Kählerness of moduli spaces of stable sheaves over non-projective K3 surfaces
aa r X i v : . [ m a t h . AG ] M a r K ¨AHLERNESS OF MODULI SPACES OF STABLESHEAVES OVER NON-PROJECTIVE K3 SURFACES
ARVID PEREGO
Abstract.
We show that a moduli space of slope-stable sheavesover a K3 surface is an irreducible hyperk¨ahler manifold if and onlyif its second Betti number is the sum of its Hodge numbers h , , h , and h , . Contents
1. Introduction 2Acknowledgements 21.1. Main definitions and notations 31.2. Main results and structure of the paper 41.3. The Beauville form and the Local Torelli Theorem 72. Limits of irreducible hyperk¨ahler manifolds 92.1. The moduli space of Λ − marked manifolds 102.2. Non-separatedness in M Z H , ( X, R ) 183.3. Deformations and K¨ahler classes 203.4. The proof of Theorem 1.15 234. K¨ahlerness of moduli spaces of sheaves 24References 29 Date : April 18, 2018.2010
Mathematics Subject Classification.
Key words and phrases. moduli spaces of sheaves; irreducible hyperk¨ahler man-ifolds; K3 surfaces. Introduction
Irreducible hyperk¨ahler manifolds are compact, connected K¨ahlermanifolds which are simply connected, holomorphically symplectic andhave h , = 1. Very few examples of them are know up to today, and allthe known deformation classes arise from moduli spaces of semistablesheaves on a projective K3 surface or on an abelian surface.In [12] we showed that if S is any K3 surface, the moduli space M µv ( S, ω ) of µ ω − stable sheaves on S of Mukai vector v = ( r, ξ, a ) ∈ H ∗ ( S, Z ) is a compact, connected complex manifold, it carries a holo-morphic symplectic form and it is of K3 [ n ] − type (i. e. it is deformationequivalent to a Hilbert scheme of points on a projective K3 surface).This holds under some hypothesis on ω and v (the K¨ahler class ω hasto be v − generic, and r and ξ have to be prime to each other: we referthe reader to [12] for the definition of v − genericity).The main open question about these moduli spaces is if they carrya K¨ahler metric: if it is so, it follows that they are all irreduciblehyperk¨ahler manifolds of K3 [ n ] − type. The answer to this question isaffirmative at least in three cases: when S is projective; when M µv ( S, ω )is a surface; when M µv ( S, ω ) parametrizes only locally free sheaves. Thislead us to the following
Conjecture 1.1.
The moduli spaces M µv ( S, ω ) are K¨ahler manifolds. Evidences are provided by the previous examples where the mod-uli spaces are indeed K¨ahler, and by the fact that their geometry issomehow similar to that of an irreducible hyperk¨ahler manifold: in[12] we show that on their second integral cohomology there is a non-degenerate quadratic form defined as the Beauville form of irreduciblehyperk¨ahler manifolds.But still, this analogy is not sufficient to guarantee that the mod-uli spaces are K¨ahler: it is known since [5], [6] and [7] that there areexamples of compact, simply connected, holomorphically symplecticmanifolds having h , = 1 which are not K¨ahler, but their second in-tegral cohomology carries a non-degenerate quadratic form, and theLocal Torelli Theorem holds.The aim of this paper is to show that the previous conjecture holdstrue under some additional hypothesis on the second Betti number of M µv ( S, ω ). Acknowledgements.
The author wishes to thank M. Toma for sev-eral useful conversations during the preparation of this work. ¨AHLERNESS OF MODULI SPACES 3
Main definitions and notations.
In this section we collect allthe definitions and notations we will use in the following.
Definition 1.2. A holomorphically symplectic manifold is acomplex manifold which carries an everywhere non-degenerate holo-morphic closed − form (called holomorphic symplectic form ). We notice that a compact holomorphically symplectic manifold isalways of even complex dimension, and a holomorphic form σ definesan isomorphism σ : T X −→ Ω X of vector bundles, where T X is thetangent bundle of X , and Ω X is the cotangent bundle of X (i. e. thedual bundle of T X ).Let X be a compact, connected complex manifold of complex dimen-sion d , and k ∈ { , ..., d } and p, q ∈ N . Definition 1.3.
The k − Betti number of X is b k ( X ) = dim C H k ( X, C ) , and the ( p, q ) − th Hodge number of X is h p,q ( X ) = dim C H q ( X, Ω pX ) . The general relation between the Betti and the Hodge numbers of X is that b k ( X ) ≤ X p + q = k h p,q ( X )for every k ∈ { , ..., d } , and the equality holds for every k if and onlyif the Fr¨olicher spectral sequence of X degenerates at the E level. Definition 1.4.
A complex manifold X is a b − manifold if b ( X ) = h , ( X ) + h , ( X ) + h , ( X ) . Definition 1.5.
A compact, connected complex manifold X is in theFujiki class C if X is bimeromorphic to a compact K¨ahler manifold. Among all manifolds in the Fujiki class C we clearly have compactK¨ahler manifolds. Moreover, a compact, connected complex manifoldin the Fujiki class C verifies the ∂∂ − lemma, and hence the Fr¨ohlicherspectral sequence of X degenerates at the E level. In particular, it isa b − manifold.If f : X −→ B is a holomorphic fibration, for every b ∈ B we let X b := f − ( b ). Let X be a compact, connected complex manifold and B a connected complex manifold. Definition 1.6. A deformation of X along B is a smooth andproper family f : X −→ B such that there is ∈ B such that X isbiholomorphic to X . PEREGO
Let now P be a property of complex manifolds. Definition 1.7.
We say that the property P is open in the analytictopology (resp. in the Zariski topology ) if for every deformation X along a connected complex manifold B , the set of those b ∈ B suchthat X b verifies P is an analytic (resp. Zariski) open subset of B ; K¨ahlerness is open in the analytic topology, but in general it is nei-ther open in the Zariski topology, nor closed. Being a b − manifold isopen in the Zariski topology. Being in the Fujiki class C is not open ingeneral.The last definitions we need are the following: Definition 1.8.
Let X be an compact, connected complex manifold. (1) The manifold X is irreducible hyperk¨ahler if it is a K¨ahlermanifold which is simply connected, holomorphically simplecticand h , ( X ) = 1 . (2) The manifold X is deformation equivalent to an irre-ducible hyperk¨ahler manifold if there is a connected com-plex manifold B and a deformation X −→ B of X along B forwhich there is b ∈ B such that X b is an irreducible hyperk¨ahlermanifold. (3) The manifold X is limit of irreducible hyperk¨ahler man-ifolds if there is a smooth and proper family X −→ B alonga smooth connected base B such that and a sequence { b n } ofpoints of B converging to , such that X b n is an irreduciblehyperk¨ahler manifold. Main results and structure of the paper.
The main result ofthe paper is the following
Theorem 1.9.
Let S be a K3 surface, ω a K¨ahler class on S . We let v = ( r, ξ, a ) ∈ H ∗ ( S, Z ) be such that r > and ξ ∈ N S ( S ) . Supposethat r and ξ are prime to each other, and that ω is v − generic. Thenthe moduli space M = M µv ( S, ω ) of µ ω − stable sheaves on S with Mukaivector v is K¨ahler if and only if it is a b − manifold. Clearly, this proves Conjecture 1.1 under the additional hypothe-sis of the moduli spaces being b − manifolds. This has an immediatecorollary: Corollary 1.10.
Let
M −→ B be any smooth and proper family ofmoduli spaces of sheaves verifying the conditions of Theorem 1.9. Theset of b ∈ B such that M b is K¨ahler is a Zariski open subset of B . In view of of Theorems 1.1 and 1.2 of [12], another immediate corol-lary is the following: ¨AHLERNESS OF MODULI SPACES 5
Corollary 1.11.
Let S be a K3 surface, ω a K¨ahler class on S and v = ( r, ξ, a ) ∈ H ∗ ( S, Z ) be such that r > and ξ ∈ N S ( S ) . Supposethat r and ξ are prime to each other, that ω is v − generic, and that M µv ( S, ω ) is a b − manifold. (1) The moduli space M µv ( S, ω ) is an irreducible hyperk¨ahler man-ifold of K3 [ n ] − type, which is projective if and only if S is pro-jective. (2) If v ≥ , there is a Hodge isometry λ v : v ⊥ −→ H ( M µv , Z ) . The case v = 0 was already treated in [12]: in this case there is aHodge isometry λ v : v ⊥ / Z v −→ H ( M µv , Z ) , and there is no need to suppose that M µv ( S, ω ) is a b − manifoldhere. For as a consequence Theorem 1.1 of [12] we already know that M µv ( S, ω ) is a K3 surface.The proof of Theorem 1.9 is an application of general results aboutcompact, connected complex b − manifolds which are holomorphicallysymplectic and limit of irreducible hyperk¨ahler manifolds. The startingpoint is the following: Theorem 1.12.
Let X be a compact, connected holomorphically sym-plectic b − manifold which is deformation equivalent to an irreduciblehyperk¨ahler manifold. Then on H ( X, Z ) there is a non-degeneratequadratic form q X of signature (3 , b ( X ) − , and the Local TorelliTheorem holds. This result is due to Guan [7], and it is a generalization of thewell-known analogue for irreducible hyperk¨ahler manifolds proved byBeauville in [1]. By Local Torelli Theorem we mean that the pe-riod map is locally a biholomorphism (as in the case of irreduciblehyperk¨ahler manifolds). We will recall the definition of the Beauvilleform and the Local Torelli Theorem in Section 2.
Remark 1.13.
In [12] we proved (see Theorem 1.1 there) that if M is amoduli spaces of slope-stable sheaves over a non-projective K3 surface(veryfing all the hypothesis of Theorem 1.9), then on H ( M, Z ) thereis a non-degenerate quadratic form of signature (3 , b ( M ) − b ( M ). Anyway, in [12] we havenot proved the Local Torelli Theorem, and here we are able to prove itonly by assuming M to be a b − manifold.In Section 3 we consider compact, connected holomorphically sym-plectic b − manifolds which are not only deformation equivalent to an PEREGO irreducible symplectic manifold, but which are moreover limit of ir-reducible hyperk¨ahler manifolds. The main result of Section 3 is thefollowing:
Theorem 1.14.
Let X be a compact, connected holomorphically sym-plectic b − manifold which is limit of irreducible hyperk¨ahler mani-folds. Then X is bimeromorphic to an irreducible hyperk¨ahler manifold(hence, it is in the Fujiki class C ). As we will see, a moduli space M verifying the hypothesis of Theorem1.9 is a compact, connected holomorphically symplectic b − manifoldwhich is limit of irreducible hyperk¨ahler manifolds. As a consequence,asking for these moduli spaces to be b − manifolds is enough to concludethat they are all bimeromorphic to a compact irreducible hyperk¨ahlermanifold.The proof of Theorem 1.14 is based on a well-known strategy al-ready used by Siu in [14] to show that all K3 surfaces are K¨alher, andby Huybrechts in [8] to show that non-separated marked irreduciblehyperk¨ahler manifolds are in fact bimeromorphic. More precisely, ifΛ is a lattice, we say that a compact complex manifold X carries a Λ − marking if on H ( X, Z ) there is non-degenerate quadratic form,and there is an isometry φ : H ( X, Z ) −→ Λ. The pair (
X, φ ) is calleda Λ − marked manifold .The set of (equivalence classes of) Λ − marked manifolds is denoted M Λ : as a consequence of Theorem 1.12, it contains the subset M s Λ of Λ − marked manifolds ( X, φ ) where X is a compact holomorphicallysymplectic b − manifold which is deformation equivalent to an irre-ducible hyperk¨ahler manifold (and whose Beauville lattice is isometricto Λ).By the Local Torelli Theorem we can give M s Λ the structure of com-plex space, in which we have a (non-empty) open subset M hk Λ of irre-ducible hyperk¨ahler manifolds. We let M hk Λ be its closure in M s Λ .Theorem 1.14 can be restated by saying that if ( X, φ ) ∈ M hk Λ , then X is bimeromorphic to an irreducible hyperk¨ahler manifold. This isthe statement we prove: the idea of the proof is that if ( X, φ ) ∈ M hk Λ ,then ( X, φ ) is non-separated from a point (
Y, ψ ) ∈ M hk Λ . A standardargument shows that X and Y have to be bimeromorphic.Theorem 1.14 is just an intermediate result on the way to the K¨ahler-ness of the moduli spaces, and it is used in Section 4 to prove that on acompact, connected holomorphically symplectic b − manifold which islimit of irreducible hyperk¨ahler manifolds, we can define an analogueof the positive cone of an irreducible hyperk¨ahler manifold. ¨AHLERNESS OF MODULI SPACES 7 Recall that if X is irreducible hyperk¨ahler and C X is the cone of real(1 , − classes over which the Beauville form is strictly positive, thepositive cone C + X is the connected component of C X which contains theK¨ahler cone of X . A result of Huybrechts shows that C + X is containedin (the interior of) the pseudo-effective cone of X .Theorem 1.14 is used to prove that on a compact, connected holo-morphically symplectic b − manifold X which is limit of irreduciblehyperk¨ahler manifolds the intersection of the pseudo-effective cone of X and of C X (which can be defined as for irreducible hyperk¨ahler man-ifolds by Theorem 1.12) consists of exactly one of the two connectedcomponents of C X : such a component is the positive cone of X , stilldenoted C + X . We then prove: Theorem 1.15.
Let X be a compact, connected holomorphically sym-plectic b − manifold which is limit of irreducible hyperk¨ahler manifolds.If there is α ∈ C + X such that (1) α · C > for every rational curve C on X , and (2) for every β ∈ H , ( X ) ∩ H ( X, Z ) we have q X ( α, β ) = 0 ,then X is irreducible hyperk¨ahler, and α is a K¨ahler class on X . The proof of this result is based on Theorem 1.14, which gives abimeromorphism f : Y X between X and an irreducible hy-perk¨ahler manifold Y . Using twistor lines for real (1 , − classes on X (which can be defined similarily to the hyperk¨ahler case thanks toTheorem 1.12) and a strategy used by Huybrechts for irreducible hy-perk¨ahler manifolds, we show that the conditions on α imply that f ∗ α is a K¨ahler class on Y . An easy argument then shows that f is abiholomorphism, and that α is a K¨ahler class on X .The last part of the paper is devoted to show that on M a class α as in the statement of Theorem 1.15 exists. This is obtained by usingthe (Hodge) isometry λ v : v ⊥ ⊗ R −→ H ( M, R )(whose existence was proved in [12]) to produce classes in C M . Bydeforming to a moduli space of slope-stable sheaves on a projective K3surface, and by using a classical construction of ample line bundles on M in this case (starting from an ample line bundle on S ), we are ableto conclude that a class as in Theorem 1.15 exists, concluding the proofof Theorem 1.9.1.3. The Beauville form and the Local Torelli Theorem.
Thestarting point of the proof of Theorem 1.9 is Theorem 1.12, which is
PEREGO due to Guan. We will not prove it here (the proof can be found in [7]),but we recall the definition of q X and the Local Torelli Theorem.1.3.1. The Beauville form on H ( X, C ) . Let X be a compact, con-nected holomorphically symplectic manifold of complex dimension 2 n .The Beauville form of X is a quadratic form on H ( X, C ) definedas follows. First, choose a holomorphic symplectic form σ on X , andassume for simplicity that R X σ n ∧ σ n = 1. For every α ∈ H ( X, C ),we let q σ ( α ) := n Z X α ∧ σ n − ∧ σ n − +(1 − n ) Z X α ∧ σ n ∧ σ n − Z X α ∧ σ n − ∧ σ n . Note that q σ ( σ + σ ) = ( R X σ n ∧ σ n ) = 1 so q σ is non-trivial. Moreover,the quadratic form q σ depends a priori on the choice of σ .1.3.2. The period map.
Let X be a compact, connected holomorphi-cally symplectic b − manifold of complex dimension 2 n , and supposethat h , ( X ) = 1. We let f : X −→ B be its Kuranishi family, and0 ∈ B a point such that the fiber X is isomorphic to X .By Theorem 1 and the following Remark 1 of [7], it follows that B issmooth, and that up to shrinking it we can even suppose that all thefibers of the Kuranishi family are holomorphically symplectic.Up to shrinking B , for every b ∈ B the fiber X b of f is a com-pact, connected holomorphically symplectic b − manifold (since beinga b − manifold is an open property). Moreover, again up to shrinking B , by the Ehresmann Fibration Theorem we can suppose that X isdiffeomorphic to X × B . In particular, this induces a diffeomorphism u b : X −→ X b for every b ∈ B , and hence an isomorphism of complexvector spaces u ∗ b : H ( X b , C ) −→ H ( X, C ) . We now let P := P ( H ( X, C )), and p : B −→ P , p ( b ) := [ u ∗ b ( σ b )] , where σ b is the holomorphic symplectic form on X b such that R X b σ nb ∧ σ nb = 1 (we notice that such a σ b is unique as h , ( X ) = 1, and hence h , ( X b ) = 1). The map p is holomorphic, and will be called periodmap of X .We let Q σ be the quadric defined by the quadratic form q σ in P , i.e. Q σ = { α ∈ P | q σ ( α ) = 0 } , and Ω σ be the open subset of Q σ defined asΩ σ := { α ∈ Q σ | q σ ( α + α ) > } . ¨AHLERNESS OF MODULI SPACES 9 As showed in [7] we have the following, known as
Local Torelli The-orem : Proposition 1.16.
Let X be a compact, connected holomorphicallysymplectic b − manifold, such that h , ( X ) = 1 . (1) The quadratic form q σ (and hence Q σ and Ω σ ) is independentof σ , and will hence be denoted q X (and similarly Q X and Ω X ). (2) Up to a positive rational multiple, the quadratic form q X isa non-degenerate quadratic form on H ( X, Z ) of signature (3 , b ( X ) − . (3) If B is the base of the Kuranishi family of X , we have that p ( B ) ⊆ Ω X , and that p : B −→ Ω X is a local biholomorphism. Using the non-generate quadratic form q X (of signature (3 , b ( X ) − C ′ X := { α ∈ H ( X, R ) | q X ( α ) > } , which is an open cone in H ( X, R ) having two connected components.Moreover, we let e H , R ( X ) := Im( { α ∈ H , ( X ) | dα = 0 } −→ H ( X, C )) ∩ H ( X, R ) , and notice that this consists exactly of de Rham cohomology classes ofreal d − closed (1 , − forms on X . We let C X := C ′ X ∩ e H , R ( X ) , which is then an open cone in e H , R ( X ) having two connected compo-nents. 2. Limits of irreducible hyperk¨ahler manifolds
This section is devoted to prove that every compact, connected holo-morphically symplectic b − manifold X which is limit of irreduciblehyperk¨ahler manifold, is bimeromorphic to an irreducible hyperk¨ahlermanifold: in other words, we prove Theorem 1.14.The proof is divided in several sections. First we construct a modulispace M Z of marked manifolds, and thanks to the Local Torelli Theo-rem we may give it the structure of a (non-separated) complex space.It will carry a period map to some period domain, which is locally abiholomorphism.Then we show that each point in the closure of the open subset of M Z given by irreducible hyperk¨ahler manifolds is non-separated froman irreducible hyperk¨ahler manifold.Adapting an argument of Siu (for K3 surfaces) and Huybrechts (forhigher dimensional irreducible hyperk¨ahler manifolds), we conclude theproof of Theorem 1.14. The moduli space of Λ − marked manifolds. In this section,we let Z be an irreducible hyperk¨ahler manifold, and we write (Λ , q ) :=( H ( Z, Z ) , q Z ) for the Beauville lattice of Z . We let P Λ := P (Λ ⊗ C ),and inside of it we let Q Λ := { α ∈ P Λ | q ( α ) = 0 } , which is the quadric defined by q , andΩ Λ := { α ∈ Q Λ | q ( α + α ) > } . The following is immediate, as X is a b − manifold and the Hodgenumbers are upper-semicontinuous. Proposition 2.1.
Let X be a compact, connected holomorphically sym-plectic b − manifold which is deformation equivalent to an irreduciblehyperk¨ahler manifold Z . Then for every p, q ≥ such that p + q = 2 we have h p,q ( X ) = h p,q ( Z ) . In particular h , ( X ) = 1 . If X is a compact, connected holomorphically symplectic b − manifold which is deformation equivalent to Z , by Proposition 2.1and Theorem 1.12 we know that H ( X, Z ) carries a non-degeneratequadratic form q X , and that there is an isometry φ : H ( X, Z ) −→ Λ.The isometry φ is called Λ − marking on X , and the pair ( X, φ ) is aΛ − marked manifold . The set of Λ − marked manifolds will be denoted M ′ Z .Moreover, we let M Z := M ′ Z / ∼ , where ( X, φ ) ∼ ( X ′ , φ ′ ) if andonly if there is a biholomorphism f : X −→ X ′ such that f ∗ ◦ φ ′ = φ .The set M Z will be referred to as the moduli space of Λ − markedmanifolds . We let M hkZ be the subset of M Z of pairs ( X, φ ) where X is an irreducible hyperk¨ahler manifold: it will be called moduli spaceof Λ − marked hyperk¨ahler manifolds .We first show that M Z has the structure of complex space (hencejustifying the name space we use for it): the following is a generalizationof Proposition 4.3 of [10], and requires the same proof. Proposition 2.2.
Let Z be an irreducible hyperk¨ahler manifold and (Λ , q ) its Beauville lattice. (1) For any ( X, φ ) ∈ M Z there is an inclusion i X : B −→ M Z ,where B is the base of the Kuranishi family of X . (2) The set M Z has the structure of smooth complex space of di-mension b ( Z ) − . (3) The subset M hkZ is an open subset (in the analytic topology) of M Z . ¨AHLERNESS OF MODULI SPACES 11 Proof.
Let X be a compact, connected holomorphically symplectic b − manifold which is deformation equivalent to Z , and f : X −→ B its Kuranishi family.Up to shrinking B we can suppose that it is a complex disk of di-mension b ( X ) − b ( Z ) −
2, and as we have seen before for every b ∈ B we can suppose that X b is a compact, connected holomorphicallysymplectic b − manifold (which is clearly deformation equivalent to Z ).Moreover, we can suppose that X is diffeomorphic (over B ) to thetrivial family X × B , and that we have a diffeomorphism u b : X −→ X b inducing an isometry u ∗ b : H ( X b , Z ) −→ H ( X, Z ). We let φ b := φ ◦ u ∗ b ,which is then a Λ − marking on X b , for every b ∈ B .It follows that for every b ∈ B we have ( X b , φ b ) ∈ M Z , so that wehave a map i X : B −→ M Z , i X ( b ) := ( X b , φ b ) . We show that i X is an inclusion. Let b, b ′ ∈ B and suppose that i X ( b ) = i X ( b ′ ). This means that ( X b , φ b ) ∼ ( X b ′ , φ b ′ ), i. e. there is abiholomorphism f : X b −→ X b ′ such that f ∗ = φ − b ◦ φ b ′ . By definition of φ b and φ b ′ , this means that f ∗ = ( φ ◦ u ∗ b ) − ◦ ( φ ◦ u ∗ b ′ ) = ( u ∗ b ) − ◦ u ∗ b ′ . Now, let σ b and σ b ′ be symplectic forms on X b and X b ′ respectively.As f is a biholomorphism, the form f ∗ σ b ′ is holomorphic symplectic on X b , and hence [ u ∗ b σ b ] = [ u ∗ b f ∗ σ b ′ ] . But as f ∗ = ( u ∗ b ) − ◦ u ∗ b ′ , this implies that [ u ∗ b σ b ] = [ u ∗ b ′ σ b ′ ]. By definitionof the period map of X , this means that p ( b ) = p ( b ′ ).But now recall that by point (3) of Proposition 1.16, the periodmap p : B −→ Ω is a local biholomorphism: up to shrinking B , for b = b ′ ∈ B we have p ( b ) = p ( b ′ ). It follows that up to shrinking B thecondition i X ( b ) = i X ( b ′ ) implies b = b ′ , and i X is an inclusion of B in M Z . This proves point 1 of the statement.To give M Z the structure of a complex space, we just need to showthat each point of M Z has a neighborhood having the structure of acomplex manifold, and that whenever two neighborhoods of this typeintersect, the corresponding complex structures glue.If ( X, φ ) ∈ M Z , the previous part of the proof suggests to view i X ( B )as a neighborhood ( X, φ ) in M Z . Now, let ( X, φ ) , ( X ′ , φ ′ ) ∈ M Z , andwe let B and B ′ be the bases of the Kuranishi families of X and of X ′ , respectively. If i X ( B ) ∩ i X ( B ′ ) = ∅ , then B ∩ B ′ is an open subsetof B and B ′ , over which the Kuranishi families coincide. This allows us to glue the Kuranishi families along B ∩ B ′ , and hence the complexstructures of i X ( B ) and i X ( B ′ ) can be glued in M Z . This shows that M Z has the structure of a complex space.We notice that as each base B of a Kuranishi family of a compact,connected holomorphically symplectic b − manifold is smooth (see sec-tion 2.2) of dimension b ( Z ) −
2, it follows that M Z is a smooth com-plex space, and its dimension is b ( Z ) −
2. This proves point 2 of thestatement.The fact that M hk Λ is open in the analytic topology is a consequenceof the fact that K¨ahlerness is an open property in the analytic topology. (cid:3) The complex space M Z has two connected components, and one canpass from one to the other by sending ( X, φ ) to ( X, − φ ).We now define the period map in this generality: we let π : M Z −→ P Λ , π ( X, φ ) := [ φ ( σ )] , where σ is a holomorphic symplectic form on X . Notice that theΛ − marking φ induces an isomorphism φ : P −→ P Λ , and as it is an isometry it induces an isomorphism φ : Ω X −→ Ω Λ . If B is the base of the Kuranishi family of X , we have π | i X ( B ) = φ ◦ p :if b ∈ B and σ b is a symplectic form on X b , we have φ ( p ( b )) = φ [ u ∗ b σ b ] = [ φ ( u ∗ b σ b )] = [ φ b ( σ b )] = π ( b ) . The first two points of the following Proposition are just a translationin this language of Theorem 1.12. For the last point: the surjectivity isTheorem 8.1 of [8]; the general injectivity is the Global Torelli Theoremof Verbistky.
Proposition 2.3.
We have the following properties: (1) the image of π is contained in Ω Λ ; (2) the map π is a local biholomorphism; (3) if M hk, Z is a connected component of M hkZ , the map π |M hk, Z issurjective and generically injective. Now, we let M hkZ be the closure of M hkZ in M Z . Using this formalism,we can state Theorem 1.14 in an equivalent way: Proposition 2.4. If ( X, φ ) ∈ M hkZ , then X is bimeromorphic to anirreducible hyperk¨ahler manifold (hence, it is in the Fujiki class C ). ¨AHLERNESS OF MODULI SPACES 13 This is the statement we will prove in the next sections.2.2.
Non-separatedness in M Z . The first result we show is the fol-lowing:
Proposition 2.5.
Let ( X, φ ) ∈ M hk, Z . Then there is ( Y, ψ ) ∈ M hk, Z such that ( X, φ ) and ( Y, ψ ) are non-separated in M Z .Proof. The statement is trivial if (
X, φ ) ∈ M hk, Z . We then supposethat ( X, φ ) ∈ M hk, Z \ M hk, Z .We let p := π ( X, φ ) ∈ Ω Λ be the period of ( X, φ ). As π |M hk, Z issurjective, there is ( Y, ψ ) ∈ M hk, Z such that π ( Y, ψ ) = p . We showthat ( X, φ ) and (
Y, ψ ) are non-separated in M Z .To do so, let U X and U Y be two open neighborhoods of ( X, φ ) and(
Y, ψ ) respectively in M Z . Up to shrinking U X and U Y , we can supposethat π ( U X ) = π ( U Y ) =: V .Moreover, by point (3) of Proposition 1.16, up to shrinking U X and U Y we can suppose that π | U Y : U Y −→ V and π | U X : U X −→ V arebiholomorphisms. Finally, as K¨ahlerness is an open property in theanalytic topology, up to shrinking U X and U Y we can suppose that U Y ⊆ M hk, Z .Now, as ( X, φ ) ∈ M hk, Z , there is a hyperk¨ahler manifold X ′ and amarking φ ′ on X ′ such that ( X ′ , φ ′ ) ∈ U X ∩ M hk, Z . We can choose( X ′ , φ ′ ) to be generic. Let p ′ := π ( X ′ , φ ′ ) ∈ V : as π | U Y : U Y −→ V is surjective, there is ( Y ′ , ψ ′ ) ∈ U Y such that π ( Y ′ , ψ ′ ) = p ′ , and as U Y ⊆ M hk, Z , we have that Y ′ is an irreducible hyperk¨ahler manifold.Hence ( X ′ , φ ′ ) and ( Y ′ , ψ ′ ) are two generic points in M hk, Z : by point(3) of Proposition 2.3 we then have ( X ′ , φ ′ ) = ( Y ′ , ψ ′ ) in M Z , so that U X ∩ U Y = ∅ , and we are done. (cid:3) This result will be the starting point of the proof of Proposition 2.4.2.3.
The proof of Theorem 1.14.
We now prove a key result in theproof of Proposition 2.4.
Lemma 2.6.
Let B be a connected complex manifold and X −→ B and Y −→ B be two smooth, proper families verifying the followingproperties: (1) for every b ∈ B the fiber Y b is an irreducible hyperk¨ahler mani-fold with a Λ − marking ψ b ; (2) for every b ∈ B the fiber X b is a compact, connected holomor-phically symplectic b − manifold deformation equivalent to anirreducible hyperk¨ahler manifolds, which has a Λ − marking φ b ; (3) there is a non-empty open subset V of B such that for each b ∈ V there is an isomorphism f b : Y b −→ X b such that f ∗ b = ψ − b ◦ φ b ; (4) for generic b ∈ V we have H , ( X b ) ∩ H ( X b , Z ) = 0 .Then V is dense in B .Proof. We choose a point 0 ∈ B and let X := X and Y := Y . Weshow that ∂V := V \ V is contained in a countable union of analyticsubvarieties of B . It has then real codimension at least 2 in B , henceit cannot separate the disjoint open subsets V and B \ V . As V = ∅ ,it follows that B = V .In order to show that ∂V is contained in a countable union of analyticsubvarieties of B , we show that if s ∈ ∂V , then Y s has either effectivedivisors or curves. This implies that s ∈ [ α S α , where α ∈ H ( Y, Z ) and S α is the analytic subvariety of B given bythose b ∈ B such that α ∈ N S ( Y b ). We proceed by contradiction: welet s ∈ ∂V , and we suppose that Y s has no effective divisors and nocurves.As s ∈ ∂V , it follows that ( X s , φ s ) and ( Y s , ψ s ) are non-separatedpoints in M Z , were Z is an irreducible hyperk¨ahler manifold amongall the Y b . We first show that X s and Y s are bimeromorphic. To showthis, let β s be a K¨ahler form on Y s and α s a closed real (1 , − form on X s whose cohomology class is in C X s .Consider a continuous family { β t } t ∈ B , where β t is a closed(1 , − form on Y t . As K¨ahlerness is an open property in the analytictopology, there is an analytic open neighborhood U of s in B such thatfor every t ∈ U the form β t is K¨ahler on Y t .Moreover, consider a continuous family { α t } t ∈ B where α t is a closed(1 , − form on X t . Up to shrinking U , and as the positivity of q X isan open property, we can suppose that for every t ∈ U the cohomologyclass of α t is in C X t .Notice that as s ∈ ∂V , the intersection of U with V is not empty,and the generic point t ∈ U ∩ V is such that f t : Y t −→ X t is abiholomorphism such that f ∗ t = ψ − t ◦ φ t . By hypothesis, we havethat X t is an irreducible hyperk¨ahler manifold with N S ( X t ) = 0. ByCorollary 5.7 of [8], this implies that the K¨ahler cone of X t is one ofthe two connected components of C X t .As the cohomology class [ α t ] of α t is in C X t , it follows that either [ α t ]or − [ α t ] is in the K¨ahler cone of X t . Up to changing the sign of α t , ¨AHLERNESS OF MODULI SPACES 15 we can then suppose that for the generic t ∈ U ∩ V the class [ α t ] is aK¨ahler class, and that α t is a K¨ahler form.In conclusion, there is a sequence { t m } m ∈ N of point of U ∩ V whichconverges to s , and such that for every m ∈ N we have a K¨ahler form α m := α t m on X m := X t m and a K¨ahler form β m := β t m on Y m := Y t m ,such that α m converges to α s and β m converges to β s .As t m ∈ V , we have an isomorphism f m : Y m −→ X m such that f ∗ m = ψ − m ◦ φ m (where ψ m := ψ t m and φ m = φ t m ). We let Γ m be thegraph of f m , and we compute its volume in X m × Y m with respect to thethe form p ∗ α m + p ∗ β m , where p and p are the projections of X m × Y m to X m and Y m respectively.We then havevol(Γ m ) = Z Y m ( β m + f ∗ m α m ) n = Z Y m ([ β m ] + f ∗ m [ α m ]) n == Z Y m ([ β m ] + ψ − m ◦ φ m ([ α m ])) n . Taking the limit for m going to infinity we getlim m → + ∞ vol(Γ m ) = Z Y s ([ β s ] + ψ − s ◦ φ s ([ α s ])) n < + ∞ . Hence, the volumes of the Γ m are bounded, so that by the BishopTheorem the cycles Γ m converge to a cycle Γ of X s × Y s with thesame cohomological properties of the Γ m ’s: namely, we have [Γ] ∈ H n ( X s × Y s , Z ), and if p and p are the two projections from X s × Y s to X s and Y s respectively, we have p ∗ [Γ] = [ X s ], p ∗ [Γ] = [ Y s ], and[Γ] ∗ γ := p ∗ ([Γ] · p ∗ γ ) = ψ − s ( φ s ( γ )) , for every γ ∈ H ( X s , Z ).Now, let us split Γ in its irreducible components: by the previousproperties we then have that either Γ = Z + P i D i where p : Z −→ X s and p : Z −→ Y s are both generically one-to-one, or Γ = Z + Z + P i D i where p : Z −→ X s and p : Z −→ Y s are generically one-to-one, but neither p : Z −→ Y s nor p : Z −→ X s are genericallyfinite. In both cases we have p ∗ [ D i ] = p ∗ [ D i ] = 0. As shown in [10](proof of Theorem 4.3), the second case cannot happen: it follows that Z is a bimeromorphism between X s and Y s .Now, recall that Y s is supposed to have no effective divisors nornon-trivial curves. It follows from this that the bimeromorphism Z isindeed an isomorphism: hence there is an isomorphism f : Y s −→ X s whose graph is Z , and X s is K¨ahler (as Y s is). Moreover, by Corollary5.7 of [8], the K¨ahler cone of Y s , and hence that of X s , is one of the components of C Y s , and the morphisms [ D i ] ∗ : H ( X s , Z ) −→ H ( Y s , Z )are all trivial (see Lemma 5.5 in [8]). In particular, we get that f ∗ =[ Z ] ∗ = [Γ] ∗ We now prove that Γ = Z . To do this, recall that the class of α s isK¨ahler, and let γ s := f ∗ α s , which is K¨ahler again on Y s . We computethe volumes on X s × Y s with respect to the K¨ahler class p ∗ α s + p ∗ γ s .We havevol(Γ) = vol( Z ) + X i vol( D i ) = Z Z ( p ∗ α s + p ∗ γ s ) n + X i vol( D i ) == Z Y s ([ Z ] ∗ α s + γ s ) n + X i vol( D i ) = Z Y s ( f ∗ α s + γ s ) n + X i vol( D i ) == Z Y s (2 γ s ) n + X i vol( D i ) , where we used that f ∗ = [ Z ] ∗ = [Γ] ∗ .Now, choose a sequence { t m } m ∈ N of points in V ∩ U converging to s ,and consider the graphs Γ m of the isomorphisms f m : Y m −→ Y m . Let-ting γ m := f ∗ m α m , the sequence { γ m } converges to γ s , and the sequence { vol(Γ m ) } m ∈ N converges to vol(Γ). Butvol(Γ m ) = Z Y m ([Γ m ] ∗ α m + γ m ) n = Z Y m (2 γ m ) n , hence this sequence converges to Z Y s (2 γ s ) n , so that vol(Γ) = Z Y s (2 γ s ) n . It turns then out that vol( D i ) = 0, hence D i = 0 and Γ = Z .It follows that f ∗ = [ Z ] ∗ = [Γ] ∗ = ψ − s ◦ φ s . But this implies that s ∈ V , contradicting that s ∈ ∂V . In conclusionif s ∈ ∂V on Y s there are either effective divisors or curves, and we aredone. (cid:3) We are now able to conclude the proof of Proposition 2.4.
Proof.
By Proposition 2.5 we know that there is an irreducible sym-plectic manifold Y , together with a marking ψ , such that ( X, φ ) isnon-separated from (
Y, ψ ) in M Z . ¨AHLERNESS OF MODULI SPACES 17 Consider
Def ( X ) and Def ( Y ), the bases of the Kuranishi familiesof X and Y respectively. Up to shrinking them, as the points ( X, φ )and (
Y, ψ ) are non-separated, the Local Torelli Theorem allows us toidentify them. Hence the Kuranishi families of X and Y are over thesame base B , and we suppose that X and Y are over the same point0 ∈ B .The non-separatedness implies that there is t ∈ B such that X t and Y t are isomorphic under an isomorphism f t such that f ∗ t = ψ − t ◦ φ t .Let V be the biggest open subset of B given by all t ∈ B verifyingthis same property. As V is open in B , and as B is open in M Z , thegeneric point t of V is such that N S ( X t ) = 0.We can then apply Lemma 2.6 to conclude that V = B . Now, noticethat if 0 ∈ V , then X and Y are isomorphic, and we are done. Hencewe can suppose that 0 / ∈ V , so that 0 ∈ ∂V . We can then apply thesame argument in the proof of Lemma 2.6 to conclude that X and Y are bimeromorphic. (cid:3) Criterion for K¨ahlerness
We now want to prove a K¨ahlerness criterion for a compact, con-nected holomorphic symplectic b − manifold X which is limit of irre-ducible hyperk¨ahler manifolds.Let us first recall a notation. In this section, X is a compact, con-nected holomorphic symplectic b − manifold X which is limit of irre-ducible hyperk¨ahler manifolds. By Theorem 1.14, we know that X isin the Fujiki class C , hence H ( X, C ) has a Hodge decomposition. Inparticular, we have e H , R ( X ) = H ( X, R ) ∩ H , ( X ) =: H , ( X, R ) . Twistor lines. If σ is a holomorphic symplectic form on X , thecohomology class of σ allows us to define a real plane P ( X ) := ( C · σ ⊕ C · σ ) ∩ H ( X, R )in H ( X, R ), which is independent of σ (as h , ( X ) = 1). If α ∈ H , ( X, R ) we let F ( α ) := P ( X ) ⊕ R · α, which is a 3 − dimensional real subspace of H ( X, R ), and we let F ( α ) C := F ( α ) ⊗ C .Now, let Z be an irreducible hyperk¨ahler manifold which is de-formation equivalent to X , and let (Λ , q ) its Beauville lattice. If φ : H ( X, Z ) −→ Λ is a Λ − marking on X (which exists by Theorem X, φ ) of M Z . As X is limit of irreduciblehyperk¨ahler manifolds, we have ( X, φ ) ∈ M hkZ .Notice that F ( α ) C is a 3 − dimensional linear subspace of H ( X, C ),hence φ ( F ( α ) C ) is a 3 − dimensional subspace of Λ ⊗ C , and P ( φ ( F ( α ) C )is a plane in P Λ . Hence P ( φ ( F ( α ) C )) ∩ Ω Λ is a curve in Ω Λ passingthrough π ( X, φ ).If B is the base of the Kuranishi family of X , the inverse image T ( α ) := π − ( P ( φ ( F ( α ) C )) ∩ Ω Λ ) ∩ B is then a curve in B , which will be called twistor line of α . Therestriction of the Kuranishi family of X to T ( α ) will be denoted κ α : X ( α ) −→ T ( α ) . For every t ∈ T ( α ) there is real (1 , − class α t on the fiber X t of theKuranishi family of X over t , and the sequence { α t } converges to α . If α is K¨ahler, then T ( α ) ≃ P , and all α t are K¨ahler on X t .3.2. Cones in H , ( X, R ) . Using the notation introduced in the pre-vious section, we have C X = { α ∈ H , ( X, R ) | q X ( α ) > } , which is an open cone in H , ( X, R ) having two connected components.If X is K¨ahler (and hence irreducible hyperk¨ahler), the K¨ahler cone K X of X (i. e. the open convex cone of K¨ahler classes on X ) iscontained in one of them: such a component is usually called positivecone of X , and denoted C + X . The other component will be denoted C − X .If N S ( X ) = 0, Corollary 5.7 of [8] gives us that K X = C + X , a fact thathas already been used in the previous sections.Theorem 1.1 of [2] tells us that if X is irreducible hyperk¨ahler, then α ∈ C + X is in the K¨ahler cone of X if and only if Z C α > C of X . Our aim is to show a similar result fora compact, connected holomorphic symplectic b − manifold X which islimit of irreducible hyperk¨ahler manifolds.As on such a manifold the K¨ahler cone could be empty, we can-not use it to define the positive cone of X . Anyway, we can use the pseudo-effective cone E X of X , i. e. the closed connected cone ofclasses of positive closed real (1 , − currents on X . If X is irreduciblehyperk¨ahler, by point i) of Theorem 4.3 of [3] we have C + X ⊆ E X . ¨AHLERNESS OF MODULI SPACES 19 Popovici and Ugarte (see Theorem 5.9 of [13]) showed that if
X −→ B is a smooth and proper family of sGG manifolds and { b n } is a se-quence of points of B converging to a point b ∈ B , then the limit ofthe pseudo-effective cones of X b n is contained in E X b , i. e. the pseudo-effective cone varies upper-semicontinuously along B .As all manifolds in the Fujiki class C are sGG manifolds (see [13]), weconclude that the pseudo-effective cone varies upper-semicontinuouslyin families of class C manifolds.We now prove the following general fact about convex cones in a realfinitely dimensional vector space. Lemma 3.1.
Let V a real vector space of finite dimension n , and let A, B ⊆ V two cones in V such that: (1) the cone A is strictly convex (i. e. it does not contain any linearsubspace of V ) and closed; (2) the cone B is open and has two connected components, each ofwhich is convex; (3) for every a ∈ B , we have either a ∈ A or − a ∈ A .Then A ∩ B is one of the connected components of B .Proof. We first notice that if B + and B − are the two connected com-ponents of B , if B + ⊆ A we have B + = B ∩ A . Indeed, if b ′ ∈ B − ∩ A ,then − b ′ ∈ B + ⊆ A . It follows that b ′ , − b ′ ∈ A , which is not possibleas A is a strictly convex cone.We are left to prove that there is a connected component of B whichis contained in A . To do so, let b ∈ B ∩ A , and let B + be the connectedcomponent of B which contains b . We show that if b ∈ B + , then b ∈ A .Consider the segment[ b , b ] := { b t := (1 − t ) b + tb | t ∈ [0 , } . Suppose that b / ∈ A , we have to find a contradiction. First, noticethat as b / ∈ A , there is t ∈ [0 ,
1) such that b t / ∈ A : indeed, if for every t ∈ [0 ,
1) we have b t ∈ A , as A is closed we would have b ∈ A .As b t and b are not in A , for every s ∈ [ t,
1] we have b s / ∈ A : indeed,as b t , b / ∈ A but b t , b ∈ B , we have − b t , − b ∈ A . As A is convex,the segment [ − b t , − b ] (whose elements are the − b s for s ∈ [ t, A . But this means that b s / ∈ A as A is a strictly convexcone.The set of those t ∈ [0 ,
1] for which b t / ∈ A surely has an infimum t ∈ [0 , t < t we have b t ∈ A , and for every t > t we have b t / ∈ A . As b t ∈ B , this implies that − b t ∈ A for every t > t .But as A is closed, these conditions give b t ∈ A (as b t ∈ A for every t < t ) and − b t ∈ A (as − b t ∈ A for every t > t ). But as A is astrictly convex cone, we get a contradiction. (cid:3) This fact will be used in the proof of the following:
Lemma 3.2.
Let X be a compact, connected holomorphic symplectic b − manifold which is limit of irreducible hyperk¨ahler manifolds. Then C X ∩ E X consists of exactly one connected component of C X .Proof. The pseudo-effective cone E X is strictly convex and closed in H , ( X, R ). The cone C X is open ans has two connected components,each of which is convex. We show that if α ∈ C X , then either α ∈ E X or − α ∈ E X (which in particular implies that C X ∩ E X = ∅ ). Once thisis done, the statement follows from Lemma 3.1.Fix an irreducible hyperk¨ahler manifold Z which is deformationequivalent to X , and we let (Λ , q ) be its Beauville lattice. Moreover,let α ∈ C X , and consider a Λ − marking φ on X (whose existence comesfrom Theorem 1.12). As X is limit of irreducible hyperk¨ahler mani-folds, we have ( X, φ ) ∈ M hkZ .Let X −→ B be the Kuranishi family of X , and let 0 be the point of B over which the fiber of X is X . By Theorem 1.14 X is in the Fujikiclass C , hence it is a sGG-manifold. This being an open condition(see [13]), up to shrinking B we can suppose that for every b ∈ B themanifold X b is sGG.Moreover, as ( X, φ ) ∈ M hkZ , there is a sequence { b n } of points of B converging to 0 over which the fiber X n is irreducible hyperk¨ahler, andwe can even suppose that N S ( X n ) = 0.Consider a sequence { α n } given by α n ∈ C X n converging to α , henceeither α n ∈ C + X n (for all n ), or − α n ∈ C + X n (for all n ). As recalledbefore, we have C + X n ⊆ E X n : it follows that either α n ∈ E X n (for all n )or − α n ∈ E X n (for all n ). By Theorem 5.9 of [13] we then concludethat either α ∈ E X or − α ∈ E X . (cid:3) The connected component of C X contained in E X will be denoted C + X and called positive cone of X , in analogy with the hyperk¨ahler case.The other connected component of C X will be denoted C − X .3.3. Deformations and K¨ahler classes.
The first result we prove isthe following:
Proposition 3.3.
Let X be a compact, connected holomorphic sym-plectic manifold in the Fujiki class C , which is limit of irreducible hy-perk¨ahler manifolds. Let α ∈ C X . ¨AHLERNESS OF MODULI SPACES 21 (1) If for every β ∈ H ( X, Z ) we have q ( α, β ) = 0 , then there is t ∈ T ( α ) such that X t is K¨ahler and either α t or − α t is a K¨ahlerclass on X t . (2) If moreover α ∈ C + X , then α t is K¨ahler.Proof. Let Z be an irreducible hyperk¨ahler manifold which is defor-mation equivalent to X , and let (Λ , q ) be its Beauville lattice. Fixa marking φ on X (which exists by Theorem 1.12), and consider thepoint ( X, φ ) ∈ M Z . As X is limit of irreducible hyperk¨ahler manifolds,we have ( X, φ ) ∈ M hkZ .We first show that for the generic t ∈ T ( α ) the fiber X t is in M hkZ .Let X −→ B be the Kuranishi family of X , and let 0 be the point of B over which the fiber X is X .As ( X, φ ) ∈ M hkZ , there is a sequence { b n } of points of B verifyingthe following properties:(1) the sequence b n converges to 0 in B ;(2) for each n the fiber X n of X over b n is an irreducible hyperk¨ahlermanifold such that H , ( X n ) ∩ H ( X n , Z ) = 0;(3) for each n there is α n ∈ C X n such that the sequence α n convergesto α .As H , ( X n ) ∩ H ( X n , Z ) = 0, up to changing the sign of α , and henceof α n , we can suppose that α n ∈ K X n for every n . We let T n be thetwistor line of α n , which is a rational curve in B passing through thepoint ( X n , φ n ).As ( X n , φ n ) converges to ( X, φ ), and as α n converges to α , we seethat the twistor lines T n converge to T ( α ). This means that if t ∈ T ( α ),there is a sequence { s t,n } of points of B such that(1) the sequence { s t,n } converges to t ;(2) for each n we have s t,n ∈ T n .As s t,n ∈ T n , and as T n is the twistor line of the K¨ahler class α n ,we see that the fiber X s t,n of the twistor family of α n over s t,n is anirreducible hyperk¨ahler manifold. As s t,n converges to t , we then seethat X t is limit of irreducible hyperk¨ahler manifolds. This means that( X t , φ t ) ∈ M hkZ .In particular, by Proposition 2.4 this implies that X t is bimeromor-phic to an irreducible hyperk¨ahler manifold Y t . Now, by hypothesis wehave q X ( α, β ) = 0 for each β ∈ H ( X, Z ). This implies that T ( α ) doesnot intersect in 0 (and hence generically) any of the hypersurfaces S β ,i. e. for a generic t ∈ T ( α ) the period of ( X t , φ t ) is generic in Ω Λ . As the periods of ( X t , φ t ) and ( Y t , ψ t ) are equal, it follows that for ageneric t ∈ T ( α ) the irreducible hyperk¨ahler manifold Y t is such that H , ( Y t ) ∩ H ( Y t , Z ) = 0, so that X t and Y t are biholomorphic.It follows that X t is irreducible hyperk¨ahler, and that K X t is one ofthe components of C X t . As α t ∈ C X t , it follows that either α t or − α t isa K¨ahler class on X t .Let us now suppose moreover that the class α is even pseudo-effective, and by contradiction that α t is not a K¨ahler class. By whatwe just proved, it follows that − α t is K¨ahler for generic t ∈ T ( α ). As K X t is contained in E X t , we then have a family − α t of pseudo-effectiveclasses converging to − α .Now, by Theorem 5.9 of [13] (which we can apply as by the previouspart of the proof the family X ( α ) −→ T ( α ) is a family of class C manifolds, and hence of sGG manifolds) we know that a limit of pseudo-effective classes along the family X ( α ) is a pseudo-effective class on X .This means that − α is a pseudo-effective class on X . As by hypothe-sis α is pseudo-effective too, it follows that α = 0, which is not possibleas q X ( α ) >
0. This shows that is α is a positive pseudo-effective classsuch that q X ( α, β ) = 0 for every β ∈ H ( X, Z ), then for a generic t ∈ T ( α ) the class α t is K¨ahler. (cid:3) We now use the previous Proposition to show the following, whichis an improved version of Proposition 2.4.
Proposition 3.4.
Let X be a compact holomorphic symplectic man-ifold in the Fujiki class C , which is limit of irreducible hyperk¨ahlermanifolds, and let α ∈ C + X be such that q X ( α, β ) = 0 for every β ∈ H ( X, Z ) . Then there is an irreducible hyperk¨ahler manifold Y and a cycle Γ = Z + P i D i in X × Y such that the following propertiesare verified: (1) the cycle Z defines a bimeromorphic map between X end Y ; (2) the projections D i −→ X and D i −→ Y have positive dimen-sional fibers; (3) the cycle Γ defines a Hodge isometry [Γ] ∗ between H ( X, Z ) and H ( Y, Z ) ; (4) the class [Γ] ∗ α is K¨ahler.Proof. Consider the family κ α : X ( α ) −→ T ( α ). By Proposition 3.3,we know that for a generic t ∈ T ( α ) the fiber X t of κ α over t is anirreducible hyperk¨ahler manifold, and that α t is a K¨ahler class on it.Let X ′ −→ T ( α t ) be the twistor family of ( X t , α t ), and notice that π ( T ( α )) is identified with an open subset of π ( T ( α t )), and that forevery s ∈ T ( α t ) the fiber X ′ s of X ′ over s is K¨ahler. ¨AHLERNESS OF MODULI SPACES 23 Restricting the twistor family X ′ to such an open subset, we thenfind two families X ( α ) −→ C and X ′ −→ C over the same base curve,which have isomorphic fibers over t , and the fibers of X ′ are all K¨ahler.We let 0 ∈ C be the point over which the fiber of X is X , and we let X ′ be the fiber of X ′ over 0.Both families are endowed with natural markings φ s and φ ′ s for each s , such that ( φ ′ t ) − ◦ φ t is induced by the isomorphism X t ≃ X ′ t . Theclass α ′ s := ( φ ′ s ) − ◦ φ s ( α s ) is a K¨ahler class on X ′ s for every s ∈ C . Inparticular the class α ′ := ( φ ′ ) − ◦ φ ( α ) is K¨ahler on X ′ .By Lemma 2.6 the points ( X, φ ) and ( X ′ , φ ′ ) are non-separatedpoints in M Z . As ( X, φ ) ∈ M hkZ and ( X ′ , φ ′ ) ∈ M hkZ , we can applyProposition 2.4 to show that there is a cycle Γ = Z + P i Y i on X × X ′ such that(1) the cycle Z defines a bimeromorphic map between X end X ′ ;(2) the projections Y i −→ X and Y i −→ X ′ have positive dimen-sional fibers;(3) the cycle Γ defines a Hodge isometry [Γ] ∗ between H ( X, Z )and H ( X ′ , Z );Notice that [Γ] ∗ α = α ′ , which is K¨ahler. (cid:3) The proof of Theorem 1.15.
We are now ready to prove The-orem 1.15, namely that if X is a compact, connected holomorphicallysymplectic b − manifold which is limit of irreducible hyperk¨ahler man-ifolds, any very general class α ∈ C + X (i. e. q X ( α, β ) = 0 for every β ∈ H ( X, Z )) such that α · C > C on X isa K¨ahler class on X , and in particular X is K¨ahler. Proof.
By Proposition 3.4, as α ∈ C + X is such that q X ( α, β ) = 0 forevery β ∈ H ( X, Z ), then there is an irreducible hyperk¨ahler manifold Y and a cycle Γ = Z + P i D i in X × Y such that the following propertiesare verified:(1) the cycle Z defines a bimeromorphic map between X end Y ;(2) the projections D i −→ X and D i −→ Y have positive dimen-sional fibers;(3) the cycle Γ defines a Hodge isometry [Γ] ∗ between H ( X, Z )and H ( Y, Z );(4) the class α ′ := [Γ] ∗ α is K¨ahler on Y .The argument used in the proof of Theorem 2.5 of [9] shows that since[Γ] ∗ α is a K¨ahler class on Y and α · C > C on X , then all the irreducible components D i of Γ which are contractedby the projection p X of X × Y to X are such that the codimension in X of p X ( D i ) is at least 2. By Lemma 2.2 of [8] it then follows that the morphisms [ D i ] ∗ : H ( Y, Z ) −→ H ( X, Z ) are all trivial. As aconsequence, we have α = [Γ] ∗ α ′ = [ Z ] ∗ α ′ .We let f : Y X be a bimeromorphism whose graph is Z . As α ′ is K¨ahler, then for every rational curve C ′ in Y we have Z C ′ α ′ > . Notice that α ′ = f ∗ α , so that we have Z C α > , Z C ′ f ∗ α > C in X and every rational curve C ′ in Y . ByProposition 2.1 of [9] it follows that f extends to an isomorphism, and α is then a K¨ahler class. (cid:3) K¨ahlerness of moduli spaces of sheaves
This last section is devoted to the proof of Theorem 1.9. Hence, welet S be a K3 surface, v ∈ H ∗ ( S, Z ) be of the form v = ( r, ξ, a ) where r > ξ ∈ N S ( S ) are prime to each other. Moreover, we let ω bea K¨ahler class on S , which we suppose to be v − generic.We want to show that if the moduli space M := M µv ( S, ω ) is a b − manifold, then it is K¨ahler. To do so, we apply Theorem 1.15 to M : we then need to prove that M is a compact, connected holomorphi-cally symplectic b − manifold which is limit of irreducible hyperk¨ahlermanifolds, and we need to provide a very generic class α ∈ C + M suchthat α · C > C in M .We will always assume that v ≥
2, as the cases v ≤ v < − M = ∅ ; if v = − M is a point; if v = 0, then M is a K3 surface by Corollary 5.3 of [12].A. The moduli space M is a compact, connected holomorphically sym-plectic b − manifold which is limit of irreducible hyperk¨ahler manifolds.The fact that M is a compact, holomorphically symplectic manifold isdue to Toma (see Remark 4.5 of [15]). The connectedness is given byProposition 4.24 of [12]. The fact that M is a b − manifold is supposedto hold true.We are then left to prove that M is a limit of irreducible hyperk¨ahlermanifolds. To do so, let S −→ B be the Kuranishi family of the K3surface S , where B is a complex manifold of dimension 20. Let B ξ ⊆ B be the subvariety of B given by those b ∈ B such that ξ ∈ N S ( S b ).Similarily, let B ω ⊆ B be the subvariety of B given by those b ∈ B such that the class ω ∈ H , ( S b , R ). Moreover, we let B ξ,ω := B ξ ∩ B ω . ¨AHLERNESS OF MODULI SPACES 25 Recall that B ξ and B ω are smooth hypersurfaces of B , and as ξ and ω are linearily independent, then B ξ and B ω intersect transversally, sothat B ξ,ω is a smooth analytic subset of B (of positive dimension). ByTheorem 3.5 of [8], the subset B pξ,ω of B ξ,ω given by those b such that S b is projective is dense in B ξ,ω .We now consider the restriction S ′ := S | B ξ,ω , together with a mor-phism S ′ −→ B ξ,ω . We suppose 0 ∈ B ξ,ω be such that S ∼ S .Recall that ω is a K¨ahler class on S : as K¨ahlerness is an open prop-erty in the analytic topology, there is an analytic open subset D ′ ⊂ B ξ,ω containing 0 and such that for every b ∈ D ′ the class ω is K¨ahler on S b . We then can consider the relative moduli space M −→ D ′ , whosefiber over b is the moduli space M b = M µv ( S b , ω ) of µ ω − stable sheaveson S b whose Mukai vector is v .As the v − genericity is an open property in the analytic topology,there is an open subset D of D ′ such that for every b ∈ D the class ω is v − generic. We then consider the restriction S D of S to D , togetherwith a morphism S D −→ D . For every d ∈ D the K3 surface S d comesequipped with a Mukai vector v = ( r, ξ, a ) and a v − generic polarization ω .As a consequence, the restriction of M to D , denoted M D is suchthat for every d ∈ D the fiber M d is a compact, connected complexmanifold (see again Proposition 4.24 of [12]). The morphism M D −→ D being submersive, it follows that the family M D −→ D is a smoothand proper family, whose fiber over 0 is M µv ( S, ω ).Now, as B pξ,ω is dense in B ξ,ω , it follows that B pξ,ω ∩ D is dense in D . Hence, for the generic point d ∈ D the fiber S d is a projective K3surface, and the fiber M d is an irreducible hyperk¨ahler manifold (seeTheorem 3.4 of [12]). Hence M µv ( S, ω ) is limit of irreducible hyperk¨ahlermanifolds.
Remark 4.1.
As a consequence of what we just proved, by Theorem1.14 we see that if M µv ( S, ω ) verifies the assumptions of Theorem 1.9,then it is in the Fujiki class C .B. A very generic class α ∈ C + M such that α · C > C of M . In order to prove Theorem 1.9, by the previous part ofthis section we just need to find such a class α on M . To do so, recallthat by [12] there is a morphism λ v : v ⊥ −→ H ( M, Z )which is an isometry (since v ≥
2) with respect to the Mukai pairingon v ⊥ and the Beauville form of M . Moreover, by the previous paragraph the moduli space M is in theFujiki class C , hence H ( M, Z ) has a Hodge decomposition, and λ v is aHodge morphism. In particular, λ v is a Hodge isometry. This remainstrue if we tensor with R , and we get a Hodge isometry λ v : v ⊥ ⊗ R −→ H ( M, R ) . We will then construct the desired class α by taking an appropriateelement of v ⊥ ⊗ R .The choice we make is the following: let m ∈ N and α m,ω := ( − r, − mrω, a + mω · ξ ) . First of all, we remark that α m,ω ∈ v ⊥ ⊗ R , as( v, α m,ω ) S = mrω · ξ − r ( a + mω · ξ ) + ra = 0 . Moreover, as ω is a real (1 , − class on S , the class α m,ω is a real(1 , − class orthogonal to v .It then follows that α := λ v ( α m,ω ) ∈ H , ( M, R ) . We prove that α is a very general class in C + M such that α · C > C on M .We start by showing that α is very general in H , ( M, R ) Lemma 4.2. If ω is sufficiently generic, then for every β ∈ H ( M, Z ) we have q M ( α, β ) = 0 .Proof. As β ∈ H ( M, Z ), there are s, b ∈ Z and D ∈ H ( S, Z ) suchthat D · ξ = sa + rb (i. e. γ := ( s, D, b ) ∈ v ⊥ ) and β = λ v ( γ ).It follows that q M ( α, β ) = q M ( λ v ( α m,ω ) , λ v ( γ )) = ( α m,ω , γ ) S == − mω · ( rD + sξ ) + rb − sa. Suppose that q ( α, β ) = 0: this is then equivalent to ω · ( rD + sξ ) = D · ξ − sam , which means that ω is on some hyperplane in H ( S, R ) associated to D . As the family of these hyperplanes is countable (since the familyof D ∈ H ( S, Z ) is countable), and as ω is sufficiently generic, we seethat q ( α, β ) = 0 for every β ∈ N S ( M ). (cid:3) We notice that we can move ω in the v − chamber of the K¨ahler coneof S where it lies without changing M (see Proposition 3.2 of [12]),hence we can always suppose that ω is sufficiently generic, and hencethat α is very generic. ¨AHLERNESS OF MODULI SPACES 27 We now show that q M ( α ) > M . Lemma 4.3. If m ≫ we have α ∈ C + M .Proof. We first prove that α ∈ C M , and then that α ∈ E M .We have q M ( α ) = q M ( λ v ( α m,ω )) = ( α m,ω , α m,ω ) S == m r ω + 2 ra − mrω · ξ. As m ≫ ω > ω is K¨ahler on S ), we then see that q M ( α ) >
0, i. e. α ∈ C M .We now have to show that α ∈ E M . To show this, consider thedeformation M −→ B ξ,ω we introduced in the previous paragraph. We let 0 ∈ B ξ,ω be the pointover which the fiber is M µv ( S, ω ). For a generic b ∈ B ξ,ω the fiber is M b = M µv ( S b , ω ) where S b is a projective K3 surface, so that the fiberis a projective irreducible hyperk¨ahler manifold.Notice that ω is still a v − generic K¨ahler class on S b , and the class α is still in C M b , and this for every b ∈ B ξ,ω . We write α := α Now, as shown in Remark 3.5 of [12], in the same v − chamber where ω lies there is a class of the form ω ′ = c ( H ) for some ample line bundle H on S b . We let α := λ v ( α m,ω ′ ). Moreover, for every t ∈ [0 ,
1] we let ω t := (1 − t ) ω + tω ′ , which is a segment contained in the v − chamberwhere ω and ω ′ are, and we let α t := λ v ( α m,ω t ). By linearity of λ v , wehave α t = (1 − t ) α + tα , and the image of the map α : [0 , −→ H , ( M b , R ) defined by letting α ( t ) := α t is a segment in C M b .Our aim is to show that α ∈ E M . As the family M −→ B is afamily of manifolds in the Fujiki class C by the previous paragraph, byTheorem 5.9 of [13] it is sufficient to show that α ∈ E M b for a generic b around 0. As for the generic b around 0 we have that M b is irreduciblehyperk¨ahler, this is equivalent to show that α ∈ C + M b .As C + M b is a convex cone and the segment [ α , α ] is contained in C M b ,to show that α ∈ C + M b it is sufficient to show that α ∈ C + M b .But now, as S b is projective we can use a general construction pre-sented in [11]: if H is a v − generic ample line bundle on S b , we canconstruct an ample line bundle L ( H ) on M µv ( S b , H ), and we have c ( L ( H )) = λ v ( α m,c ( H ) ). As M µv ( S b , ω ) = M µv ( S b , H ) (as ω and c ( H )are in the same v − chamber, by [12]), the class λ v ( α m,ω ′ ) is an ample class on M µv ( S b , ω ). It then lies in the K¨ahler cone of M b , and hence in C + M b . (cid:3) In conclusion, we have shown that up to choosing m ≫ ω sufficiently generic, the class α is a very generic class in C + M . We areleft to show that α · C > C in M . Lemma 4.4. If m ≫ and ω is sufficiently generic, we have α · C > for every rational curve C on M .Proof. Let [ C ] ∈ H n − , n − ( M, Z ), and let β C ∈ N S ( M ) be the dualof [ C ], so that α · C = q M ( α, β C ) . We then just need to prove that q M ( α, β C ) > C on M .Let S −→ B be the Kuranishi family of S , and let 0 ∈ B be suchthat S = S . We let B C be the subset of B of those b ∈ B suchthat β C ∈ N S ( S b ), i. e. C is a rational curve on S b . Consider theintersection D C := D ∩ B C , which is an analytic subset of D , whosegeneric point d is such that S d is a projective K3 surface.We let M C be the restriction of the relative moduli space M −→ D to D C . Notice that for every d ∈ D C we have the class α ∈ C M d andthe rational curve C on M d . As the intersection product of α with C is constant along D C , it is sufficient to show that q M d ( α, β C ) > d ∈ D C .As β C ∈ N S ( M d ), there are s, b ∈ Z and ζ ∈ N S ( S d ) such that γ := ( s, ζ , b ) ∈ v ⊥ and β C = λ v ( γ ). As λ v is an isometry, we have q M d ( α, β C ) = q M d ( λ v ( α m,ω ) , λ v ( γ )) = ( α m,ω , γ ) S d . It is then sufficient to show that ( α m,ω , γ ) S d > ω ′ on S d which isin the same v − chamber of ω and such that for every η ∈ N S ( S d ) wehave ω · η = ω ′ · η . Then we have α m,ω ′ ∈ v ⊥ , and( α m,ω , γ ) S d = ( α m,ω ′ , γ ) S d . It is then sufficient to show that ( α m,ω ′ , γ ) S d > ω ′′ in a neighborhood of ω ′ in the amplecone of S d , and let p ∈ N and H an ample line bundle on S d such that pω ′′ = c ( H ). As we can choose m ≫
0, we can suppose that m = m ′ p for some very big m ′ ∈ N . As H is v − generic we have that λ v ( α m ′ ,c ( H ) )is the first Chern class of an ample line bundle, so that λ v ( α m ′ ,c ( H ) ) · C > . ¨AHLERNESS OF MODULI SPACES 29 It follows that ( α m ′ ,c ( H ) , γ ) S d > . Now, notice that α m ′ ,c ( H ) = ( − r, m ′ rc ( H ) , a + m ′ c ( H ) · ξ ) == ( − r, m ′ prc ( H ) /p, a + m ′ pξ · c ( H ) /p ) == ( − r, mrω ′′ , a + mω ′′ · ξ ) = α m,ω ′′ . Hence we get ( α m,ω ′′ , γ ) S d > ω ′′ in a neighborhood of ω ′ , thisimplies that ( α m,ω ′ , γ ) S d ≥ . As we saw before, this implies that α · C ≥ C in M . But as β C ∈ N S ( M ) and α · C = q M ( α, β C ), and as we knowthat α is very generic by Lemma 4.2, it follows that α · C = 0. Inconclusion we have α · C >
0, and we are done. (cid:3)
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