on Xl, (PI - (IXI = 4 is in the ideal generated by

z i 1 Z irdz i +, A dZ i

- A dz,

for {il...., il,1) = (1,...,1- 1). Clearly such elements ( can be lifted to k forms ( on V which vanish on Xi, i * 1. Thus, when X = (A + ,

(IXi=(pi foralli.

At the remaining stages, we use the above notation and inductive hypotheses as well. Thus,

X = (')IIXl, [PI = (X/)[P] IIy[P- 1] x = (X')11X1, X~ ~ -

and hence

0 - (al)O2~P-1 - (a) [ (a P)*k-)P] ()Uk 0, p ? 0

is exact, and leads to morphisms of the appropriate Cech complexes which are exact up to sign. Taking the long exact cohomology sequences, we obtain, for

p 2 25

5cs ((a._ )* ukft-l]) -*'K((a. )* Ukt]) _* CP ()* Uk(, I]

0, p > 2 0

by the inductive hypotheses, and for p = 1 we obtain QC^((a)*E2( )=

coker( 4k -* a*2EA) = 0, by the proof of the first part. 2) Let (N, D) be the double complex of (1.3). Reversing the order of taking

cohomology in the proof of (1.4), we obtain a new spectral sequence "E"p q with E2 term



80 ROBERT FRIEDMAN

By 1), G5QQY' q= 0, i > 0, and C(ff' )= q 2 /rql. Hence "E2 = "Em . There-

fore, 5Cd(2 X/TX) = 0, i > 0, and

d~d (XgX )= C.

3) Using the resolution of S2q/Txq by ( q, 8) and taking hypercohomology, we obtain a spectral sequence abutting to Hk(X; UI/Xq ) whose E1 term is

Er s = Hs(X[r]; &2q[r]) =* Hr+s(X, 0 q/T )

By [GS, (4.8)], (DqE' s degenerates at the E term, and the result is an associated graded space of H*(X; C). Thus if the spectral sequence

F p, q = Hq(X; P/TxP) ) HP +q(X; C)

does not degenerate at the F1 term, i.e. if there exists a nontrivial differential somewhere in the spectral sequence, this would force

dim H*(X; C) < dim H*(X; C),

which is absurd. 4) This follows from the explicit description of the Hodge filtration given in

[GS, ? 4].

Remark (1.6). The corresponding spectral sequence whose E1 term is Hq(X; i2p) = HP+q(X; C) does not in general degenerate at the E1 term.

Next, we study the torsion part Tx = Tx of 2. First assume X is described locally via (1.5.1). Thus

Tx -5X>a*ax x

Define

i = Z ..2i Zdzi.

Letting X1 = {zj = 0), a component of X, we see clearly that q9iX1 = 0 for all j, and hence r(q(j) E Tx. On the other hand, if r(Efi dzi) E Tx, Ef dzi vanishes on each X1, and hence z jf, i = j. Thus, the r(pi) generate Tx over Ox

There is one obvious relation 1 1

1 1

hence Er(qpi) = 0 in Tx. Since 021 = (021Ax)/(EZpi) . Ox, this is the only relation. These local calculations can be described more intrinsically. First, we give

some notation which will be used throughout.




Definition (1.7). Let Di = Xi (U j1,Xj) be the variety with normal crossings defined (locally) by {zi = 0, z1 zi = 0). Define D, the double locus to be Xsing = U iDi. Finally, let D = liDi C X, a divisor with normal crossings in X.

Now notice that

jAi

Thus, there is an induced map

a * I D,/i (? I, TX,

where

ID,= ideal sheaf of Di in Xi,

Ix, = ideal sheaf of Xi in X.

Here, IDi/I2 is the conormal bundle of the (singular) divisor Di C Xi. Moreover, Ix is generated locally by the single element zi. Of course, zi is a zero divisor in ex so that Ix is not an invertible sheaf. This can be remedied as follows.

LEMMA (1.8). X, IDi and IXjID are locally free on Di and D, respectively.

Proof In fact, IXI1Dj = ?X ? OD, = IXIJD,1 where JD, = (Zi, Z1 ...

i... z) is the ideal sheaf of Di in X. Hence it suffices to prove that f Zi xE IXJD= f E JD , for all f E Ox.

But if fzi = gzi, g E JD' then f = g + h, where h E Annex(zi). Since Anne (zi) = (z1 * z * *D, f E JD as desired. A similar argument works for

'XIlD

Definition (1.9). Let

?D(_ X) = (ID,/I2t)

??(It/ ID),

?D(( X) = (4IX 1 I4D) ()D (IX2/IX2ID) CED ... ?D(IX/IXI ID)'

and let ED(X), the infinitesimal normal bundle, be the line bundle on D dual to ?D(- X)*

PROPOSITION (1.10). 1) OD (- X) and ?D( - X) are locally free of rank one on Di and D, respectively.

2) There is a natural exact sequence

0 > ?D( X) -> eiOD(- X) -> TX ->?0

Proof 1) is a restatement of (1.8), and 2) is simply the intrinsic reformula- tion of the local computations preceding (1.7).



82 ROBERT FRIEDMAN

The remainder of this section is devoted to a discussion of the meaning of ?D(X).

LEMMA (1.11). Suppose that X c V is a divisor (globally) in a smooth (n + 1)-fold V. Then

?D(X) = ?V(X)ID,

where C)v(X) has the usual meaning. A similar statement holds for CD (- X).

Proof This is immediate from the definitions.

COROLLARY (1.12). Suppose r: -> A is a proper flat map from the smooth (n + 1)-fold 9C to A = {z E C: IzI < 1), with 7T-'(0) = X scheme-theoretically (i.e., i: -> A is a semi-stable degeneration). Then CD(X) = eD.

Proof: Since ef(X) -6, this follows from (1.11).

Definition (1.13). If 9D(X) = OD X is d-semi-stable.

Hence, if X is a reduced fiber in a one-variable degeneration of smooth n-folds, such that the degeneration has smooth total space, X is d-semi-stable. This is the typical situation arising from Mumford's semi-stable reduction theorem [Ke]. The central theme of this paper is to what extent the converse is true.

To illustrate these concepts somewhat, consider the example of a "tetrahedron", X = 4 planes in general position in P3. The double locus D is then the union of the six lines comprising the "edges" of X, and D is singular along the four vertices V1, V2, V3, v4 which are the triple points of X. Using the exact sequence

0 eD eD fJDID ?

and the local computation

lengthc (eD,vi/eD,Vi =2,

one obtains Pa(D) = hl(?D) = 3. In fact, a little argument shows that Pic(D) (C*)3 X Z6. Thus, given a class t E Pic(D), the projection of t to Z6 represents topological obstructions for t to be trivial (essentially just the triple point formula ([P, p. 39]), whereas the component in (C*)3 represents a more subtle analytic obstruction.

By (1.11), ?D(X) = ?D(4), the restriction of e p(4) to D C P3. Thus, X is quite far from being d-semi-stable. Indeed, the total space of the general pencil of quartics

9= {(z, t) E P3 X A: tFO(z) + G(z) = 0),




where Fo(z) = z0z1z2z3, say, and {G(z) = 0) defines a general quartic, is singular at the 24 points {G(z) = 0) n D. For general G, these singularities are ordinary double points. After a single blow-up, these points become F0 = P1 x P1, which may be contracted along either ruling smoothly; this has the effect of blowing up a point pi E {G(z) = 0) n D = (P..1.., P24) in one of the planes it lies in, then gluing back along the proper transform of the line. One checks easily that in the resulting smooth threefold 9, X is replaced by X with

eD(X) = ?D(4 - 1 .. P24 = eD

Remark (1.14). Additionally, one can prove the following: 1) lp P3(4)1 cuts out the complete linear series I(eD(4)1 on D; hence, if one

wants to blow up points as indicated above to make X d-semi-stable, the only possible choices are 24 points cut out by a quartic in P3. Generically, any such choice will occur as a central fiber in a semi-stable degeneration.

2) 24 points on D vary with 24 moduli. Since Pa(D) = 3, there are three conditions they must satisfy to be cut out by a quartic, and since Aut(X) is 3-dimensional, we end up with 18 moduli. As we shall see in Section 5, this number is, not coincidentally, one less than the number of moduli of quartic K3 surfaces.

2. Deformation theory

After recalling the theory of deformations of a complex space (see [LS], [Pa], [FK] for more details), the general formalism is applied to study varieties with normal crossings, and in particular d-semi-stable varieties.

Let X be a reduced local complete intersection, locally embedded as such in a smooth variety V. The conormal sequence is then

0 IX/I TIX x 0,

a short locally free resolution of S 1 Its dual is

0 -> Tx vx Nx/v

where Tx` = sheaf of derivations of X, 9v = tangent bundle to V, and NX/V Home (IX/I2, ex) is the normal bundle to X in V. Recall that, locally, Tx is the cokernel of this map:

Tx = coker(ev NxIv)

Hence Tx measures the failure of the normal sequence to be exact. More intrinsically, under our assumptions

Tx = Exte (0x 5 ex).



84 ROBERT FRIEDMAN

The global candidate for a Zariski tangent space to moduli is in our case,

Tx = Ext' (Q2k, ex) (global Ext).

The local to global spectral sequence for Ext yields a spectral sequence with E2 term

E2P q = HP(X, Extq(E4, ex)) = ExtP?+q (Qx ex) for p + q = 1, this is simply an exact sequence

0 -> H?(Tx0) Tx -> H0(T' H2(TX).

Remark (2.1). The terms in this exact sequence have geometric meaning. First, the space H'(Txo) is, by analogy with the smooth case, the tangent space to the set of all "locally trivial" deformations of X, i.e. those which are locally a product (where locally means with respect to the topology of the total space). Hence H'(Tx0) is naturally a subspace of Tx. Next, given a class in Tx' its image in H0(Tx) measures how the singularities of X deform infinitesimally.

Remark (2.2). There are natural Kodaira-Spencer maps in this situation [Pa]. Given 7: -> A a flat proper map with V- '(0) - X, the Kodaira-Spencer map 6 is a map

0: A, 0 ->Tx

There is also a local analogue, which is obtained by composing 6 with the projection Tx -> H0(Tx); it describes the change in the singularities of X in the family i: -> A.

Alternatively, the conormal sequence

0 IX/IX > Q X > > 0

1p

ex defines an extension of Si by e ; hence classes in Ext'(021 ex) = Tx and H0(Ext'(S1 , ex) = H0(Tx). (The sheaf map I /I2 -ex _21X is injective because X is reduced and hence generically smooth.) An easy calculation shows that the extension classes are, up to a non-zero scalar, the image of d/dt under 0.

Now we specialize to the case where X C V. an open neighborhood of 0 E Cn+l, is defined by {f(z) = ... z = 0). Thus locally IX/IX is generated by fmod f2, and

dfmod f2 = .z z d z.dzi (E j Ix. 1




Dualizing, we obtain

TX = e/(z1 ... Z^ *i zli = 1525... I)=(D (D =Xsing).

This identification is local; the global result is

PROPOSITION (2.3). There is a canonical isomorphism

TX - D(X)

where eD(X) is defined in (1.9).

Proof Without making any choices, we have

T1 (IX/Ix 2) '$ C)D=6D (iX~al" OD = eD(X).

COROLLARY (2.4). If X is d-semi-stable,

TX1 - e D.

In particular, if X satisfies the hypotheses of (1.12), TX -D.

The following result is a partial converse.

PROPOSITION (2.5). Suppose that X is d-semi-stable and r: - A a deformation of X such that the Kodaira-Spencer class 6(d/dt) generates Tx at all points. Then D is smooth near X.

Proof. It suffices to show that 02 is locally free of rank = dim DC. Suppose we show that Ski1x is locally free. As X is reduced,

rank f1x = dimX + 1 = dim'D.

Given p e X, choose a small neighborhood U of p and sections (pq,..., 5(Pn+1 of 01 which restrict to a basis of S1 Ix (n = dim X). By Nakayama's lemma,

are a basis for 01 near X; hence S2 is locally free at p, which is (2.5). The proof that S1 Ix is locally free proceeds in two steps.

Step I: Ext'(02k1I, (Jxe) = 0 for all i ? 1.

Proof: For i = 1, consider the coboundary map 6 in the Ext sequence arising from the short exact sequence

(2.5.1) ? O 1 t-x ?:

6: Hom(ex, ex) -> Ext'(0 , ex) Ext'(02x5, ex) Extl(ex, ex) = 0.

Now 6(Id) = local extension class of (2.5.1) in Ext'(02k, ex) (see, e.g. [GH, p. 722]). Thus, if this generates Ext'(012k, ex) everywhere (cf. (2.3)), Ext'(02Ix5, Ox) = 0.



86 ROBERT FRIEDMAN

For i > 1, the Ext sequence of (2.5.1) shows that

Ext'(Q'l I, eX) )- Ext'(Q , eX)

As Q2 has (locally) a short free resolution, all of these groups vanish for i > 1.

Step II: Ext'(02kIx, 6JY) = 0 for every coherent sheaf WYand every i ? 1.

Proof It follows as in Step I that Ext'(SkI x, CY) = 0 for every WYand i ? 2. For i = 1, we can choose, locally, a resolution of 6J of the form

0 __> - _> en--> 6jY--> 0 X

where 9 is coherent. Then the Ext sequence yields

0 = Ext'(021 %J en) -> Ext'(02Ix5, 5 -> EXt2(g2lX, I ) = 0,

proving Step II. The assertion of Step II now suffices to prove that S 1 x is locally free.

Indeed, write S1 Ix locally as a quotient

0 -*- 6A--> eI--> Qk 1 X--> 0.

Since Ext'(01 Jxk 6A) = 0. the extension splits locally; hence 21j x is locally free, q.e.d. for Proposition (2.5).

Remark (2.6). The proof actually shows that 9 is smooth along X, in the general case, precisely at those points where the extension class, viewed as a section in H0(eD(X)), does not vanish.

Remark (2.7). The discrepancy between d-semi-stability and the situation of (1.12) arises from two sources:

1) The map Tx-> H0(Tx) = HO(eD) might fail to be surjective. The obstruction to surjectivity is the map H0(Tx1) -> H2(Tx). It would be illuminat- ing to find an example where this map is nonzero.

2) Even if we do have e E Tx which generates Tx everywhere, e merely corresponds to a deformation of X over the dual numbers, and there is no guarantee that e comes from an actual family.

The remainder of this section deals with various technical results valid for any variety with normal crossings.

LEMMA (2.8). Let ID be the ideal sheaf of D = Xsing in X. Then

Ext'(ID, &X) = 0, i ? 1.

(Since ex is not regular, this does not imply that ID is locally free.)

Proof We write down a resolution of ID X as an x, X module. For simplicity of notation, assume that X is the surface {Z1z2z3 = 0), x = 0, R = ex x =




C{Z1, z2, Z3)/(Z1Z2Z3) and IDx = (Z1Z2, zlz3, z2z3). Consider

* * *eReJeReJeRJA->REDREDR-R R R-IDCJ R->?

defined by

e(f1, f2, f3) = Z2Z3fl + ZlZ3f2 + Z1Z2f3,

T1(g91 g2, g3) = (Z1g1 Z2g2, Z33),

cp2(h1, h2, h3) = (z2z3h1, ZlZ3h2, z1z2h3),

T3 = T1i T4 = T2. etc. Clearly e is onto ID' If e(f1, f2, f3) = 0. then zi f1 (lift to Ctzl, z2, Z3),

which is a UFD); hence the sequence is exact at the first stage, and the other stages are similar.

Applying HomR(, 1R) to this resolution gives

where Aj =', satisfies

4O(fP, f2, f3) = (ZlAf, Z2f2, z3f3),

+2(9 9g2, g3) = (Z2Z3g1, ZlZ3g2, ZlZ293).

etc. Hence the dual sequence is also exact, and thus

ExtR(ID, R) = 0, i ? 1.

LEMMA (2.9). The natural map 01-> 02 induces an isomorphism

(0x = double dual of 20.) Proof Since 01 is locally free on X - Xsing5 any section of ker(Q2 -x> Q) is

supported onXsing, hence c Tx. On the other hand, the dual of any ex-module has no sections which vanish on an open set, whence

= ker( 0 --> Q

It therefore suffices to prove that S -++1. If locally X C V where V is smooth, from the conormal sequence

o -> 01IX > - > ->

it suffices to show that S41 lx = -* l 1. Now the normal sequence is



88 ROBERT FRIEDMAN

and hence 0 ---> -o lx ID 0 is exact. Dualizing yields

Qvl X --,> 2x Ext'(ID, OX) = 0,

which is what we had to show. We can sketch a more geometric argument for (2.9). If a: X -> X is the

normalization, a straightforward argument shows that Tx0 is the sheaf of vector fields on X, tangent to D with the obvious compatibilities, and vanishing at the triple points (in the surface case).

Because of the tangency requirement, one might expect TO to be a* Q 2(log D), but a calculation in local coordinates shows that Tx0 and

_

a* ((log D) pair to a* (n. In fact, again via local coordinates, a calculation shows that the subsheaf of a * Q2 (log D) which pairs with Tx0 to ex consists precisely of those sections of a * Q which map to 0 under

a*Qt -- (aj)*Q]L;

by (1.5), this kernel is S /x. The final technical result of this section is a duality statement. If X has

normal crossings, the local to global spectral sequence for Ext*(Ql, ex) is a long exact sequence:

(at):-> H'(Txo) -->Ext'(21 ex) -->H'-'(Txl) -->H'+'(Txo) *** this follows because Ext'(21, ex) = 0, i * 0, 1.

Now Serre duality (see [RR] for the analytic case) asserts that

Ext'(1, ex)* = Hn-i?(Q X x)

where xx is the dualizing sheaf of Serre and Grothendieck, locally free in this case. Thus, from the short exact sequence

0-? Tx ? Xx -x 2 ?X

Xx - (&2k/Tx) ? WX -> 0,

we obtain

(/3): ) * H~( Tx ?> cx) H- ?ic (o Wx H x x ) Hn() -i(n2-/Tx ? (A))

LEMMA (2.10). The sequences (a) and (/X) are naturally dual to each other.

Proof (/3) is dual to ... -* Ext'(91k/Tx, Ex) -- Ext'(9, Ox) - Ext i(x, ex)

and hence it is enough (modulo a straightforward commutativity check) to prove

i) Ext'(91 1Tx, Ox)-Hi(Txo),

ii) Ext'(Tx, Ox) - H'-'(Txl)




Statements i) and ii) follow in turn from

i)' EXt (9 1/Tx, EOx) = 0, i > 0,

ii)' Ext'(x, Ox) = 0, i * 1, and

Ext (1(X, Ox ) >Ext 1( Tx, x ),

by the Ext spectral sequence. Moreover, plainly i)' ii)', since Ext'(9 k, E) = 0, i > 2, and Hom(Tx, Ex) = 0. ii)' follows from an explicit resolution of Tx. We let X = {Z1z2z3 = 0) and R= C{Z1, z2, z3)/(zlz2z3) as before. Consider

3 3R3 3 P2 ' 1 --> --- --* R3 ? R --- R - T~ -> 0

where

4J(ej, e2, e3) = e1z2z3 dzl + e2z1z3 dz2 + e3z1z2 dz3

42(f1, f2, f3, h) = (zjf1 + h, z2f2 + h, z3f3 + h),

43(g1, g2, g3) = (Z2Z3g1, ZlZ3g2, Z1Z2z3,O), etc.

Exactness requires only straightforward checking, and we leave it to the reader to show directly from the resolution that Ext'(Tx, x) = 0, i * 1. For i = 1, consider

33 pR 43

3 3 P2 '3 P1 *

R3~- R

R3 ?D R R T~ 0

I1 _ _ _ _ _ _ _ _ _ _ {992 {Pi o~~~~~~~~~~~~~~

O sR - s11>21 0

where p1(el, e2, e3) = elZ2Z3 dZl + e2ZlZ3 dz2 + e3Z1Z2 dz3, viewed as an element of 9ThLx = R dzl ? R dz2 ? R dz3, and p2(f1, f2, f3, h) = h, where we identify R with IX/IX and 1 E R with z1z2z3mod I 2 Let J = imt'a2 C (R3 ? R) v; hence

I = 4(Z1c1, Z2c2, Z3c3, (c' + c2 + c3)): -i E R).

Claim. The map R -* (R3 ? R)V defined by 1 ER ~-4 (0,0,0,1) defines an isomorphism

R/(z1Z2, zlz3, Z2Z3) -kert43/Imt42.

Proof Clearly z1z2 -* (0,, 0, z1z2) c Jif we set

(EJ c25 3) = (0,0, z1z2), and similarly for zlz3, z2z3.

Conversely, (0,0 ,0 , h) E J =* h = c1 + c2 + c3 with c1 a multiple of z2z3 and so



90 ROBERT FRIEDMAN

on, so that

R/(Z1Z2, ZlZ3, Z2Z3) -* ker t43/ is infective.

Now kert43 = {(Zg1, Z2g25 Z3g3, h): gi, h e R). Mod J, this is generated by (0,0 ,0 , 1), proving the claim and thus the required isomorphism

Ext'(Ql, Ex) A Ext'(rx, 5X

Remark (2.11). The dualizing sheaf cxx which may be defined intrinsically or via the usual adjunction formulas, has the following description in the case of normal crossings. Any line bundle on X = U Xi is a collection of line bundles Li on Xi, together with gluing isomorphisms

Ti j: Li I Xi n Xi-- Lji Xi n Xi,

For the dualizing sheaf,

Li= CxjXi = Kxi + Di = QXi(Di)

The gluing maps are as follows: let Dij= Xi n X1, Tij = the triple locus X[3] n

Dij. Then by adjunction, there are canonical isomorphisms

LijDij = Kx, + DilDij = KDj + Tij = LjlDij.

We conclude this section by applying the formalism developed above to the tetrahedron X described at the end of the previous section. By (1.11) and (2.3), TX= eD() and h0(D, eD(4)) = 22; since HO(Op3(4)) HO(OD(4)) (1.14), all of the classes in H0(Tx) can be realized via quartic pencils. Since X is a degenerate K3 surface, and there exist nonalgebraic K3 surfaces, we expect an interesting locally trivial deformation of X as well.

Using (2.10) and the fact that cox = 6Ox (by the adjunction formula), we obtain

H'(Txo) -

H'(EX/Tx)*.

From

? --* 91 1Tx --+ a*QL --+ a*2t -a+ ? - x X D




we obtain 4 6

O-* Hi( 2/TX) - Hi(P2, up ) H6 Pi, p ) H (0X /TX) ?

C4 >C6.

A direct calculation shows that the map C4 C6 has precisely a one-dimensional kernel, whence H'(Txo) C.

Now Tx = Ext2(E21, OX) = 0 since H'(Tx) = 0 (direct calculation) and H2(TO) - HO(Q2/Tx)* = 0 by arguments as above. Hence all obstructions vanish and all first order deformations come from actual deformations.

Notice that ho(Txo) = h2(21/TX) = 3, and so dim Tx- dim Tx + dimTx2 = 3 - 23 =-20

and - 20 = X(es), where S is a smooth K3 surface. This is to be expected [Pa, p. 175].

We must still describe the nontrivial locally trivial deformation of X. Let Y = X1 U X2 U X3, where X =X1 U X2 U X3 U X4, and let D = Y n X4 c Y. D' = Y n X4 c X4. Hence D D' is a triangle. Now, four planes in P3 have no moduli, but the isomorphism D- D' does, in the following sense: D has 003 automorphisms-indeed, Aut?(D) = (C*)3, but only oo2 of these are induced by the inclusion D_ D' c P2 (as the automorphisms of P2 fixing 3 lines = automorphisms of P2* fixing 3 points). Choose then an arc of nonequivalent isomorphisms qt: D > D' (nonequivalent modulo the projective ones, that is), and define

Xt= YIIX4/x 9Pt(X).

It can be shown that (Xt) is a nontrivial deformation of X.

Remark (2.12). For "generic" t, there does not exist any nontrivial line bundle on Xt (much less an ample one). Here generic means outside of a countable dense subset. Indeed, the exponential sheaf sequence on Y looks like

0 = H1(-y) -H'( *) H2(YZ) -H2(Y) = 0,

where the vanishing of H'(Oy) and H2( @Y) follows easily from the exactness of 0 - y - a * > (al)*Oym --* (a2)*Ey[2] 0

if we use the fact that the dual graph r of Y is contractible. A standard argument (cf. the proof of (1.4)) shows that H2(Y, Z) = Z (in fact, H2(Y, Z) is generated by OY(1) = CO-, viewed as a class in H2).



92 ROBERT FRIEDMAN

Now Xt carries an invertible sheaf L < there exist invertible sheaves L1 and L2on Y and X4 and an isomorphism

'Pt (2ID') Li ID.

Clearly, this is possible only if

t*(6 p2(1)1D') X9 (Cy(- I)ID)

is torsion in Pic(D) = C*, and hence for only countably many t. The surface X is singled out among the xol surfaces Xt as the unique surface

such that

(t- (6 y (I)lD) = 3 collinear points on D'.

Remark (2.13). The surfaces Xt have a mixed Hodge structure of the form

o= 2(r) C W2

where F is the dual graph of Xt. By the theory of extensions of mixed Hodge structures described in [C], the corresponding extension of Hodge structures is classified by an element of J1W1, which in this case is simply the generalized Jacobian J(D). The extension class can be shown to be

t*(C p2(1)1D') X (Oy(- I)ID),

and from this one can deduce a Torelli type theorem for the family {Xt).

3. The abstract log complex

Let v: 9- A be a semi-stable degeneration with 7v- '(0) = X (1.12). Recall the construction of the relative log complex [DI (3.3)], [S]: 21/A(log X) is defined by the exact sequence

0 -*S21(0) S21 (log X) -+ S21 /(log X) -O 0,

and we set

Q /A(log X) = A 12/,(log X),

with the induced differentiation. The importance of 2i /f(log X) in various questions connecting the geometry of X to that of the general fiber (limiting mixed Hodge structure, Clemens-Schmid exact sequence, etc.) should be clear from the references above. In this section, we show that

Al = 923C/A(log X)Ix




can be intrinsically defined for any d-semi-stable variety X; the applications to deformation theory will be discussed in the next section.

LEMMA (3.1). There is a natural map r (to be constructed in the course of the proof)

r: Q' x/(log X)Ix = Al a*912x(log )

Proof Recall that, if {zl,..., Zn+ l) are local coordinates on X such that r is defined by

t = 7Z(z) = Zl Zk

then the sections of /f(log X) are of the form

dz1 dzk =- a + 0+ k + ak+ldZk+l + + an+ldZn+fl

Z1 Z, kZk

modulo the relation Ek dZi/zi = v* dt/t = 0 (where the a are holomorphic functions).

Moreover, D = lIDi C II Xi, where, if locally Xi is defined by {zi = 0), Di is defined by {z1 ...z^i Zk = 0). Define r(co) to be the k-tuple (pi) E

eDk 2X (log Di ), where

dz dzk T = (a1-_ ai) ? + *.? + (ak-ai) + ak+ldzk+l + + an+ldzn+1,

Z1 Zk

where we may think of the dzi/zi term as being omitted or as having a 0 coefficient.

Plainly r(E dzi/zi) = 0, so that r does indeed define locally a map Al a* 2 (log D). We must show that r is independent of the choices made and therefore globalizes. Consider those choices of coordinates (z1, .., Zk) such that

t = Zl ... Zk

is fixed. Following a suggestion of the referee, we define the map r alternatively as

the composition of the maps

/1 a(log X) -* 02 (log X) -*t A co



94 ROBERT FRIEDMAN

and Q4j(log X) a.*. 21(log D) via residue. Since

dt k dz dz. dexfiA) = ----, if co= a + ajdzj, resx -A co t Zi j~k J j> k

dz dz __ _ _z_

res a , j izA -+ ai d(zj A i )~ +~ ( a iZ A

j<i jki

= p0

in the previous notation. Hence r is independent of the choice of Z1, .., Zn.

Notice that if we replace t by u t = t', u(O) * 0, then

dt' *du + * dt tf U 7Tt

as ur*u is constant on each Xi, the value of r is unaffected. Hence r is well defined. Since, at a smooth point of X,

r: 21 Za (log X )x I2x -+

is an isomorphism, r does indeed induce an inclusion of sheaves, q.e.d.

(3.1) implies that an intrinsic construction of Al is equivalent to an intrinsic characterization of Im(r). One condition is clear: write

dz.- mi= bijz-+ E bijdzj,

ji<$k i jPk+1 j? k

where

((Pi5 .. k) E a* 2'(log D).

Then, if we set pi{pi) = biJXi n Xj, 1 < i, j < k, the pij are residue maps, hence invariantly defined. Clearly on Im(r) we must have

(3.1.1) Pij + Pji =0 and Pij+ Pjk= Pik onXinXjnXk

For curves, (3.1.1) characterizes Im(r) completely and identifies Al with those differentials qp on X having at worst simple poles at those points p and q of X such that a(p) = a(q) is a double point and satisfying

respqp + resqqp = 0.

Thus, Al is just Wxx the sheaf of dualizing differentials on X (2.11).




For dim X > 1 (3.1.1) alone is not sufficient. We must add conditions that, roughly, match up the b 1's, j > k. To do this intrinsically, it will be necessary to choose a nonvanishing section of the infinitesimal normal bundle of X (which is what the choice of the coordinate t amounts to). We always assume that (3.1.1) holds for the forms (q(i).

Consider first the case where X = X1 U X2 has no triple points, and let

= b(z) + E bk(z) dZk on X, Z2 k>3

bl(z) + E bl(z) dZk on X2; k>3

(3.1.1) amounts to the assertion b + b' = 0 on Xi n X2 = D. Now a section s12 Of ED(- X) = ND/X1 ?&ND/X2 leads to a natural restric-

tion on the choice of the zi's; we should assume

ZlZ21D = S12 E H0(D, OD(- X))

With this choice, an allowable change of coordinates must be of the form

Z1 - Z1,

=X'z2 Z2 = 2

with XX' = 1 on D; hence dX/X + dX'/X' = 0 on D. In this case, Ek?3bk dZk is replaced by

k3 bkdzk + b( Xd)

and k 23b dZ kby Ek23 bk dZk + b'(- dAX/X). Since, on D, b'(- dX/X) = b'(dX'/X') =-b(dX'/X'), the condition

(3.1.2)' EbkdzklD= E b dzkID k>3 k>3

is well defined, and, by the proof of (3.1) (compare also (1.5) 1)), is clearly the condition that (TP1, 2) is in the image of r.

In the general case X = {Z1 ... Zk = 0), choose a section s of HO((9D(-X)); we may assume that the coordinates zi satisfy

1 ZZk = s E H0((9D(- X)) Let (wi) e a *21(log D) satisfy (3.1.1), say

qi = Z

bil z1 j+ Z bil dzj.

I~i I lI>k 1<k



96 ROBERT FRIEDMAN

Consider the conditions

(3.1.2) a) bir- - i S xi = bir -bjsI XnX

bir bisxinx, = -brslxtnx,; is jr, s < kandr, s # i, j,

b) b = b I > k + 1.

Clearly, if ((pi) E Im(r), ((pi) satisfies (3.1.2). To check the invariance of (3.1.2) under changes of coordinates, consider first the effect of replacing z1 by XZ1, Z2 by X'z2, with XXI'D = 1. Thus,

qi= E b dzl + E bil dzl li l >k l<k

dz~ dX dX fx1 = L zl + E bi, dzi + bil-A + bi2A if i,5 j 1, 2.

l>k 1<k

If i, j 1, 2, then

bil A + bi2 AX x (bil - bi2) AX (bjl - bd2) dX j1? /X ~x xqnx1 (b1-b2 X xqnx'

so that (3.1.2) a) and b) are clearly preserved. We leave the cases i or j = 1 or 2 to the reader. (The case i = 1, j = 2 involves (3.1.1).) Next, consider a coordinate change in Zk?1..., Zn If the new coordinates are z1,..., Zk, Zk+<? ..., 5n+1, then

cp = Z b1 dz1 + E b'l dzl = b'Zl + E b'l dzt; l<k l>k l<k 1>k t=l t l*i l=i

it is easy to check that (3.1.2) for (bi'l} is equivalent to (3.1.2) for (bil}. We leave it to the reader to formulate and check the appropriate compatibilities for (3.1.2) as we pass from ak(X[k]) to ak-l(X[k-1]). (We are being sketchy here because, in the next section, we will present an alternative argument that (3.1.2) is intrinsic.)

Note that, throughout the preceding discussion, it does not suffice to have trivializations of the bundles

OD(- X)lxnx = ex x, (X_ - Xi - T) (T= a2(X[2]=

we need the gluings

exnx,(-Xi - Xj- T)- Oxnx( X j-Xi -T)5




obtained by viewing

Z, * z .Z *z H z, t<k

tFi, j,l

as a local section of either bundle. A compatible system of sections of the OX xf - X) under these gluings is, of course, just a section of ?D(- X). In the compact case, however, sections of OEXt - X) IT will differ by constants, and this suffices to state (3.1.2).

Next, we claim that (3.1.1) and (3.1.2) characterize Im(r). For, if we define A' locally by

Al = {(qgi) E a*i2x(log D ): ((i) satisfies (3.1.1) and (3.1.2)), then Im(r) C Al. Using the maps pij of (3.1.1), we define

p = E pij: Al' -> ker((aJ)*Ox[1] -> (a X[2] i<j

By (1.5), 1) for the case k = 0, pIIm(r) is surjective. Thus given (4i) C Alx after subtracting off (qi) E Im(r), we may assume that (4i) is actually holomorphic. But, by (3.1.2) b),

fi- j = 0 on (Xi n X,) - T, hence in Xi n Xr

so that by (1.5) 1) for the case k = 1, (4') C Im(r) (and in fact (4i) is a section of &21/Tx). Summarizing, we have

THEOREM (3.2). If X is a compact d-semi-stable variety, there is an intrinsically defined subcomplex

Ax c a* 2,-(Iog D ) such that Ax is locally free and agrees with i2-9/,(log X) IX if 77: A lS is any semi-stable degeneration with central fiber X.

Proof. The preceding discussion defines Al intrinsically; note that it is irrelevant if we replace s C H0(D, DD( -X)) by a scalar multiple or if D fails to be connected. Set

Ax = exterior complex on Ak.

Locally at any point of X, we may always assume (and will freely do so in what follows) that

X c 9Xisdefinedby{z,1.. Zk = 0),

where s = .. Zk E HO(OD(- X)) is the given section of HO(O D(- X)); thus t = Z Zk realizes X locally as a fiber in 9 -> lA, and in this local model

AX qC/A(logX)Ixviarasin (3.1).



98 ROBERT FRIEDMAN

Hence Ax is locally free and a subcomplex, by the analogous statements for i21 ,7(log X) I x and the last statement is clear.

Remark (3.3). Anx = xx This follows from (2.11) or from a corresponding local statement for i2nf7(log X) I x.

Definition (3.4). Set LH*(X) = H*(X; Ax). If X is a central fiber in a semi-stable degeneration, LH*(X) is the "limiting" cohomology of X.

The remainder of this section is devoted to discussing what amounts to a "partial" mixed Hodge structure on LH*(X). First note that A' is naturally filtered:

VOAl = /x C V1A1 = Al thus

VLA1k/VOAkx ker((aj)*Qj --> (a X[2-

and VoAkX ker(a**21 -> (a1)* 21[1]).

Define V AX = Amx n W, (a * 2 x (logLD))X

where W1 is the weight filtration [DI], [GS], [S]. Now sections of Ax are of the form T [1 @@9 gk) with

IdI dz' dzr Ti ~AI, where ~= A reflZ

1 #1=1 (\ Z!)Z IC {1..k

and Wie 2e I

Moreover, the W[ satisfy, among other relations, the following:

(3.5.1) W = (-1)#I'wj on X n X1 (i, j I),

(3.5.2) 8((.of)) = 0, where 8 = restriction of (wI) to X [,

followed by Cech coboundary S2- X[,+t,.

Indeed, (3.5.1) and (3.5.2) follow from (3.1.1).

PROPOSITION (3.5).

VA"/V,-A -mx ker((ai)* 2Xmj(T 1 -) * 2Xml+1 ).

Proof. By abuse of notation, we will henceforth omit all (al)*'s. Via Poin- care residue [DI], [GS], [S], there is an isomorphism




By (3.5.1), this isomorphism plus a choice of global ordering of the components of X give a map using (3.5.2):

V Am/Vx/ AX -" ker(s2xT' -2X[I+l) C

Locally on X, we may assume that X C 9 as in the proof of (3.2). Applying Poincare residue on X, we have

Wli29(log X)/WI1~E2M(log X) 2X[1-1].

Via the map r in (3.1), there is a natural diagram commutative up to sign (the commutativity check is left to the reader):

W12M(log X) VI I

P.R. I P.R.

Hence, the image of V1A'x in &27 jT1 contains 8(s2X[NT-1'); by (1.5) 1) this is ker 8: X 1 2 X['+l], q.e.d.

COROLLARY (3.6). The following sequence is exact:

--- V AMX/ V1A u XU] u XIt1 Xtt 0 -~~~_jMX--- ?27 ---> ~2XMj1+k -> ~2MIj2]>

Hence (s2 m[7?1l] 8) is a resolution of V1 Amx/V1 VA'.

Replacing V1Ax by V-'AX, we obtain a decreasing filtration VAx of A-, hence a spectral sequence

E1 mi1 = H k(VAX/v-1Ax) => Hm(Al) = LHm(X).

In turn there is a spectral sequence

pq = Hq(i2XJl? ) - Hql(X[l+P] C) = 1,

where p + q = m. Similarly, there are spectral sequences

E- ~= HW(VjAx/VVLAx) =: Hm(x)

-q Hq(i2lP) q E-1,m+I p + q = m.

Remark (3.7). If the Xi are compact K~ihler manifolds, the V filtration on LH*(X) carries a natural W-filtration on gr'LH*(X), arising from the spectral



100 ROBERT FRIEDMAN

sequences E and F. These filtrations are closely related to the limiting mixed Hodge structure of X (compare [S]).

Remark (3.8). By (3.6), the complex

(VA- /V,-,A- d)

is, up to a shift, a resolution of the constant sheaf C on a1X11]. In particular

E-1,m+1 = H k-1(a X111 C

in such a way that the FL" q spectral sequence is the E1 term of the usual Mayer-Vietoris spectral sequence.

4. The log complex and deformation theory

This section is concerned with technical results relating to the local study of obstructions to deformations. Quite generally, the graded module

Tx = ED Tk

has a structure of a sheaf of graded Lie algebras; we make explicit calculations in the case of normal crossings. It turns out that the local calculations tie in closely with the abstract log complex discussed in the preceding section. There is an induced bracket on T* which will be used in the next section.

Let S = C(z1,..., Zn), f= Z1 * Zk E S and R = S/f. S. Denote Tk0 - TR, where X = the germ defined by f = 0. The recipe for computing the Lie bracket on TRO ED TR is as follows [Pa]:

Define S -1 SO = free rank one S-modules with basis elements e 1, eo, and

s= S- ID So.

We see that S is a graded S-module by identifying SO = S (eo -> 1) and letting SO act on S-1 as S. Define

do0: 5 '

by do(e-1) = f * eo, and d0 is a degree one map. Hence (S, d0) is a differential graded algebra.

A derivation of degree k of S is a C-linear map A: s -> S, homogeneous of degree k, satisfying, for a homogeneous,

.p(af3) = p)(a) . /3 + (- I)k degcta ()

Let Derk( ) be the set of all such maps; it is an S-module, with do E Der'(5),




and

Der(S) e Derk(5) = Der-'(S) e Dero(S) e Der'(5). k

Der( S) has a graded Lie algebra structure; if qy T2 are homogeneous elements of Der( S), define

[T'1T2] =1 = - (- 1)deg 1 deg c2

Define D: Der(S) -> Der(S) by D = [, do]. Then D2 = 0 and D is compatible with [., .] by the Jacobi identity. Hence ~iHi(Der S, D) has a graded Lie algebra structure.

PROPOSITION (4.1). H'(Der S, D) is naturally TR.

This is proved in [Pa] in full generality, and will follow from the calculations below (except for the naturality).

We use the following notation: If 4a, b e Homc(Sa, SOb) deg 4a, b = a + b, we write

4ab(g ea) = P, b(g)eb, Pa, b e Homc(S S ). (4.2) The following are straightforward calculations: 1) 40, -1 e Der-'(S) pO' - 1 is a derivation of S. 2) 40 o + 41 - 1e Der?( S) pO 0 is a derivation of S and there exists a

c e S with p11 - (g) = g c + pO O(g), in which case c = p1 _ 3) 1 e0 Der'(S) p1 0 is S-linear. 4) D4,_1 = A00+ A, ,where

Po, o(h) = f po, - (h), Pi, _ (h) = pO, - (f h).

5) D(4o 0 + A1 - 1) = ip1,0 .where

P1,0(h) = p00(f h )-fpj1_1(h) (S-linear!).

Choosing local coordinates z, . ... , Zn, we can rewrite (4.2) 1)-5) as follows:

Der (d)=Sz = e a d

Dero(S) = Sn e S = E S e D S, dz, Der'(5)= S;

D: Der-(S) > Der0(S) is given by

D(Lai 8) = (Efai a 5 Yai f)



102 ROBERT FRIEDMAN

and D: Der0(5) -> Der'(5) by

D(zbj c) = bi cf.

In particular,

ker(D: Dero -> Der')/Im(D: (Der' -> Dero) = ker(Der(S)/f. Der(S) R-

(where c(Ebi d/dzi) = Xbi df/dzi) = TRO by Section 2, and coker(D: Dero Der') = S/(f, df/dzi) = TR.

PROPOSITION (4.3). Let t generate TR. Then the map

[, ]:T-TRO TR

is R-linear and surjective.

Proof Choose {l o which equals ( mod(f, df/dzi). Then p1 0(h) = u h, where u is a unit. The Lie algebra structure on Der(S ) looks like

[4oo + ,, -1. iO](ge-1) = p0,0 a pl,o(g) * eo - p1,0 pl, p (g) * e

clearly linear in 4 = 0, o + 1 -1 e Der0, by (4.2) 3). From p1 1-(h) = hp, -,(1) + po 0(h) = h * c + po 0(h) we obtain

PooPo(g) - P1,0p1,-(g) = pO0(ug) - up1,,g)

= pO,(ug) - up00(g) - guc

= g(po,(u) - c u) =g biaz -c -.

Now there exists c with

P1 1(1) = c, 4 5 kerD '* cf= Zbi df;

in our case, since kf = EkZi df/8Zk, C =1 occurs. Moreover, we can take b1 to be in the maximal ideal for all i, whence

, b _ - u is a unit, hence invertible, q.e.d.

The relevance of (4.3) to deformation theory is connected with the following exact sequence: fix ( as an everywhere generating section of Tx and consider

0 > SX TX TX 0

an exact sequence of coherent sheaves, where Sx is by definition the kernel of




[, A]. Taking cohomology, we have

H'(Txo) -> H'(TX ) -> H2(SX).

Hence, if H2(Sx) = 0, H'(Txo) 0 H0(Tx) H'(Tx). Now, if X is a surface, H2(Sx) is Serre dual to H0(SvX 0 Wx), so that Sx arises naturally also.

PROPOSITION (4.4). If X is d-semi-stable and a, Sx are as above, then, in the notation of (3.2),

S-AlXI Proof. As in Section 3, choose a system of coordinates {z1,..., zn+1) in

Cn"' near O so that X = {z1 ... Zk= O} C an open neighborhood of 0, and t corresponds to 1 E C{z)/(z1 ... * i ... zn+l); in particular, under the identification TX = ?D(X), we may assume

Z1 *.. Zk = (=- H (ed(- X)).

With the notation of this section,

= O + +1, = (EP Id/dzl, y is D closed

1~~~~~~~~~~~~~ LA- - Y f

(4.4.1) zl,B for all I = 1,. .. ., k and y =EA

Moreover, with t as in (4.4),

[4 (] = O Y - Omod(f5 d,) k md ~df)

1- mod ( . zdz

Recall (2.9) that Tx _ a* Q:L under the natural pairing. If we denote by TV(ID) the image of TxO ? ID in Tx?, then clearly

TX?(ID) C SXCT

with equality on a dense open set; hence dually

TX C Cx (TX(I)

Claim . ( Tx?( ID))- Va*2 X(lgD)

Indeed, by the exact sequence 0?-* Q - TX X ID -T TO(ID) 0,



104 ROBERT FRIEDMAN

Q some torsion sheaf, we obtain

Hom(Tx (ID), OX) = Hom(TX ? ID, OX) = Hom(TX, Hom (ID X OX))

= Hom (Txo, a* ?;) (by the resolution used in (2.8)),

and thus

Hom(TXO(ID) ?9x) = a*Homnl?(a*T?, x )5 which implies the claim by the explicit description of a*Txo (4.4.1).

Let

(hi) ( Ebil dzllzl + E bil dzj) a* *x(log D ). 1<k 1>k lmi

Then ((Ti), i) = (hi), where

hi=Lb1b - Lbil+ l l<k l>k l=-i

the condition that (qp) E Sx amounts to

h lt)= hjltX

for all choices of Pi such that 4 Ee Sx. Thus, choosing /3I = 0, 1 < k, we see that

bil bjiix,nx 5 I>k;

for r s < k, choose r =-3s 0, /3 = 0,1 r, s, to obtain

bir-b Is1x~1 = br -b18I n 5, is j * r, s

and similarly for the remaining conditions of (3.1.1) and (3.1.2). Since Sx is generated by 4''s with a choice of /3 as described above, these necessary conditions for hi | ( are also sufficient, and thus Sx = Al~ q.e.d.

Note that (4.4) gives another proof that (3.1.2) has intrinsic meaning for any d-semi-stable variety.

We may give a geometric interpretation to the bracket [v, t] E H'(Txl) as follows:

PROPOSITION (4.5). If X is d-semi-stable and ( E H0( ?D(X)) is a nowhere vanishing section, then

{v E H'(T0): [v, f] = o) c H'(Tx)

is the set of locally trivial first order deformations of X for which ?D(X) remains trivial.

Proof. Given t and v = (vet) e H1(Ti), let 9 be the corresponding first order deformation. For a sufficiently small open cover {U1a}) of X, we may choose




coordinates (zr) of Ua so that

(Ua 9, Ua) C Cn+ 1 x Spec C[e]/2

is defined by z ...z = 0 and z . ZkIDnIu = . 1. Let SPa define the isomorphism

Ua n UE U, n U1 U, n Uf ED

U, n Ufl

corresponding to the product realization (9ffu 0E e eu; thus Ta"(Zi) =

(1 + ehi)z.. Thus i e H'(OD), and if it is a coboundary ja - j, after modifying zl, say, by (1 + eja)z", the relation HkZl |Ua n up n D = HkZ I Ua n U14 n D gives a trivialization of ?D(X) independent of E.

But if v = (vet) = EkV afl d/8z + En+ lVa d/za, the associated derivation of Cu cup satisfies

v(z'u) = ku,8 so that OP = h-e8z

By the calculation of (4.3), however, k V a, k

[V, fl=Z =a = ht ID' 1 Zi D 1

concluding the proof of (4.5).

5. Semi-stable K3 surfaces

The results of the preceding section are applied to complete the classification of semi-stable degenerations of K3 surfaces. The local structure of the versal deformation is described in detail.

We begin with a summary of the recent classification of degenerations of K3 surfaces, due to Kulikov and Persson-Pinkham:

THEOREM (5.1) [K], [PP]. Let 7r: 9- A be a semi-stable degeneration of K3 surfaces with all components Xi of X = g-V (e) algebraic. Then there exists a birational isomorphisrn p: 9 -' 9X' and a commutative diagram

P

\A

such that (i) p is an isomorphism 9C - (0) - T)-1(). (ii) ST': 9X' -* A' is semi-stable. (iii) KS = 9cf.



106 ROBERT FRIEDMAN

THEOREM (5.2) [P]. With 9 A as in (5.1) and satisfying (iii), X = - (o )

is one of the following:

Type I: X is a smooth K3 surface.

Type II: X = X1 U ... U Xn is a chain of surfaces with X1, Xn rational, Xi elliptic ruled, 2 < i < n - 1, and with the double curves in Xi two disjoint sections of the ruling, 2 < i < n - 1. The dual graph F of X triangulates a line segment with endpoints corresponding to X1, Xn.

Type III: X = UXi, where each Xi is rational and the double curves Di C Xi are cycles of rational curves. Moreover the dual graph F triangulates S2.

(5.2.1) In all cases, by the adjunction formula, the double curves Di C Xi are anticanonical divisors and oxx = ex

THEOREM (5.3) (Miranda-Morrison [SAGS]). In the Type III case, after a further sequence of birational modifications (of types I and II [SAGS]), we may assume that every smooth component of every double curve Di has self-intersection - 1 on either component of X on which it is defined, and that the singular components of double curves (i.e. those "cycles" which consist of a single nodal rational curve) have self-intersection 1 on the normalization of the singular component on which they are double curves.

THEOREM (5.4) [SAGS]. If X = i'-1(0) is a central fiber in a Type III degeneration of K3 surfaces, then dim Aut(X) = 0, i.e. H0(X, Tx0) = 0.

Definition (5.5). Let X be a compact surface with normal crossings. X is a d-semi-stable K3 surface if

1) X is d-semi-stable, 2) wX = OX,

3) X satisfies one of the descriptions of Types I-III in (5.2). X will be called of Type I III depending on which description it satisfies. If X is of Type III and satisfies the conclusions of (5.3), X will be said to be in - 1 form.

Remark (5.6). Both (5.3) and (5.4) are true without the existence of 9 A,

that is, for an abstract d-semi-stable K3 surface, where modifications of Types I and II are to be performed on the abstract surface X (as opposed to the threefold 9).

Our main result will state that every d-semi-stable K3 surface is in fact the central fiber in a semi-stable degeneration. First, some preliminary lemmas:




LEMMA (5.7). If X is a d-semi-stable K3 surface,

H0(X, s,/Tx) = H'(X, ?x) = 0.

Proof. From 0 -- /x a Q2L -* a t2'- 0,

we obtain 0 H- -(Q HO(X, HoX ) - Ho(D, DO).

If X is Type III, H0(X, Q4) = 0, and in the Type II case, the map H0(X, s) -)

H0(D, -) is easily seen to be injective. Similarly, we have

? (9> X -- a * Oi -- a * OL --

a*CT ?->0

whence X((x) = X(F) = 2. Since h0(?x) = 1 and h2((x) = ho(oxx) = 1, h1((x) = 0.

LEMMA (5.8). 1) H 2(Tx) = 0. 2) Tx'- FH?( TXl) 3) Tx = H'(Tx) = H1(?D).

4) All locally trivial deformations of X are unobstructed.

Proof: H2(Txo)* = Hom(Tx?, wX) = H?(Q4) = H?(Q/lx) = 0. 2), 3) follow from the Ext spectral sequence, and 4) is immediate from 1).

LEMMA (5.9). The map H'(TX) ? H0(TX) -* H'(TX) is surjective

Proof: Choosing a section E e H0(Tx) which generates Tx everywhere, we have the exact sequence (4.4)

H'( Txo) "H'( Tx ) -->H(x)

with H2(Sx)* = Hom(Sx, wx) = H0(SX) = H?0(Al'). Thus, we must prove that H0(AlA) = 0. We prove this only in the Type III case; the Type II case is similar and simpler.

Claim 1. We may assume X is in - 1 form. Indeed, a straightforward check shows that H0(Al A) is invariant under modifications of Types I and II. Alterna- tively, one can put X into - 1 form, smooth it via (5.10) below, and deduce the vanishing of H0(Vx) from the existence of the smoothing.

Claim 2. X contains a component Xi with double curve Di = U 'Cj such that the C1 are independent as cohomology classes in H2(Xi) = H'(Xj, )

An easy exercise in Euler's theorem [SAGS] shows that there exist Xi, Di as above with e < 5 (with X always in - 1 form). Thus, it suffices to show that, for such an X1 and for any C., there exists an exceptional curve Ej g Di with



108 ROBERT FRIEDMAN

E* C = 1, for in this case, if EakCk= 0 in H2, E1 Ck =Sk and hence

a j=( akCk) Ej = 0 for allj

But each Xi dominates P2, and one can simply make a case-by-case argument. Alternatively, one can make a case-by-case argument using simple facts on ruled surfaces. As a sample, here is e = 4:

Let Di = C1 + C2 + C3 + C4, arranged cyclically (Ci. Ci +1 0). We prove that C1 meets an exceptional curve. For C2 + C3 is a fiber of a ruling for which C1, C4 are sections. As C2 = C4= 1, a glance at the description of anti-canonical cycles consisting of three components on a minimal rational ruled surface shows that either C1 or C4 must meet an exceptional curve * C2, C3. If the curve meets C4, examine the reducible fiber of which it is a part to find another-I component of the fiber which meets C1, q.e.d. for Claim 2.

Proof of (5.9). Using (3.6), we have an exact sequence

0 = H0(VOA'x) -* H0(Ax) -* H0(VAx/VOAkx) H'(VoAx)

H?(OX[1]) Hl(2 ) t

where the bottom arrow is induced by 1 E H0(?c) the cohomology class of C, C any double curve.

Now

H0(VlAlx/VOA'x) = ker(HO(?x[l] H- (OX[21).

If a jkCjk e ker 6, Cjk =xi n Xk (where we identify Cjk with 1 e H?(6CIM5 then ai= - a1 = 0, where i is as in Claim 2. Using the triple point condition

2ajkCjk e ker(H0(?X[1] -> H0(?X[2]), we get

ajk =O if Xin Xjn Xk #X 0,

as then aij = aik = 0. Thus, for any surface X with X n X* 0, if Cik,

C11 are the two curves in the anti-canonical cycle adjacent to Cij, then ajk = a,, = 0. But the remaining curves are independent in cohomology, already on Xi; hence ajk = 0 for all k. Continuing in this way, we see that ajk = 0 for all j and k; hence 8 is injective and H0(A' ) = 0, q.e.d. for (5.9).

THEOREM (5.10). Let X be a d-semi-stable K3 surface, and assume that D is connected. Then X is smoothable, and thus a fiber in a semi-stable degeneration. More precisely, the versal deformation space near X looks like V1 U V2, where

1) V1, V2 are germs of smooth subvarieties of Tx; 2) V1 is a smooth divisor in TX, corresponding to locally trivial deformations

of X;




3) dim V2 = 20, and the surfaces corresponding to points of V2- V1 are smooth K3 surfaces;

4) V1 and V2 meet transversally along a 19-dimensional subvariety of V2,

which corresponds to those locally trivial deformations of X for which ED(X) remains trivial.

Proof Choose e E Tx which maps onto 1 E H0(Tx) = C; this is possible by (5.8), 2). Replacing e by e + x, x E H'(Txo) (using [ T, : Tx -X Tx Tx2induced from the local bracket), we replace

[e, e] by [e + x, e + x] = [e, e] + 2[x, e] + [x, x].

But [x, x] E H2(Txo) = 0, and using (5.9), after a suitable choice of x we may assume that [e, e] = 0; here we use the fact that as H2(Tx0) = 0, [, ]: Tx ? Tx

- Tx2 induces [ ]: H'(Tx) ? H0(Tx) -* H'(Tx) as in Section 4. Define

W, = H 1 (T?) c Tx, a hyperplane,

W2= {v E Tx: [v, e] = 0)

= (x + Xe: X E C, x E W1, [x, e] = 0).

Thus W, n W2 = {x E W,: [x, e] = 0), a hyperplane in W2, and dimW2 = dim Tx - dim Tx2, by (5.9).

Now by the theorem of Douady-Grauert-Palamodov-Forster and Knorr [Do], [G], [Pa], [FK], there exists a germ of a holomorphic map f: Tx' - Tx2, defined in a neighborhood of 0, with f(O) = 0 and without linear terms, so that f '(0) is the base space of a versal deformation of X (even semi-universal). Moreover, the quadratic term of f is 1/2[v, v] [Pa]. On the other hand, the base of versal deformation of X, f 1(0), must contain the smooth divisor V1 corresponding to locally trivial deformations. Let {g = 0) define the reduced germ of V1, so that the differential (dg)o is a nonvanishing linear form defining W1. Then glf, and we may write f = g h, where again h: (Tx,0) -* (Tx2,0) is the germ of a holomorphic map.

Set L(v) = linear term of h; it follows that

(dg)o(v)L(v) = 2[v, v] = l(v)L(v),

say, where 1(v) is the linear form (dg)o(v). Note that Im L contains all brackets [v, v], v E Tx.

Since [e, e] = 0 and 1(e) * 0, L(e) = 0. But now, for v = x + Xe e W2, L(v) = L(x) and, as [x + Xe, x + Xe] = 0,

0 = I(x + Xe)L(x) = Xl(e)L(x);



110 ROBERT FRIEDMAN

for X * 0, this forces L(x) = 0, hence W2 C kerL. By (5.9), L: Tx - Tx is surjective, so that dim ker L = dim W2; hence kerL = W2. We have now shown that h is of maximal rank at 0 and that upon setting V2 = (h(v) = 0), f '(0) = V, U V2, where V, is a divisor, V2 is smooth, V, and V2 meet transversally, and dim V2 = dim Tx - dim Tx. Using [Pa, p. 146], we have

2(wx)) = 10X(Ox) + X(Tx*),

0 = 2 10 + dim Tx - dim Tx + dim Tx.

Since dim Tx = 0 (5.4), dim V2 = 20. By (2.5), the points of V2 - V, all correspond to smooth surfaces. To check

that they are K3 surfaces, it is enough to verify this for m9: -> A, any one variable semi-stable degeneration with central fiber X and general fiber Xt. For small values of t, q(Xt) = 0 as H'(Ox) = 0, by the semi-continuity theorem. Moreover,

H1(X, w7,AIX) = H1(X, wx) = H1(X, ex) = 0,

so that (RO7T*6XX,)0 = H?(X, ox) = C. Thus there is a section of W6XA over 7T-1 of some neighborhood of 0 whose restriction to X is the nowhere-vanishing section of ox. Therefore, its restriction to Xt must also be nowhere-vanishing for t small, so that Kx -0 and Xt is a K3 surface.

We have proved all of the assertions of (5.10) except for the last statement of 4), which follows from Remark (4.5).

Renark (5.11). In the Type II case with D not connected, the statement and proof of (5.10) go over with minor modifications. Let d be the number of components of D. Then there exist 2d independent linear forms Z1 ..., Zd, W1,..., Wd on Tx so that, locally, the map f: Tx Tx = Cd is defined by

f(v) = (Z1W1,..., ZdWd).

Here, (zi = 0) defines the hypersurface of deformations of X which are locally trivial around the ith component Ei of D, and (wi = 0) defines the locus of deformations for which ?D(X)IE remains trivial; the locus (wi = 0, zi = 0) corresponds to those deformations which smooth out Ei.

Let X = U Xi satisfy 3) of (5.5), along with 1)' All components of D satisfy the triple point formula [P. p. 39] and all

components Di e I-Kx 1. Thus, X is combinatorially a d-semi-stable K3 surface. Indeed, that 2) of (5.5) follows combinatorially from 3) is a consequence of the next lemma.



SMOOTHINGS OF VARIETIES i11

LEMMA (5.12). If X satisfies 3) of (5.5), where X is Type III,

Pic(x) -->Ker(NS(X-) -->NS(D)). In particular, Wxx- 0.

Proof: If (Taf) is a I-cocycle with values in 0 *, the map

d log{ 9pop dq=,9 Af ,~ Ta/3

gives a well defined map H1(06) H2(X, ?, ) = H2(X; C). By a diagram chase, the following commutes:

0 = H'(0x) WOHO( o) pH2(X; Z) (exponential sheaf sequence)

dlog\ H2(X; C)

11 H2(X; &U)

(where we use the fact that H2(X; Z) is torsion free). Hence H 1(0) c F1H2(X;C). Now F1H2(X) n WOH2(X) = 0 and WOH2(X) = W1H2(X) since the Xi and Xi n X are rational. Therefore

H'(0*) C F'(H2(X)/W1H2(X)) = H2(X)/WlH2(X)

= Ker(H2(X) -- H2(D)).

Remarks (5.13). 1) An alternate proof uses the hypercohomology spectral sequence associated to the resolution of 0 X,

0*0 * * >0 0 eX > X D T

and the fact that H'( IF I; C*) = 0, to conclude that

this proof also works in the Type II case. Note that both proofs use the fact that H2(X; Z) is torsion-free.

2) One can describe completely the image of H'(0*) in ker(H2(X) H2(D)) and the appropriate extension of mixed Hodge structures along the lines of (2.13).

Combined with (5.13), the next result guarantees that any combinatorial possibility does indeed arise as a Type III degeneration.

LEMMA (5.14). Let X be of Type III and satisfy 1)' and 3), and let D9. = D- T. the smooth part of Xi n X1. Suppose that D is smooth for all I]U ,i



112 ROBERT FRIEDMAN

(i, j) (i.e. Dij is not a nodal rational curve). Then there exists a choice of isomorphism pi j: D P. c Xi DP. c X where ii j, the extension of pi j to Di j, fixes the triple points, and so that

LIXi/ - (xcE Dij - Tij(x3))

is d-semi-stable.

In other words, by modifying the gluing data (ij), we can make X d-semi-stable.

Proof By the triple point formula, for a given (i, j), either (DW2)x or ( * 0. Let di = deg(DW2)x, and assume this * 0.

Let Gij = (divisors of degree 0 on DP0)/(div(f)) where the f are functions on Dij, not 0 or x at the triple points tj and t2, and such that f(tj) = f(t2). Then Gij = C*, and Pico(D) is clearly generated by the images of the Gij, under the obvious inclusion Gij C Pico(D).

Given (pij e Auto(DPj), suppose we modify (pij by A E Auto(DP). From the exact sequence

0 -H -9 H o(C* ) >H o Ho H'(07,) ?6 ,

we see that Pic D is determined via the gluing data from H0O0*/O,). Now OD(- X) is defined by trivial bundles ?D , plus the gluings defined by using

ZiZJZk E Ho(ODi(- Xi - Xij- T))

as a generating local section. Replacing X by the variety with normal crossings obtained by modifying the gluing along Di jby X E Auto(WDj) _ C* (canonically), we replace ?D(- X) by the bundle which has gluing data, at a triple point tijk5

ZiZjZk E Ho(Dik( Xi - Xk - T),

ZiZjZk E H (ODi(

XJ- Xk - T),

XA ZiZJZk E Ho( OD(- Xi -X - T),

and similarly at the other triple point tij which lies on Dij. Since

(D/D )tiJk (C= ) /C*

where the (C*)3 has basis (eip, ejk, eik) and the C* by which we take the quotient is the diagonal subspace, we see that the effect on ?D(X) is to multiply the ei1 component at tijk by X and the corresponding component at tij by A-'. Up to a power of 2 this is exactly the action of Gij on Pic?(D), which proves (5.14).




Remark (5.15). 1) In the Type II case X = X1 U ... U X , suppose that all double curves have non-zero self-intersection on every component. Modifying the isomorphisms p: Di C Xi -> Di C Xi,, by some X c Auto(Di), we change

OD(X)IDi by

NDi/Xi (A )Di+Xi+

If deg NDl/x * 0. any element of Pico(Di) is of this form, whence X can be made d-semi-stable. If however all double curves have self-intersection 0, it is not enough to modify the gluings; one must also modify the surfaces Xi. Since any rational surface S with K' = 0 has an exceptional curve, which can be flipped by a Type I modification, we may reduce to the first case; hence all combinatorial possibilities occur.

Similarly, in the Type III case, with some components containing irreduc- ible double curves, with self-intersection 0, we must modify the components and not just the gluings. However, as above, it is easy to see that all combinatorial possibilities arise.

2) The combinatorics of polygonal decompositions of S2 into polygons of < 6 sides, such that exactly three edges meet at every vertex, is depressingly involved. See [SAGS] for a preliminary discussion.

COLUMBIA UNIVERSITY, NEW YoRK, N.Y.

REFERENCES

[C] J. CARLSON, The obstruction to splitting a mixed Hodge structure over the integers I, University of Utah preprint (1979).

[DI] P. DELIGNE, Equations Differentielles d Points Singuliers Reguliers, Springer Lecture Notes 163 (1970).

[DII] , Theorie de Hodge III, Publ. Math. I.H.E.S., France, 44 (1975), 6-77. [Do] A. DOuADY, Le probleme des modules locaux pour les espaces C-analytiques compacts,

Ann. E.N.S. 7 (1974), 569-602. [FK] 0. FORSTER AND K. KNORR, Konstruktion Verseller Familien Kompakter Komplexer

Raume, Springer Lecture Notes 705 (1979). [G] H. GRAUERT, Der Satz von Kuranishi fur kompakte komplexe Raume, Inv. Math. 25

(1974), 107-142. [GH] P. GRIFFITHS AND J. HARRIS, Principles of Algebraic Geometry, New York, John Wiley

(1978). [GS] P. GRIFFITHS AND W. SCHMID, Recent developments in Hodge theory: a discussion of

techniques and results, in Discrete Subgroups of Lie Groups, Bombay, Oxford University Press (1973).

[KE] G. KEMPF, F. KNUDSEN, D. MUMFORD, AND B. SAINT- DONAT, Toroidal Embeddings I, Springer Lecture Notes 339 (1973).

[K] V. KULIKOv, Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izvestiya 11 (1977), 957-989.



114 ROBERT FRIEDMAN

[LS] S. LICHTENBAUM AND M. SCHLESSINGER, The cotangent complex of a morphism. Trans. A.M.S. 128 (1967), 41-70.

[PA] V. PALAMODOV, Deformations of complex spaces, Russian Math. Surveys 31 (1976), 129-197.

[P] U. PERSSON, Degenerations of Algebraic Surfaces, Memoirs of the A.M.S. 189 (1977). [PP] U. PERSSON AND H. PINKAM, Degenerations of surfaces with trivial canonical bundle,

Ann. of Math. 113 (1981), 45-66. [RR] J. RAmis AND G. RUGET, Complexes dualisants et theoreme de dualit6 en geometrie

analytique complexe, Publ. Math. I.H.E.S. 38 (1970), 77-91. [R] H.- J. REIFFEN, Das Lemma von Poincar6 fuir holomorphe Differentialformen auf komple-

xen Raumen, Math. Zeit. 101 (1967), 269-284. [SAGS] The Birational Geometry of Degenerations: proc. Harvard Summer Algebraic Geometry

Seminar, to appear in Progress in Mathematics. [S] J. STEENBRINK, Limits of Hodge structures, Inv. Math. 31 (1976), 229-257.

(Received April 9, 1982)

Note added in Proof: U. Persson and H. Pinkham have constructed examples where the coboundary map of (2.7.1) is nonzero, and have shown that many of these examples are not smoothable.



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