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APPEARED IN BULLETIN OF THE

arXiv:math/9201267v1 [math.CV] 1 Jan 1992RESEARCH ANNOUNCEMENTAPPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 113-118LIFTING OF COHOMOLOGY AND UNOBSTRUCTEDNESSOF CERTAIN HOLOMORPHIC MAPSZiv RanAbstract. Let f be a holomorphic mapping between compact complex manifolds.We give a criterion for f to have unobstructed deformations, i.e.

for the local modulispace of f to be smooth: this says, roughly speaking, that the group of infinitesimaldeformations of f, when viewed as a functor, itself satisfies a natural lifting propertywith respect to infinitesimal deformations. This lifting property is satisfied e.g.

when-ever the group in question admits a ‘topological’ or Hodge-theoretic interpretation,and we give a number of examples, mainly involving Calabi-Yau manifolds, wherethat is the case.One of the most important objects associated to a compact complex manifoldX is its versal deformation or Kuranishi familyπ: X →Def(X);this is a holomorphic mapping onto a germ of an analytic space (Def(X), 0) (theKuranishi space) with the universal property that π−1(0) = X and that any suf-ficiently small deformation of X is induced by pullback from π by a map uniqueto 1st order. In general, Def(X) is singular and even nonreduced; in case Def(X)is smooth, i.e.

a germ of the origin in CN, we say that X is unobstructed. In ananalogous fashion, a holomorphic mappingf : X →Yalso possesses a versal deformation, which in this case is a diagram˜f : X−→YցւDef(f)Received by the editors August 27, 19901980 Mathematics Subject Classification (1985 Revision).

Primary 32G05, 32J27; Secondary32G20, 14J40Supported in part by NSF and IHESc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2ZIV RANwith a similar universal property. Again we say that f is unobstructed if Def(f) issmooth.Now in [R3], we gave a criterion which deduces the unobstructedness of a compactcomplex manifold X from a lifting property (in particular, deformation invariance)of certain cohomology groups associated to X; this implies in particular the unob-structedness of Calabi-Yau manifolds, i.e.

K¨ahler manifolds with trivial canonicalbundle KX (theorem of Bogomolov-Tian-Todorov [B, Ti, To]), as well as that ofcertain manifolds with “big” anticanonical bundle −KX. In this note we announcean extension of our criterion to the case of holomorphic maps of manifolds anddiscuss some applications, mainly to maps whose source is a Calabi-Yau manifold.1.

GeneralitiesGiven a holomorphic mapf : X →Yof complex manifolds, we defined in [R1] certain groups T if, i ≥0, which are relatedto deformations of f; in particular, T 1f is the group of 1st-order deformations of f.For our present purposes, it will be necessary to consider the corresponding relativegroups T i˜f/S, which are associated to a diagram˜f : X−→YցւSwith X/S, Y/S smooth (we call such a map ˜f an S-map, or a deformation of f ).In the notation of [R1, R2], we haveT i˜f/S = Exti(δ1, δ0)where δ0 : f ∗OY →OX , δ1 : f ∗ΩY/S →ΩX /S are the natural maps. As in [R1], wehave an exact sequence(1.1)0 →T 0˜f/S →T 0X /S ⊕T 0Y/S →Hom ˜f(ΩY/S, OX )→T 1˜f/S →T 1X /S ⊕T 1Y/S →Ext1˜f(ΩY/S, OX ) →· · ·where T iX /S = Hi(TX /S), TX /S being the relative tangent bundle and similarly forT iY/S, Hom ˜f(·, ·) = HomX ( ˜f ∗·, ·) and Exti˜f(·, ·) are its derived functors.Now put Sj = Spec C[ε]/(εj).

Our main general result, which is an analogue formaps of a result given in [R3] for manifolds, is the followingTheorem-Construction 1.1. Suppose given Xj/Sj, Yj/Sj smooth and fj : Xj →Yj an Sj-map, for some j ≥2, and let Xj−1/Sj−1, Yj−1/Sj−1, fj−1 : Xj−1 →Yj−1be their respective restrictions via the natural inclusion Sj−1 ֒→Sj.

Then(i) associated to fj is a canonical element αj−1 ∈T 1fj−1/Sj−1;(ii) given any element αj ∈T 1fj/Sj which maps to αj−1 under the natural restric-tion map T 1fj/Sj →T 1fj−1/Sj−1, there are canonically associated to αj deformationsXj+1/Sj+1, Yj+1/Sj+1 and an Sj+1-map fj+1 : Xj+1 →Yj+1, extending Xj/Sj,Yj/Sj and fj : Xj →Yj respectively.The proof is analogous to that of Theorem 1 in [R3] and will be presentedelsewhere. In view of this theorem it makes sense to give the following

LIFTING AND UNOBSTRUCTEDNESS3Definition 1.2. A map f : X →Y is said to satisfy the T 1-lifting property if forany deformation fj : Xj/Sj →Yj/Sj of f and its restriction fj−1: Xj−1/Sj−1 →Yj−1/Sj−1, the natural mapT 1fj/Sj →T 1fj−1/Sj−1is surjective.Abusing terminology somewhat, we will say that T 1f is deformation-invariant ifthe groups T 1fj/Sj are always free Sj-modules and their formation commutes withbase-change.

Note, trivially, that whenever T 1f is deformation-invariant, f satisfiesthe T 1-lifting property.As an easy consequence of Theorem 1.1, we have thefollowingCriterion 1.3. Suppose f : X →Y is a map of compact complex manifolds sat-isfying the T 1-lifting property (e.g.

T 1f is deformation-invariant); then f is unob-structed.Remark 1.4. Various variants of this criterion are possible, e.g.

for deformations ofmaps f : X →Y with fixed target Y . In the special case that f is an embedding,with normal bundle N, we obtain that the Hilbert scheme of submanifolds of Yis smooth at the point corresponding to f(X) provided H0(N) satisfies the lift-ing property (e.g.

is deformation-invariant). Also, the converse to Criterion 1.3 istrivially true, though we shall not need this.2.

ApplicationsUnless otherwise specified, all spaces X, Y considered here are assumed smooth.Theorem 2.1. Let X be a Calabi-Yau manifold and f : Y ֒→X the inclusion ofa smooth divisor.

Then f is unobstructed and moreover the image and fibre of thenatural map Def(f) →Def(X) are smooth.Proof. In this case we may identify T 1f with H1(T ′) where T ′ is defined by the exactsequence(2.1)0 →T ′ →TX →NY/X →0,and it will suffice to prove deformation invariance of H1(T ′).Now identifyingTX ∼= Ωn−1X, NY/X ∼= Ωn−1Y, n = dim X, we may write the cohomology sequence of(2.1) as0 →Hn−1,0(Y ) →H1(T ′) →Hn−1,1(X)f ∗→Hn−1,1(Y ) · · · .As Hn−1,0(Y ) and ker(f ∗) are both deformation-invariant, so is H1(T ′), hence fis unobstructed, and since moreover the former groups are the respective tangentspaces to the fibre and image of Def(f) →Def(X), the latter are smooth.Q.E.D.A similar argument can be used to reprove a recent theorem of C. Voisin [V] (seeop.

cit. for examples and further results):

4ZIV RANTheorem 2.2 (Voisin). Let X be a K¨ahler symplectic manifold, with (everywherenondegenerate) symplectic form ω ∈H0(Ω2X), and f : Y →X a Lagrangian embed-ding, i.e.

f ∗ω = 0 and dim Y = 12 dim X. Then f is unobstructed and the imageand fibre of the natural map Def(f) →Def(X) are smooth.Proof.

In this case we may identify TX ∼= ΩX, NY/X ∼= ΩY , and we may argue asin the proof of Theorem 2.1 (note that this property of being Lagrangian is open).Next we consider deformations of fibre spaces f : Xn →Y m with X Calabi-Yau(i.e. f is a flat map whose fibres are reduced and connected).

Note that for a fibrespace f, its general fibre is clearly a Calabi-Yau manifold. Also, it follows easilyfrom the sequence (1.1) that Def(f) ֒→Def(X).

When R1f∗OX = 0, the morphismDef(f) →Def(X) is an isomorphism by a theorem of Horikawa [H], hence in thatcase unobstructedness of f follows from that of X.We will consider here twoextreme cases: namely m = n −1 and m = 1.Theorem 2.3. Let f : X →Y be an elliptic fibre space (i.e.

general fibre ellipticcurve) with X Calabi-Yau. Then f is unobstructed.Proof.

Using the usual exact sequence (1.1) and Criterion 1.3, it suffices to provethe deformation invariance ofker(H1(TX)α→H0(Y, R1f∗OX ⊗TY )) .Now by relative duality we haveR1f∗OX ∼= ω−1X/Y ∼= ωY ,hence we may identify α with the push-forward map (or “integration over the fibre”)Hn−1,1(X) →Hn−2,0(Y ),and in particular ker α is deformation-invariant. (Note that we have Def(f) ∼=Def(X) whenever α = 0, e.g.

Hn−2,0(Y ) = 0, which holds whenever Hn−2,0(X) =0. )Theorem 2.4.

Let f : X →C be a fibre space from a Calabi-Yau manifold to asmooth curve. Then f is unobstructed.Proof.

Note that for any fibre Y of f we haveh0(OY (Y )) = h0(OY ) = 1,and it follows that the scheme Div0(X) parametrizing reduced connected effectivedivisors of X is smooth and 1-dimensional locally at the point corresponding to Y .Consequently if we denote byp: Z →Div0(X)the universal family and q: Z →X the natural map, then we have in fact a 1-1 correspondence between morphisms f : X →C as above and smooth compactconnected 1-dimensional components C ⊂Div0(X) such that q|p−1(C) is an iso-morphism. Now it follows from Theorem 2.1 and its proof that for any smooth fibreY of f, the locus D′ ⊂Def(X) of deformations over which Y extends is smoothand independent of Y .

It follows that almost all, hence all, of C as component ofDiv0(X) in fact extends over D′, hence so does f, so that D′ = Def(f), provingthe theorem.In the intermediate cases, we have only much weaker results:

LIFTING AND UNOBSTRUCTEDNESS5Theorem 2.5. Let f : X →Y be a smooth morphism and assume either(i) KX is trivial; or(ii) KX/Y is trivial.Then Def(f) →Def(Y ) has smooth fibres.Proof.

We will prove (ii), as (i) is similar.It suffices to prove the deformationinvariance of H1(TX/Y ), where TX/Y is the relative (vertical) tangent bundle. Nowwe haveTX/Y ∼= Ωn−1X/Y ⊗K−1X/Y ∼= Ωn−1X/Yn = dim(X/Y ) .By relative Hodge theory, H1(Ωn−1X/Y ) is a direct summand of Hn(f −1OY ), and itwill suffice to prove the deformation invariance of the latter.We have a Lerayspectral sequence(2.2)Hp(Y, Rqf∗f −1OY ) ⇒Hn(f −1OY ) .However Hp(Y, Rqf∗f −1OY ) = Hp,0(Y, Rqf∗CX) is a direct summand of Hp(Y, Rqf∗CX),hence the degeneration of the Leray spectral sequence of CX implies that of (2.2),hence the deformation invariance of Hn(f −1OY ).AcknowledgmentI am grateful to P. Deligne for some helpful comments concerning [R3], and tothe IHES and Tel-Aviv University, in particular Professor M. Smorodinski, for theirhospitality.Added in proofThe above ideas are pursued further in the author’s preprints, Hodge theory andthe Hilbert scheme (September 1990) and Hodge theory and deformations of maps(January 1991).References[B]F. A. Bogomolov, Hamiltonian K¨ahler manifolds, Dokl.

Akad. Nauk SSSR 243 (1978),1101–1104.[H]E.

Horikawa, Deformations of holomorphic maps. III, Math.

Ann. 222 (1976), 275–282.

[R1] Z. Ran, Deformations of maps, Algebraic Curves and Projective Geometry (E. Ballico andC.

Ciliberto, eds. ), Lecture Notes in Math., vol.

1389, Springer-Verlag, Berlin, 1989. [R2], Stability of certain holomorphic maps, J. Differential Geom.

34 (1991), 37–47. [R3], Deformations of manifolds with torsion or negative canonical bundle, J. AlgebraicGeom.

(to appear). [Ti] G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifoldsand its Peterson-Weil metric, Math.

Aspects of String Theory (S. T. Yau, ed. ), pp.

629–646,World Scientific, Singapore, 1987. [To] A. N. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≥3), (Calabi-Yau) manifolds, preprint IHES, November, 1988.[V]C.

Voisin, Sur la stabilit´e des sous-vari´et´es Lagrangiennes des vari´et´es symplectiques holo-morphes, Orsay, preprint, April, 1990.Institut des Hautes ´Etudes Scientifiques, Paris, FranceCurrent address: Department of Mathematics, University of California, Riverside, California92521

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