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The Moser isotopy for holomorphic symplectic and C-symplectic structures

A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Moser's isotopy theorem for families of C-symplectic structures and list several applications of this result. We prove that the degenerate twistorial deformation associated to a holomorphic Lagrangian fibration is locally trivial over the base of this fibration. This is used to extend several theorems about Lagrangian fibrations, known for projective hyperk\"ahler manifolds, to the non-projective case. We also exhibit new examples of non-compact complex manifolds with infinitely many pairwise non-birational algebraic compactifications.


Introduction
Let X be a complex manifold. Recall that a holomorphic 2-form Ω ∈ H 0 (X, Ω 2 X ) is symplectic if it is closed and non-degenerate at every point of X. Viewing Ω as a section of the bundle Λ 2,0 X, we note that one can uniquely recover the complex structure of X from the form Ω. Namely, the subbundle T 0,1 X is the kernel of Ω and T 1,0 X is its complex conjugate. We see that the complex 2-form Ω alone can be used to encode both the complex structure and the symplectic structure of X. Abstracting the properties of Ω that are necessary to reconstruct the complex structure, we arrive at the notion of a C-symplectic form, see Definition 2.1.
A similar point of view on holomorphic symplectic structures appeared in the work of Hitchin [Hi] and was further explored in [Ve] and [BDV]. One advantage of this point of view is that we can describe deformations of both the complex and symplectic structures on X simultaneously.
Our aim is to investigate the properties of C-symplectic structures further. Recall that the ordinary real symplectic structures admit a simple local description. Namely, by Darboux theorem, any real symplectic manifold M is locally symplectomorphic to an open subset in a vector space with the standard symplectic form. Moreover, any compact Lagrangian submanifold N ⊂ M admits a neighbourhood that is symplectomorphic to a neighbourhood of the zero section in T * N with its canonical symplectic form, see e.g. [MS,section 3]. To prove these statements one can use an idea that goes back to Moser [Mo] and construct an isotopy that maps one symplectic form to another.
In the present paper, we prove some analogues of Moser's isotopy theorem for C-symplectic structures (section 2.3) and discuss their applications (section 3). It turns out that it is useful to apply Moser's isotopy to holomorphic Lagrangian fibrations. Given such a fibration π : X → S over a projective base S, we consider a degenerate twistorial deformation of X (introduced in [Ve]), which is a family of holomorphic symplectic manifolds parametrized by C (see section 2.2 for details). Such families have been classically studied in the case when X is a K3 surface, see e.g. the discussion in [Hu2], where they are called Brauer families. In [Mar] these families are studied in the case when X is of K3 [n] -type.
Our main observation is the following. Given a hyperplane section D ⊂ S, we consider its open complement U ⊂ S. If X t is the fibre of the degenerate twistorial deformation of X over the point t ∈ C, it still admits a holomorphic Lagrangian fibration over S. Let X U,t be the preimage of U in X t . We prove in Theorem 3.1 that X U,t are isomorphic as complex manifolds for all t ∈ C. This has several implications.
First, we construct examples of complex manifolds that admit infinitely many structures of a quasi-projective variety that are pairwise non-birational, see Corollary 3.4. Examples of complex manifolds that admit several non-isomorphic algebraic structures are well known, see e.g. [Ha,Chapter VI,§3] and [Je]. One may also compare our examples with those arising from non-abelian Hodge theory of Hitchin-Simpson. Recall that de Rham and Betti moduli spaces are isomorphic as complex analytic varieties (the Riemann-Hilbert correspondence), but not as algebraic varieties, see e.g. [Si,Proposition 9].
Next, we observe that any fibre of a holomorphic Lagrangian fibration admits a Zariski-open neighbourhood that has a structure of a quasi-projective variety, see Corollary 3.5. In particular, any Lagrangian fibration is a Zariski-locally projective morphism. This strengthens a result of Campana [Ca] who proved analogous statement in the analytic topology. Using the same idea, we show in section 3.4 that any hyperkähler manifold admitting a Lagrangian fibration contains an open dense subset that is Stein. This is related to a question of Bogomolov about the existence of "Stein cells" in compact complex manifolds, see [B2]. We use quasi-projective neighbourhoods to generalize to the non-projective setting the results of Matsushita [Ma3] about higher direct images of the structure sheaf, see Corollary 3.10. This shows how to obtain Matshushita's theorem in full generality and in elementary way, without using more complicated techniques of Saito [Sa].
In section 3.6 we consider a holomorphic symplectic manifold that admits a bimeromorphic contraction such that the contraction centre is mapped to a point. We use Moser's isotopies to show that such contraction centres are rigid. More precisely, in a family of such contractions the contraction centres of nearby fibres are isomorphic, and moreover the isomorphisms are induced by symplectomorphisms of open neighbourhoods of the contraction centres, see Theorem 3.11. The Moser isotopy method which we describe can be considered as an analytic version of the work [KV], which deals with formal deformations of holomorphic symplectic manifolds, not necessarily compact. In [KV] it was shown that any deformation of a holomorphic symplectic manifold with H >0 (X, O X ) = 0 is trivial in the formal category, as long as the de Rham cohomology class of its holomorphic symplectic form does not change. This theorem, which has many useful applications, is in many aspects deficient. Indeed, a formal deformation of a Stein manifold or an affine manifold is always trivial. Using the Moser isotopy for C-symplectic structures, we can prove a local analytic version of this formal deformation theorem, Corollary 2.6.
2. C-symplectic structures 2.1. Definition and main properties. We recall the definition of a C-symplectic structure from [BDV].
(2) The kernel of Ω has rank 2n at all points of X. Let the subbundle T 0,1 X ⊂ T C X be the kernel of Ω and T 1,0 X its complex conjugate. Then T C X = T 1,0 X ⊕ T 0,1 X.
(3) The subbundle T 1,0 X is closed under the Lie bracket. This defines an integrable almost-complex structure on X. Therefore any manifold with a C-symplectic form admits an intrinsically defined complex structure. (4) The complex structure operator I ∈ End(T X) can be expressed as follows: Here we view the 2-forms ω j as morphisms T X → Λ 1 X defined by contracting ω j with the tangent vectors. (5) With respect to the above complex structure, the form Ω is holomorphically symplectic.
2.2. Lagrangian fibrations. We will be interested in certain families of C-symplectic structures that arise from holomorphic Lagrangian fibrations. Let X be a complex manifold of dimension 2n and S a normal complex analytic variety of dimension n. Let π : X → S be a proper surjective holomorphic morphism with connected fibres. Let S • ⊂ S be the maximal open subset over which the morphism π is smooth. Recall that by Sard's theorem S • is non-empty and dense in S. We denote by π • the restriction of π to X • = π −1 (S • ). Assume that Ω ∈ Λ 2,0 X is a holomorphic symplectic form. We will call π as above a holomorphic Lagrangian fibration if for any fibre F of π • we have Ω| F = 0. Since the fibres are n-dimensional, the latter condition means that F is Lagrangian. It is known that F is a complex torus, see e.g. [DM,Theorem 2.6].
Remark 2.2. The base S is smooth in all known examples of Lagrangian fibrations that are of interest for us. However this condition is not really necessary, so we do not include it in the definition. We will need to use C ∞ differential forms on S, and in the case of singular S we define them as in [Va].

Theorem 2.3 ([BDV]
). In the above setting, the 2-forms Ω t have the following properties.
(1) Ω t is a C-symplectic form on X for any t ∈ C.
(2) If I t denotes the complex structure corresponding to Ω t , then π is a holomorphic Lagrangian fibration from (X, I t ) to S for any t ∈ C.
Proof. Part (1). It is clear that Ω t is closed. We will show that Ω n+1 t = 0 and Ω n t ∧ Ω n t = Ω n ∧ Ω n for any t ∈ C. It is enough to check both equalities on the dense open subset X • .
Part (2). To prove that π is a holomorphic map from (X, I t ) to S it is again enough to restrict to the dense open subset X • . Using the splitting of the tangent space at arbitrary x ∈ X • as above, we write Since e l ∈ ker dπ, we see that dπ(T 0,1 It X) ⊂ T 0,1 S, hence dπ is complex-linear and π is holomorphic. To prove that it is a Lagrangian fibration note that for any fibre F ⊂ X • we clearly have Ω t | F = 0.
In the case when X is a compact hyperkähler manifold, it has been shown in [Ve] that the complex manifolds (X, I t ) form a smooth complex analytic family over the complex line C which we denote by X . This family is called degenerate twistorial deformation of X. The terminology can be explained as follows. Inside the period domain D the usual twistor conic can degenerate into the union of a pair of projective lines. Removing the point of intersection of those lines, we get a disjoint union of two affine lines. The periods of the fibres of the degenerate twistorial deformation form one of such affine lines. The two lines are exchanged by complex conjugation, i.e. one obtains the family parametrized by the second affine line if one replaces the holomorphic form Ω with Ω in the construction described above.

Moser's isotopy.
Recall the idea of Moser's isotopy theorem: given a family of real symplectic forms ω t parametrized by t ∈ [0, 1] such that the cohomology classes [ω t ] ∈ H 2 (X, R) do not depend on t, one constructs a flow of diffeomorphisms ϕ t such that ω 0 = ϕ * t ω t , see [Mo,Theorem 2]. Our aim is to prove a similar statement for C-symplectic forms. However, no strict analogy is apparent, because certain additional cohomological obstructions arise, and one needs to introduce some further conditions to construct the isotopy.
We work in the following setting: X is a manifold of real dimension 4n, Ω t ∈ Λ 2 C X is a family of C-symplectic forms for t ∈ [0, 1]. When talking about families of tensor fields, we always assume that the dependence on t is continuous and, if necessary, sufficiently differentiable. We denote by I t the complex structure corresponding to Ω t and by X t = (X, I t ) the corresponding complex manifold. If we use the (p, q)decomposition with respect to I t , the corresponding bundles have subscript I t , e.g. Λ p,q It X. We denote by L V the Lie derivative along a vector field V . Lemma 2.4. In the above setting assume that there exists a family of complex Applying the inverse of this isomorphism to −α t , we get the vector field V 1,0 t , and its real part is the desired real vector field.
In the setting of the above lemma, it remains to integrate the family of vector fields V t in order to obtain isotopies that deform Ω 0 into Ω t .
Theorem 2.5 (Moser's isotopy, version I). In the above setting, assume additionally that H 1 (X t , O Xt ) = 0 and [Ω t ] = [Ω 0 ] ∈ H 2 (X, C) for all t ∈ [0, 1]. In this case: (1) There exists a family of vector fields V t such that d (1) We need to find a family of 1-forms α t that satisfies the conditions of Lemma 2.4. By our assumption t + ∂∂f t = dβ t and α t is of type (1, 0) as required. By Lemma 2.4 we obtain a family of vector fields V t .
(2) Since X is compact, we can integrate V t to a flow of diffeomorphisms ϕ t , t ∈ [0, 1] with ϕ 0 = id. We compute: This result can be applied to compact hyperkähler manifolds, which satisfy H 1 (X, O X ) = 0, but in that case it easily follows from Bogomolov's version of Kuranishi-Kodaira-Spencer theory, [B1]. On non-compact manifolds, the assumption H 1 (X, O X ) = 0 is often quite restrictive. Moreover, on non-compact manifolds one needs some additional information to guarantee that the flow ϕ t exists globally on X. Without such information, integrating the vector field V t in a neighbourhood of a compact subset, one can deduce from Theorem 2.5 the following local statement.
Corollary 2.6. Let π : X → ∆ be a smooth family of holomorphic symplectic manifolds (not necessarily compact) over the unit disc, trivial as a family of C ∞ manifolds. Denote by X t = π −1 (t) a fibre, and by Ω t ∈ H 0 (X t , Ω 2 Xt ) its holomorphic symplectic form. Using the C ∞ trivialization to identify cohomology groups of the fibres, assume that the cohomology class of Ω t does not depend on t ∈ ∆, and H 1 (X t , O Xt ) = 0. Let K ⊂ X t0 be a compact subset. Then there exists an open neighbourhood U ⊂ ∆ of t 0 ∈ ∆, and an open subsetŨ ⊂ π −1 (U ), with K ⊂Ũ , with the following property. The setŨ is locally trivially fibred over U , with all fibresŨ ∩ π −1 (t), t ∈ U isomorphic as holomorphic symplectic manifolds.
It turns out that one can prove a useful version of Moser's theorem in the situation when the family Ω t comes from a Lagrangian fibration.
Theorem 2.7 (Moser's isotopy, version II). Let π : X → S be a holomorphic Lagrangian fibration in the sense of section 2.2. Denote by Ω the holomorphic symplectic form on X and assume that α ∈ Λ 1,0 S. Let Ω t = Ω + tπ * (dα) for t ∈ [0, 1] be the family of C-symplectic forms on X. Then there exists a family of diffeomorphisms ϕ t of X such that ϕ 0 = id and ϕ * t Ω t = Ω 0 for t ∈ [0, 1]. 1 Proof. As we recalled in section 2.2, the 2-forms Ω t are C-symplectic and π is a holomorphic Lagrangian fibration on (X, I t ). It follows that π * α is in Λ 1,0 It X for any t ∈ [0, 1]. Hence we can apply Lemma 2.4 with α t = π * α and obtain a family of vector fields V t .
We need to integrate V t to a flow of diffeomorphisms. Recall that by the construction in the proof of Lemma 2.4 we have V t Ω t = −α t . Let F be a smooth fibre of π and W ∈ T 1,0 F some locally defined vector field. Then Ω t (V t , W ) = −α t (W ) = −(π * α)(W ) = 0. Since F is Lagrangian and W arbitrary, the last equality implies that V t is tangent to F . So dπ(V t ) = 0 at all points of smooth fibres, hence everywhere on X, because smooth fibres are dense.
We conclude that integral curves of V t are contained in the fibres of π. Let now F be an arbitrary fibre. By our assumptions about Lagrangian fibrations, F is compact. This implies that any integral curve of V t that starts at a point of F can be extended for all t ∈ [0, 1]. Since this is true for all fibres, we get a well-defined flow of diffeomorphisms ϕ t , t ∈ [0, 1] and the rest of the proof goes as in Theorem 2.5.

Applications
We apply Theorem 2.7 to Lagrangian fibrations on compact hyperkähler manifolds of maximal holonomy, that is, compact simply connected Kähler manifolds X with H 0 (X, Ω 2 X ) spanned by a symplectic form Ω. Further on, we shall call these manifolds just "hyperkähler manifolds". We let 2n = dim C X. Assume that π : X → S is a holomorphic Lagrangian fibration in the sense explained in section 2.2. For the discussion of basic properties of such fibrations we refer to [Ma1], [Ma2] and [Hu1]. It is explained in [AC, footnote on page 53] that in this setting the base S is always projective.

A degenerate twistorial deformation is Zariski locally trivial over the base.
We fix a projective embedding ι : S ֒→ P m , let ω F S ∈ Λ 1,1 P m be the Fubini-Study form and η = ι * ω F S . Setting Ω t = Ω + tπ * η, t ∈ C, we get a family of C-symplectic structures that form a degenerate twistorial deformation of X. We use the notation X t = (X, I t ) as above.
Let H ⊂ P m be an arbitrary hyperplane and U = S \ H its open complement in S. We denote by X U,t the preimage of U in X t and X U = X U,0 .
Theorem 3.1. For any t 1 , t 2 ∈ C the complex manifolds X U,t1 and X U,t2 are isomorphic.
Proof. It is enough to prove that X U,t1 is isomorphic to X U,0 . For this we shall use Moser's isotopy theorem 2.7.
We may choose local coordinates (x 0 : . . . : x m ) on P m such that H is given by x 0 = 0. In the open affine chart x 0 = 0 the Fubini-Study form is given by Note that U is the intersection of P m \ H with S. We let α be the restriction of the 1-form −it 1 ∂ log(1 + m j=1 |x j | 2 ) to U . Then α ∈ Λ 1,0 U and dα = t 1 ω F S | U . We can apply Theorem 2.7, obtaining a flow of diffeomorphisms ϕ t , t ∈ [0, 1], such that ϕ * 1 Ω t1 | π −1 (U) = Ω| π −1 (U) . Then ϕ 1 defines an isomorphism of complex manifolds Remark 3.2. The restriction of the Fubini-Study form to S represents the cohomology class of a hyperplane section, so this restriction is not an exact form. Therefore to apply Moser's lemma it is necessary to have H = ∅ in the above theorem. This condition is not redundant, because the fibres X t are typically not pairwise isomorphic to each other.
Denote by Aut s (X/S) the following sheaf of abelian groups in the Zariski topology on S. For an open subset U ⊂ S the sections of Aut s (X/S) over U are complex analytic automorphisms of the complex manifold X U that commute with π and preserve the symplectic form Ω| XU . Since a general fibre of π is a complex torus, such automorphisms form an abelian group.

Corollary 3.3. For any t ∈ C the complex manifold X t can be obtained as the twist of X by a 1-cocycle with values in Aut s (X/S).
Proof. Keeping the notation as above, we cover S by m + 1 affine charts U i and denote by ϕ i : X Ui ∼ → X Ui,t the isomorphisms constructed in Theorem 3.1. Then define the 1-cocycle in question. Below we consider several applications of the above theorem.

A complex manifold with several algebraic structures.
Recall that small deformations of compact Kähler manifolds remain Kähler ( [KS]). This implies that in the above setting for t sufficiently close to zero the manifolds X t are hyperkähler. Using the projectivity criterion of Huybrechts ([Hu1,Proposition 26.13]), we can determine for which t the manifolds X t are projective. Assume that two projective fibres X t1 and X t2 are not birational. Note that X U,t1 and X U,t2 are both quasi-projective. Theorem 3.1 shows that X U,t1 and X U,t2 are isomorphic as complex manifolds. But they can not be isomorphic (or even birational) as algebraic varieties, otherwise X t1 and X t2 would be birational, contrary to our choice.
Starting from a K3 surface X one can produce a sufficiently general degenerate twistorial line in the moduli space of X, and make sure that infinitely many fibres X ti are projective and pairwise non-birational. This gives us the following statement.
Corollary 3.4. There exists a complex K3 surface X, a Zariski-open subset X ⊂ X and infinitely many complex quasi-projective varieties Y j that are pairwise nonbirational and such that Y an j are isomorphic to X as complex manifolds. Proof. For a complex K3 surface X, let V R = H 2 (X, R) for R = Z, Q, R and C. Let q be the intersection form on V Z . Let D = {x ∈ PV C | q(x) = 0, q(x,x) > 0} be the period domain. We assume that π : X → P 1 is a Lagrangian fibration and denote by ℓ ∈ V Z the primitive cohomology class [π * ω F S ].
Let W ⊂ V Q be a 3-dimensional subspace that contains ℓ, and denote by S the set of all such subspaces. This set is countable. Let D W = D ∩ P(W ⊥ ⊗ C) and D + W be the union of all degenerate twistorial lines in D ℓ that pass through the points of D W . Since q| W ⊥ is non-zero, the dimension of D W is 17, and the dimension of D + W is not bigger than 18. Hence D + W is of positive codimension in D ℓ , and D • ℓ = D ℓ \ ∪ W ∈S D + W is non-empty. Deforming X we may assume that its period p ∈ D • ℓ .
Step 2. We consider the degenerate twistorial line through [Ω] and ℓ. Cohomology classes of C-symplectic forms on this line are of the form [Ω t ] = [Ω] + tℓ. Recall that V Z is the even unimodular lattice of signature (3,19). Denote by H(d) the rank two hyperbolic plane with intersection form multiplied by d. By [PS, Appendix to §6, Theorem 1], there exists a primitive embedding of lattices H(d) ֒→ V Z , unique up to an automorphism of V Z . Conjugating such an embedding by an automorphism of V Z , we find for arbitrary d a primitive sublattice Step 3. We consider the fibres X t d of the degenerate twistorial family. These fibres are complex K3 surfaces, in particular they are Kähler. By our construction, NS(X t d ) contains Λ d . The rank of NS(X t d ) cannot be bigger than 2 by our choice of p made in Step 1. Hence NS(X t d ) ≃ Λ d . Since Λ d contains positive elements, X t d are projective. The discriminants of Λ d are different for different d, hence X t d are pairwise non-birational. Now it remains to set X = X U,0 , Y j = X U,tj , where U ≃ A 1 ⊂ P 1 , and apply Theorem 3.1.

Local projectivity of Lagrangian fibrations.
We will sharpen the result of Campana [Ca] about local projectivity of Lagrangian fibrations. Let π : X → S be a Lagrangian fibration with X compact hyperkähler and S projective. It was shown in [Ca] that any fibre F of π admits an open analytic neighbourhood V, such that π| V is a projective morphism.
Corollary 3.5. In the above setting, F admits a Zariski-open neighbourhood V that is a quasi-projective variety and such that π| V is a projective morphism.
Proof. Since S is projective, we can find a Kähler form η ∈ Λ 1,1 S whose cohomology class is rational. Let ℓ = [π * η] ∈ H 2 (X, Q). As before, we denote by Ω the holomorphic symplectic form on X.
Consider the degenerate twistorial deformation of X determined by η. Using the density of H 2 (X, Q) in H 2 (X, R), we can find a class x ∈ H 2 (X, Q) with q(x) > 0 and t = −q([Ω], x)/q(ℓ, x) arbitrary close to zero. Then for the fibre X t of the degenerate twistorial family we have [Ω t ] = [Ω] + tℓ, x ∈ [Ω t ] ⊥ , hence NS(X t ) contains an element with positive BBF square. Since |t| is small, X t is Kähler, and hence projective by Huybrechts's criterion [Hu1,Proposition 26.13]. Pick a hyperplane section H of S that does not contain π(F ), let U = S \ H and V = π −1 (U ). Then apply Theorem 3.1.
Remark 3.6. This corollary gives, in particular, an alternative proof of the well known fact that all fibres of π are projective varieties.
Remark 3.7. If X is not projective, it can not be Moishezon, because being Kähler and Moishezon implies being projective. Using the open subsets V from Corollary 3.5, we get an example of a compact complex manifold that admits a covering by Zariski-open quasi-projective subsets, but is not Moishezon.

Stein cells in hyperkähler manifolds with Lagrangian fibrations.
It was asked by Bogomolov (see [B2]) whether any compact complex manifold X admits a "Stein cell". By this one means an open subset V ⊂ X that is a Stein manifold, and such that the complement X \V is "small". Ideally the complement of a Stein cell should be a finite CW complex whose dimension is strictly smaller than the dimension of X. If X is projective, one can take for V the complement of an ample divisor, which is an affine variety, hence Stein. For non-algebraic varieties the question is more subtle, and no general construction is known, even conjecturally. The condition on the complement of a Stein cell stated above is quite restrictive, and it is debatable how realistic it is. We observe here that if one weakens this condition, then Theorem 3.1 provides a way to construct Stein cells in arbitrary (possibly non-algebraic) hyperkähler manifolds admitting a Lagrangian fibration.
Corollary 3.8. Let π : X → S be a Lagrangian fibration with X compact hyperkähler and S projective. There exists an open subset V ⊂ X that is Stein and dense in the analytic topology.
Proof. Consider the degenerate twistorial deformation of X as in the proof of Corollary 3.5 and let X t be a projective fibre. Let H ⊂ S be a hyperplane section, U = S \ H and ϕ : X U → X U,t the isomorphism of complex manifolds constructed in Theorem 3.1. Let Z = π −1 (H) ⊂ X t . For any ample divisor A ⊂ X t the divisor mA + Z is ample for m big enough.
Remark 3.9. In the notation of the proof above, note that the closure of ϕ −1 (A ∩ X U,t ) ⊂ X U may not be a complex-analytic subset of X. So the complement of V constructed above may be more complicated than a CW complex.
3.5. Higher direct images of the structure sheaf. Let π : X → S be a Lagrangian fibration with X compact hyperkähler and S a smooth projective variety. In the case when X is projective, Matsushita has shown in [Ma3] that R j π * O X ≃ Ω j S . One can deduce the same result for non-projective X using the theory of Saito [Sa]. We show how to deduce the non-projective case directly from [Ma3] without using the complicated techniques from [Sa].
Corollary 3.10. In the above setting R j π * O X ≃ Ω j S for all j. Proof. We let η and ℓ be as in the proof of Corollary 3.5, and we pick a projective fibre X t in the degenerate twistorial deformation of X as in that proof. We let ω ∈ Λ 1,1 X be a Kähler form and L ∈ Pic(X t ) be an ample line bundle.
Let π • : X • → S • be the smooth part of the fibration π, as in section 2.2. We recall from [Ma3, section 2] the construction of the isomorphism → Ω 1 X • induced by the symplectic form Ω. Compose this isomorphism with the natural surjection Ω 1 X • → Ω 1 X • /S • and note that this composition vanishes on the subbundle T X • /S • , because the fibres of π • are Lagrangian. It therefore induces an isomorphism (π , and the bundle in the middle carries a polarized variation of Hodge structures, the polarization being induced by the restriction of the Kähler class [ω] to the fibres of π • . Using the polarization, we get an isomorphism From the construction we see that the isomorphism (3.1) depends on the class of the polarization [ω] ∈ H 2 (X, R). Recall, however, that for any fibre F of π • the restriction map H 2 (X, R) → H 2 (F, R) has rank one and its kernel is the orthogonal complement of ℓ, see [Ma4,Lemma 2.2]. The polarization on the fibres therefore depends only on the value of q(ℓ, [ω]), and not on [ω] itself. We can rescale ω so that q(ℓ, [ω]) = q(ℓ, c 1 (L)), where L is the ample bundle on X t chosen above. Then [ω] and c 1 (L) induce the same polarization on the VHS (R 1 π • * C) ⊗ O S • and thus the same isomorphism (3.1).
We need to show that (3.1) extends over the discriminant locus of π. We choose an arbitrary hyperplane section of S in some projective embedding and denote by U ⊂ S its open complement. We let U • = U ∩ S • and X U = π −1 (U ). Theorem 3.1 implies that the complex manifolds X U and X U,t are isomorphic. Applying the results of [Ma3] to X t and using the line bundle L to define the polarization, we get an isomorphism R 1 π * O XU ≃ R 1 π * O XU,t ≃ Ω 1 U . When restricted to U • , this isomorphism coincides with the restriction of (3.1), because of the remark about polarizations in the previous paragraph. Therefore we can extend (3.1) to an isomorphism over U . Since U was arbitrary, we conclude that (3.1) extends to an isomorphism R 1 π * O X ≃ Ω 1 S . To finish the proof, we use the isomorphisms Λ j (R 1 π * O X ) ≃ R j π * O X from [Ma3]. These isomorphisms may be checked locally on S, so we again replace X U by X U,t as above and apply the results of [Ma3].
3.6. Deformation invariance of contraction centres. Consider a smooth family π : X → ∆ of holomorphic symplectic manifolds over the unit disc. Assume that Z ⊂ X is a subvariety that is flat over ∆ and such that π| Z is proper, i.e. the fibres Z t are compact subvarieties of X t . Assume that for every t ∈ ∆ the subvariety Z t can be contracted to a point. More precisely, there exists a flat family π ′ : Y → ∆ and a proper morphism ρ : X → Y such that π ′ • ρ = π and ρ t : X t → Y t is a bimeromorphic morphism with exceptional set Z t , and such that ρ t (Z t ) is a point in Y t . We call Z t the contraction centre of ρ t .
We would like to use Moser's isotopy to show that all Z t are isomorphic, the isomorphisms being induced by symplectomorphisms of some open neighbourhoods of Z t . Every Z t is a deformation retract of its open neighbourhood in X t , see e.g. [Qu,Proposition 3.5]. By shrinking X we may assume that every fibre Z t is a deformation retract of X t . Moreover, since every point in Y t admits a Stein neighbourhood, by shrinking Y we may assume that all Y t are Stein spaces. Therefore we are reduced to the following statement.
Theorem 3.11. Let X be a manifold with a family of C-symplectic structures Ω t ∈ Λ 2 C X, t ∈ [0, 1] and corresponding complex structures I t . Denote X t = (X, I t ) and assume that Z t ⊂ X t is a family of compact complex subvarieties such that: (1) Z t is a deformation retract of X; (2) X t admits a bimeromorphic morphism ρ t onto a Stein space Y t such that the exceptional set of ρ t is Z t .
Proof. Since X t are holomorphic symplectic manifolds, Y t are holomorphic symplectic varieties in the sense of Beauville [Be]. It is known [Be,Proposition 1.3] that the singularities of Y t are rational, and since Y t are Stein, H j (X t , O Xt ) = 0 for all j > 0. As before, we denote by [Ω t ] the cohomology class of Ω t in H 2 (X, C). We know from [Ka,Corollary 2.8] that [Ω t ]| Zt = 0. Since Z t is a deformation retract of X, we have [Ω t ] = 0 for all t. Hence we can apply part (1) of Theorem 2.5 and obtain a family of vector fields V t .
Since X is not compact, it is not possible, in general, to integrate V t to a flow of diffeomorphisms. However, since Z 0 is compact, we can integrate V t in an open neighbourhood U 0 of Z 0 for small t. More precisely, there exist diffeomorphisms ϕ t of U 0 onto some open subsets of X integrating V t and defined for t sufficiently close to zero. Note that ϕ * t Ω t = Ω 0 | U0 and ϕ t (Z 0 ) is a compact complex subvariety of X t . Since X t is contractible onto a Stein space Y t with contraction centre Z t , the only positive dimensional compact subvariety of X t is Z t . Hence ϕ t (Z 0 ) = Z t , and since all Z t are compact, we can extend the flow for all t ∈ [0, 1], after possibly shrinking U 0 . Then we define U 1 = ϕ 1 (U 0 ) and ϕ = ϕ 1 .