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SHAFAREVICH–TATE GROUPS OF HOLOMORPHIC LAGRANGIAN

FIBRATIONS

ANNA ABASHEVA, VASILY ROGOV

Abstract. Consider a Lagrangian fibration π : X Ñ Pn on a hyperkähler manifold X .There are two ways to construct a holomorphic family of deformations of π over C.The first one is known under the name Shafarevich–Tate family while the second oneis the degenerate twistor family constructed by Verbitsky. We show that both familiescoincide. We prove that for a very general X all members of the Shafarevich–Tatefamily are Kähler. There is a related notion of the Shafarevich–Tate group X associatedto a Lagrangian fibration. The connected component of unity of X can be shown tobe isomorphic to CΛ where Λ is a finitely generated subgroup of C and C is thoughtof as the base of the Shafarevich–Tate family. We show that for a very general X ,projective deformations in the Shafarevich–Tate family correspond to the torsion pointsin X

0. A sufficient condition for a Lagrangian fibration X to be projective is existenceof a holomorphic section. We find sufficient cohomological conditions for existence of adeformation in the Shafarevich–Tate family that admits a section.

Contents

1. Introduction 12. Preliminaries 42.1. Lagrangian fibrations 42.2. Shafarevich–Tate groups 52.3. Degenerate twistor deformations 63. Shafarevich–Tate group of a Lagrangian fibration 73.1. Structure of Shafarevich–Tate groups: first steps 73.2. Shafarevich–Tate family 83.3. Shafarevich–Tate twists are symplectic 93.4. Degenerate twistor deformations as Shafarevich–Tate twists 103.5. The period map of a Shafarevich–Tate family 114. Connected component of unity of a Shafarevich–Tate group 124.1. The sheaf Γ 124.2. The connected component of unity of Shafarevich–Tate groups 135. Kählerness and projectivity properties 155.1. M-special Hodge structures 165.2. Kählerness of degenerate twistor deformations 175.3. Algebraic points in Shafarevich–Tate families 186. Sections of Lagrangian fibrations 206.1. Obstruction for existence of a section 206.2. Hard Lefschetz type theorems for higher direct images of QX 226.3. The discrete part of Shafarevich–Tate groups 24Bibliography 25

1. Introduction

A compact connected Kähler manifold X is called a hyperkähler manifold if it is sim-ply connected and H0pX,Ω2

Xq is generated by a holomorphic symplectic form σ. For aLagrangian fibration π : X Ñ Pn we can associate a certain group called the Shafarevich–Tate group.

1

http://arxiv.org/abs/2112.10921v2

2 ANNA ABASHEVA, VASILY ROGOV

Definition. The Shafarevich–Tate group1X of a Lagrangian fibration is the abelian

group H1pPn, Aut0XPnq where Aut0XPn is the connected component of unity of the sheafof vertical automorphisms of X over Pn.

Choose an affine open cover Pn “ ŤUi. Denote Ui X Uj by Uij. A class s P X can be

represented by a Čech cocycle with coefficients in Aut0XPn. That is to say, for every pairof indices i, j we are given an automorphism sij of π´1pUijq that commutes with π. Let usreglue the manifolds π´1pUiq by the automorphisms sij . We obtain a complex manifoldXs

equipped with a holomorphic projection πs : Xs Ñ Pn. The fibers of πs are isomorphic tothe fibers of π : X Ñ Pn. This new manifold is called the Shafarevich–Tate twist of X bys P X. Markman studied Shafarefich-Tate twists of Lagrangian fibrations on manifoldsof K3rns-type in [Markman]. Our work was notably influenced by his paper.

Define the sheaf Γ of finitely generated abelian groups by the exact sequence

0 Ñ Γ Ñ π˚TXPn Ñ Aut0XPn Ñ 0

This exact sequence induces the long exact sequence of cohomology groups. It can beproven to be right exact.

H1pPn,Γq Ñ H1pπ˚TXPnq Ñ X Ñ H2pPn,Γq Ñ 0

The holomorphic symplectic form σ onX induces an isomorphism π˚TXPn – Ω1

Pn [deCRS,Lemma 2.3.1]. Thus, H1pπ˚TXPnq – H1,1pPnq “ C. Let X0 be the image of H1pπ˚TXPnqin X. We conclude that X

0 is isomorphic to a quotient of C by the finitely generatedabelian group impH1pPn,Γqq.

In [Ver15] the author associates to a Lagrangian fibration π : X Ñ Pn a family ofdeformations over C called the degenerate twistor family. The construction goes as follows.Let α be a closed p1, 1q-form on Pn. There exists a unique complex structure Iα on X

such that the form ω ` π˚α is a holomorphic 2-form on pX, Iαq [Ver15, Thm. 3.5]. Themanifold pX, Iαq is the degenerate twistor deformation of X with respect to the form α.Note that the construction of degenerate twistor deformations has differential geometricflavour while the construction of Shafarevich–Tate twists is of complex analytic nature.Nevertheless, these two constructions turn out to yield the same result.

Theorem A (Theorem 3.8). Pick a class s P H1pπ˚TXPnq. Consider the twist Xs ofπ : X Ñ Pn by the image of s in X. Let α be a closed p1, 1q-form on Pn representing thesame class in H1,1pPnq – H1pπ˚TXPnq as s. Then the complex manifolds Xs and pX, Iαqare isomorphic as fibrations over Pn.

Are Shafarevich–Tate twists of a hyperkähler manifold hyperkähler themselves? Weprove that indeed, they admit a holomorphic symplectic form (Subsection 3.3) but thequestion of Kählerness turns out to be trickier. We need to introduce a notion of M-specialhyperkähler manifolds which is motivated by [Markman, Def. 1]. A hyperkähler manifoldX is called M-special if the subspace H2,0pXq ` H0,2pXq Ă H2pX,Cq contains a rationalclass. A very general hyperkähler manifold is not M-special.

Theorem B (Theorems 5.11, 5.20). Let π : X Ñ Pn be a Lagrangian fibration on a notM-special projective hyperkähler manifold X. Then every Shafarevich–Tate twist Xs ofX by an element s P X

0 is a hyperkähler manifold. Moreover, the set of s P X0 such

that the twist Xs is a projective manifold forms a non-empty torsor over the group oftorsion points of X0.

1It is sometimes called analytic Shafarevich–Tate group in the literature in order to distinguish it fromarithmetic Shafarevich–Tate group.

SHAFAREVICH–TATE GROUPS OF HOLOMORPHIC LAGRANGIAN FIBRATIONS 3

In order to prove Theorem B, we first study the sheaf Γ. It turns out that Γ b Q isisomorphic to R1π˚Q (Proposition 4.4). The Leray spectral sequence helps us to computesome cohomology groups of R1π˚Q. We derive from that a description of the group X

0

solely in terms of the Hodge structure on H2pX,Zq (Corollary 4.8). As we have alreadyseen, the group X

0 is isomorphic to CΛ for a finitely generated abelian group Λ. Itturns out that X is not M-special if and only if Λ is a dense subgroup of C. We use thatthe set of twists of X that are Kähler manifolds is open and Λ-invariant to conclude thatall Shafarevich–Tate twists of X are Kähler. The statement about projectivity follows bycarefully applying Huybrechts’ criterion [Huy01, Thm. 3.11].

In Section 6 we study whether there exists a Shafarevich–Tate twist Xs of π : X Ñ Pn

that admits a holomorphic section. We show that obstructions to existence of such atwist lie in H2pPn,Γq “ XX0 (Corollary 6.2). Assume that the fibers of π are reducedand irreducible. We show that the group H2pPn,Γq b Q can be embedded into H3pX,Qq(Corollary 6.9). This implies the following theorem.

Theorem C (Theorem 6.3). Let π : X Ñ Pn be a Lagrangian fibration on a compacthyperkähler manifold over a smooth base. Assume that the following holds:

‚ the fibers of π are reduced and irreducible;‚ H3pX,Qq “ 0;‚ H2pPn,Γq is torsion-free.

Then there exists a unique s P X0 such that πs : Xs Ñ Pn admits a holomorphic section.

In particular, the fibration π can be deformed to a fibration with a holomorphic sectionthrough the Shafarevich–Tate family.

At the core of the proof of Theorem C is a Hard Lefschetz-type result for fibers ofLagrangian fibrations. We believe that it may be of independent interest and state ithere.

Theorem D (Corollary 6.6). Let π : X Ñ Pn be a Lagrangian fibration on a projectivehyperkähler manifoldX. Then there exists a sheaf N on Pn such that the sheaf R2n´1π˚QX

decomposes into the direct sum

R2n´1π˚QX » R1π˚QX ‘ N.

Theorem D allows us to prove that the differential d2 : H0pR2π˚Qq Ñ H2pR1π˚Qq inthe Leray spectral sequence for QX vanishes. The arguement is inspired by the one usedby Deligne to prove that the Leray spectral sequence of a smooth projective submersiondegenerates on the second page [Del].

The paper is organized as follows. The notions of a Shafarevich–Tate group and adegenerate twistor deformation, as well as basic facts about Lagrangian deformations arerecalled in Section 2. The main result of Section 3 is a refined version of Theorem A. Wealso prove that Shafarevich–Tate twists are holomorphic symplectic in Subsection 3.3 andstudy the period map for the family of Shafarevich–Tate twists in Subsection 3.5. Section4 is concerned with the description of the group X

0 in terms of topological invariantsof X. Its results are used in Section 5 to prove Theorem B. In Section 6 we discussobstructions to existence of sections and prove Theorem C. Theorem D is proved there asan intermediate result.

Acknowledgments. We are deeply grateful to Misha Verbitsky, Dmitry Kaledin,Rodion Déev, and Andrey Soldatenkov for stimulating conversations and encouragement.A.A. thanks Giulia Saccà for her interest and for pointing out to us the papers [deCRS]and [Markush96]. A.A. is also extremely grateful to Raymond Cheng for reading a draft


of this paper. A.A. very much appreciates his reasonable and helpful suggestions. V.R.thanks Olivier Benoist for his insights about the content of Section 6. The work on thisproject started during our stay at the summer school ”Algebra & Geometry–2021” inYaroslavl, Russia organized by the Laboratory of Algebraic Geometry of Higher School ofEconomics and Yaroslavl State Pedagogical University. We are grateful to the organizersfor their hospitality and inspiring atmosphere. A.A. was supported in part by SimonsFoundation.

2. Preliminaries

2.1. Lagrangian fibrations. Let X be a hyperkähler manifold. Fix a holomorphic sym-plectic form σ on X. Consider a map π : X Ñ B to a normal base B of dimension halfof that of X. A map π : X Ñ B is called a Lagrangian fibration if it is surjective withconnected fibers and the restriction of σ to every smooth fiber vanishes. We always as-sume that the base B of a Lagrangian fibration π is isomorphic to Pn. It is the case in allknown examples (see e.g. [HM, Subection 1.2]). We refer the reader to [Huy01] for basicson hyperkähler manifolds and Lagrangian fibrations.

Throughout this paper, we write X for a compact hyperkähler manifold, σ for a holo-morphic symplectic form on X and π : X Ñ B for a Lagrangian fibration. For b P B wedenote the fiber of π over b by Fb. The subset of the regular values of π is denoted by B˝.This is a Zariski open subset of B and D :“ B zB˝ is known to be a divisor [HO, Prop.3.1].

Smooth fibers of a Lagrangian fibration π : X Ñ B are complex tori [Markush86]. Onecan show that they are abelian varieties even if X is not projective [Cam].

Let X be a hyperkähler manifold. Then H2pX,Zq carries a non-degenerate symmetricbilinear form q : H2pX,Zq b H2pX,Zq Ñ Z called the Beauville–Bogomolov–Fujiki form[Beauv, Fu85]. It satisfies the following properties:

(1) The Hodge decomposition

H2pX,Cq “ H2,0pXq ‘ H1,1pXq ‘ H0,2pXqis orthogonal with respect to q. The restriction of q to H2,0pXq ‘ H0,2pXq isa positive Hermitian form and the restriction of q to H1,1pXq has the signaturep1, h1,1pXq ´ 1q;

(2) (Fujiki formula) There exists a non-zero constant cX such that

qpω, ωqn “ cX

ż

X

ω2n

Let η “ π˚rHs P H2pX,Zq be the pullback of the class of a hyperplane. It follows fromthe Fujiki formula that qpη, ηq “ 0.

Proposition 2.1. ([O, Section 2]) Let π : X Ñ B be a Lagrangian fibration, η P H2pX,Zqas above. Then the restriction map H2pX,Qq Ñ H2pFb,Qq has rank 1 for every smoothfiber Fb “ π´1pbq Ă X. Moreover, the kernel of this map is precisely

ηK :“ tv P H2pX,Qq |qpv, ηq “ 0u.Corollary 2.2. With the assumptions of Proposition 2.1, there is a Kähler class rωs onX that restricts to an integral class on every smooth fiber Fb.

Proof: By Proposition 2.1 the restriction map H2pX,Qq Ñ H2pFb,Qq, for any b P B˝,factors through the one-dimensional quotient H2pX,QqηK. This vector space is generatedby an integral class. Every Kähler class rωs on X restricts to a non-zero class on a fiber.Therefore, the class rωs restricts to an integral class after appropriate scaling.


Lemma 2.3. ([deCRS, Lemma 2.3.1]) Let π : X Ñ B be a Lagrangian fibration. Considerthe isomorphism σ : Ω1

X

„ÝÑ TX induced by the holomorphic symplectic form σ. Then themap σ sends π˚Ω1

B Ă Ω1

X isomorphically to TXB :“ kerpdπ : TX Ñ π˚TBq.It follows that the sheaves Ω1

B and π˚TXB are isomorphic. In particular, the sheafπ˚TXB is locally free.

Assume that X is projective. Let ω be a closed p1, 1q-form on X that represents arational class rωs P H1,1

Q pXq. The contraction of ω with holomorphic vector fields definesa map

ω : π˚TXB Ñ R1π˚OX .

We get the following maps by taking the exterior powers of ω:

rωi : ΩiB – Λipπ˚TXBq Ñ ΛiR1π˚OX Ñ Riπ˚OX , i “ 1, . . . , n.

Proposition 2.4. ([Mats, Thm. 1.2]) Let π : X Ñ B be a Lagrangian fibration on aprojective hyperkähler manifold X over a smooth base B. Then the map

rωi : ΩiB Ñ Riπ˚OX .

is an isomorphism for every i “ 1, . . . , n. In particular, the sheaves Riπ˚OX are locallyfree.

Remark 2.5. The assumption of projectivity of X can be dropped, as was shown in [SV].Let rωs be a Kähler class on X as in Corollary 2.2. Then the restriction of the isomor-

phism π˚TXB Ñ R1π˚OX to B˝ is given by the contraction of ω with holomorphic vectorfields [SV, Cor. 3.7].

Corollary 2.6. The following cohomology groups are isomorphic:

H1pB, π˚TXBq » H1pB,Ω1

Bq » H1pB,R1π˚OXq » C.

Proof: The sheaves π˚TXB, Ω1

B, and R1π˚OX are isomorphic by Lemma 2.3 and Propo-sition 2.4. The last isomorphism holds since h1,1pPnq “ 1.

2.2. Shafarevich–Tate groups. Let π : X Ñ B be a surjective holomorphic map ofcomplex manifolds. Consider the sheaf AutXB of vertical automorphisms of X over B.This is a sheaf of groups on B defined as

AutXBpUq :“ tϕ : π´1pUq Ñ π´1pUq | π ˝ ϕ “ πu.Let us define the subsheaf Aut0XB Ă AutXB as follows. It consists of those verti-cal automorphisms ϕ whose restriction to every fiber Fb lies in Aut0pFbq. Pick a classs P H1pB,Aut0XBq. We can represent it by a Čech cocycle sij for an open cover B “ Ť

Ui.We are given an automorphism sij : π

´1pUijq Ñ π´1pUijq for each pairwise intersectionUij :“ Ui X Uj . Let us glue a new complex manifold as

Xs :“ğ

π´1pUiqO

x P π´1pUiq „ sijpxq P π´1pUjq.

The manifold Xs is equipped with a natural fibration πs : Xs Ñ B. The isomorphismclass of the fibration πs : Xs Ñ B depends only on the class of s in H1pB,Aut0XBq. Themanifold Xs is called the twist of X by the class s P H1pB,Aut0XBq.

Suppose from now that a general fiber of π is a complex torus. Let B˝ Ă B be the setof values of π for which the fiber π´1pbq is a complex torus.

Lemma 2.7. Let π : X Ñ B be as above. Then the sheaf Aut0XB is a sheaf of commutativegroups.


Proof: The sheaf Aut0XB is a sheaf of commutative groups at least over B˝ Ă B. Indeed,for any point b P B˝ the group Aut0pFbq is a complex torus. In particular, Aut0pFbq iscommutative. Let g, h P Aut0XBpUq be two local sections of Aut0XB. Then rg, hs|B˝XU istrivial. Therefore, rg, hs is itself trivial, because B˝ X U is dense in U .

Definition 2.8. ([FM, Ch. I, Section 1.5.1]) The group H1pB,Aut0XBq is called theShafarevich–Tate group of the fibration π : X Ñ B and is denoted by X.

See [DG, K, Markman] for applications of Shafarevich–Tate groups in study of ellipticand Lagrangian fibrations.

2.3. Degenerate twistor deformations. The main references for this Subsection are[Ver15, BDV, SV]. Let X be a hyperkähler manifold. Fix a holomorphic symplectic formσ P H0pX,Ω2

Xq. The key observation here is that the complex structure ofX is determinedby the form σ, viewed as a smooth C-valued 2-form on X. Namely, the complex structureis uniquely determined by the subbundle of p0, 1q-vectors T 0,1X Ă TX. This subbundlecan be recovered as ker σ|TXbC.

We can characterize all complex-valued smooth 2-forms on X that can be realized asholomorphic symplectic forms for some complex structures.

Definition 2.9. ([BDV]) Let M be a smooth manifold of real dimension 4n. Consider asmooth complex-valued 2-form σ P ΓpΛ2T ˚M b Cq. It is called a c-symplectic structureif the following hold:

‚ dσ “ 0;‚ σn`1 “ 0;‚ σn ^ σn ‰ 0 at each point of M .

Proposition 2.10. ([Ver15, Prop. 3.1]) Let σ be a c-symplectic structure on M . Define

T 0,1M :“ ker σ Ă TM b C

and T 1,0M :“ T 0,1M . Then the following holds:

‚ TM “ T 0,1M ‘ T 1,0M ;‚ rT 0,1M,T 0,1Ms Ď T 0,1M .

Equivalently, the operator Iσ : TM Ñ TM that acts as?

´1 on T 1,0M and as ´?

´1 onT 0,1M is an integrable complex structure on M . The form σ is holomorphic symplecticwith respect to Iσ.

Lemma 2.11. ([Ver15, Thm. 1.10]) Let M be a complex manifold (not necessarily com-pact) and σ a holomorphic symplectic 2-form on M . Consider a proper Lagrangian fibra-tion π : M Ñ S over a complex manifold S. Fix a closed 2-form α on S of Hodge typep2, 0q ` p1, 1q. Then the form σα :“ σ ` π˚α is a c-symplectic structure. Moreover, theprojection π : M Ñ S is holomorphic with respect to the complex structure Iα :“ Iσα

. Itis a Lagrangian fibration with respect to the holomorphic symplectic form σα.

Suppose that the base S of a Lagrangian fibration satisfies the condition H1pS,OSq “ 0.The next two statements will show that the isomorphism class of the complex manifold(M , Iα) depends only on the cohomology class of α.

Lemma 2.12. ([SV, Thm. 2.7]) Let pM,σq be a holomorphic symplectic manifold andπ : M Ñ S a proper Lagrangian fibration. Assume that H1pS,OSq “ 0. Let α be an exact2-form of type p2, 0q ` p1, 1q on S. Consider a family of c-symplectic structures

σt :“ σ ` tπ˚α.


Then there exists a flow of diffeomorphisms ϕt on M preserving the fibers of π such thatfor each t

ϕt : pM, I0q Ñ pM, Itqis a biholomorphism.

Corollary 2.13. Let π : M Ñ S be as in Lemma 2.12. Consider two cohomologous 2-forms α and α1 of Hodge type p2, 0q ` p1, 1q on S. Let I and I 1 be the complex structureson M induced by the c-symplectic forms σ ` π˚α and σ ` π˚α1 respectively. Then thereexists a biholomorphism ϕ : pM, Iq Ñ pM, I 1q preserving the fibers of π.

Proof: Follows by applying Lemma 2.12 to the holomorphic symplectic manifold pM, Iqequipped with the holomorphic symplectic form σα :“ σ ` π˚α. Indeed, in this case theform σ ` π˚α1 can be written as σα ` π˚dβ for some 1-form β.

Definition 2.14. ([Ver15, Def. 3.17]) Let π : X Ñ B be a Lagrangian fibration and α aclosed 2-form on B of type p2, 0q ` p1, 1q. Consider the family of c-symplectic structuresσt :“ σ ` tπ˚α, t P C. Let It be the associated family of complex structures on X.The degenerate twistor family Xdeg.tw of pX, πq is the following manifold. Its underlyingsmooth manifold is X ˆ C and the almost complex structure is defined as

pItwqpx,tq :“ It ‘ IC,

where x P X, t P C, and IC is the standard complex structure on C.

One can show that the almost complex structure Itw from Definition 2.14 is integrable[Ver15, Thm. 3.18]. In other words, Xdeg.tw is a complex manifold. The projectionXdeg.tw Ñ C is holomorphic and Xdeg.tw is the total space of a holomorphic family ofcomplex holomorphically symplectic manifolds. It is endowed with a holomorphic fibrationΠdeg.tw : Xdeg.tw Ñ B ˆ C that restricts to a fiber Xt Ă Xdeg.tw as π.

3. Shafarevich–Tate group of a Lagrangian fibration

3.1. Structure of Shafarevich–Tate groups: first steps. Let X be a compact hy-perkähler manifold and π : X Ñ B a Lagrangian fibration, B » Pn. Recall that theShafarevich–Tate group X of π is defined to be H1pB,Aut0XBq (Definition 2.8).

Consider the exponential map π˚TXB Ñ Aut0XB. Define the sheaf Γ by the short exactsequence

0 Ñ Γ Ñ π˚TXB Ñ Aut0XB Ñ 0.

It induces the following long exact sequence:

(1) H1pB,Γq Ñ H1pB, π˚TXBq Ñ X Ñ H2pB,Γq Ñ 0.

Indeed, the sheaf π˚TXB is isomorphic to Ω1

B by Lemma 2.3. Hence,

H2pB, π˚TXBq “ H1,2pPnq “ 0.

WriteĂX “ ĂXpX, πq :“ H1pB, π˚TXBq.

Of course, the group ĂXpX, πq is (non-canonically) isomorphic to C (Corollary 2.6). How-ever, we will use this notation when we want to emphasize its relation to X.

Lemma 3.1. The sheaf Γ :“ kerpπ˚TXB Ñ Aut0XBq is a sheaf of finitely generatedtorsion-free abelian groups.


Proof: The restriction of the sheaf Γ to B˝ is a local system of torsion-free abeliangroups of rank n “ 1

2dimX. For every open U Ă B the restriction map

(2) H0pU,Γq Ñ H0pU X B˝,Γqis injective. Indeed, an element of H0pU,Γq is a vector field on π´1pUq. If a vector fieldvanishes on an open subset of π´1pUq, then it vanishes on π´1pUq. The group on theright-hand side of (2) is torsion-free of finite rank, hence so is the group on the left-handside.

Lemma 3.1 implies that the cohomology groups of Γ are finitely generated abeliangroups. Let X

0 denote the image of the map ĂX “ H1pB, π˚TXBq Ñ X from the longexact sequence (1). The group X fits into the short exact sequence

(3) 0 Ñ X0 Ñ X Ñ H2pB,Γq Ñ 0.

Being a quotient of ĂX “ C by a subgroup, the group X0 inherits a structure of a

connected topological group. Endow X with the translation invariant topology such thatX

0 is the connected component of unity of X. View H2pB,Γq as a discrete topologicalgroup. Both maps in the exact sequence (3) are continuous maps of topological groups.

The following provides useful intuition for Shafarevich–Tate groups.

Remark 3.2. Let B be a complex manifold. Let Y :“ B ˆ Cˆ. Let Aut0Y B Ă AutY B

be the subsheaf consisting of the automorphisms that lie in the connected component ofunity. Then Aut0Y B “ O

ˆB and H1pB,Aut0Y Bq “ PicpBq. Total spaces of twists by classes

in H1pB,Aut0XBq are holomorphic principal Cˆ-bundles over X. They are in one-to-onecorrespondence with holomorphic line bundles.

Thus, the Shafarevich–Tate group X serves as an analog of the Picard group, itssubgroup X

0 is an analog of Pic0pBq, and XX0 may be thought of as the Néron–Severi group NSpBq.

3.2. Shafarevich–Tate family. Consider an element s P ĂX. The twist Xs of X by s

is defined to be the manifold X rss where rss is the image of s under the map ĂX Ñ X

(Subsection 2.2). The twist Xs is a complex manifold endowed with a holomorphic mapπs :“ πrss to B.

Proposition 3.3. There exists a holomorphic family of complex manifolds

XXT Ñ ĂX – C

such that the fiber over s P ĂX is Xs. This family is endowed with a holomorphic fibration

ΠXT : XXT Ñ B ˆ ĂX.

and the map ΠXT restricts to πs on each fiber Xs.

Proof: Set rB :“ B ˆ ĂX and rX :“ X ˆ ĂX. Let rπ :“ πˆ id : rX Ñ rB be the projection.For any affine covering B “ Ť

i Ui define rUi :“ Ui ˆ ĂX. Of course, rB “ Ťi

rUi is an affinecovering of rB.

Let v P ĂX be a non-zero class. Represent it by a Čech cocycle vij on some open coveringUij. Let rvij be the pullback of vij to rUij . Define a 1-cocycle on rB with coefficients inrπ˚T rX rB as

wij :“ tĂvij P rπ˚T rX rBp ĂUijq


where t is the coordinate on ĂX – C. The twist of rπ : rX Ñ rB along the cocycletϕiju :“ texppwijqu is the desired family XXT. In other words, the manifold XXT isobtained as

XXT “ğ `

π´1pUiq ˆ C˘

O`π´1pUiq ˆ C

˘Q x „ ϕijpxq P

`π´1pUjq ˆ C

˘.

Definition 3.4. The family XXT Ñ ĂX constructed in Proposition 3.3 will be referredto as the Shafarevich–Tate family.

3.3. Shafarevich–Tate twists are symplectic. A twist of a Lagrangian fibration byan element of X is a priori only a complex manifold. We will see in this Subsection thatit is a holomorphic symplectic manifold.2

Lemma 3.5. Let pM,σq be a (not necessarily compact) holomorphic symplectic manifoldand π : M Ñ S a proper Lagrangian fibration on M over a smooth base S. Consider anautomorphism ϕ P H0pS,Aut0MSq. Then

ϕ˚σ ´ σ “ π˚α

where α is a closed holomorphic 2-form on S.

Proof: The statement is local on S so we can always shrink S if necessary. For S smallenough, one can realize ϕ as exppvq for a vertical holomorphic vector field v. There existsa holomorphic 1-form β on S such that ιvσ “ π˚β (Lemma 2.3). Let ϕt denote the flowof v. Then

ϕ˚σ ´ σ “1ż

0

dpϕ˚t σqdt

dt “1ż

0

ϕ˚t Lvσdt “ d

ż1

0

ϕ˚t pιvσqdt “ d

1ż

0

ϕ˚t π

˚βdt “ π˚dβ.

The last identity holds because π ˝ ϕt “ π. Thus ϕ˚σ ´ σ “ π˚α where α “ dβ.

Theorem 3.6. Let π : X Ñ B be a Lagrangian fibration on a compact holomorphic sym-plectic manifold X. Denote by Aut0,σ

XB the subsheaf of Aut0XB consisting of σ-symplectic

automorphisms. Then the inclusion Aut0,σ

XB Ñ Aut0XB induces an isomorphism of coho-

mology groups H ipB,Aut0,σXBq Ñ H ipB,Aut0XBq for i “ 0, 1.

Proof: We will only prove that the map H1pB,Aut0,σXBq Ñ H1pB,Aut0XBq is surjec-

tive. The proof of injectivity of this map, as well as the proof of the isomorphismH0pB,Aut0,σ

XBq » H0pB,Aut0XBq follow the same pattern and are left to the reader.Consider a class ϕ P H1pB,Aut0XBq. It can be represented by a Čech 1-cocycle

ϕij P Aut0XBpUijq for an open covering B “ ŤUi. Lemma 3.5 implies that ϕ˚

ijσ´σ “ π˚αij

for a closed holomorphic 2-form αij on Uij .The collection of forms tαiju defines a Čech 1-cocycle on B with coefficients in Ω2

B. AsH1pB,Ω2

Bq vanishes, the cocycle tαiju is exact. Consequently, there exist holomorphic2-forms βi on Ui such that αij “ βj |Uij

´βi|Uij. The form αij is closed for every i, j, hence

dβi|Uij“ dβj|Uij

. That means that the forms dβi glue to a globally defined holomorphic3-form. There are no non-trivial holomorphic 3-forms on B “ Pn, hence dβi “ 0 for everyi. Therefore, the forms βi are exact.

2A twist of a Lagrangian fibration might not be Kähler. That is why we use the term holomorphicsymplectic instead of hyperkähler.


We can find a holomorphic 1-form γi on Ui such that βi “ dγi. Let vi be the verticalvector field on π´1pUiq such that ιviσ “ π˚γi and ψi be the flow of vi. Define Ăϕij to bethe automorphism

rϕij :“ ϕij ˝ ψi|Uij˝ ψ´1

j |Uij.

A direct computation shows that rϕ˚ijσ “ σ. At the same time trϕiju define the same class

in H1pB,Aut0XBq as tϕiju. This proves the surjectivity of the map

H1pB,Aut0,σXBq Ñ H1pB,Aut0XBq.

Corollary 3.7. Let π : X Ñ B be a Lagrangian fibration and πs : Xs Ñ B be the twistof π by an element s P X. Then Xs is a holomorphic symplectic manifold and πs is aLagrangian fibration.

Proof: Every element s P X can be represented by a Čech cocycle ϕij P Aut0,σ

XBpUijq(Theorem 3.6). The manifold Xs is obtained by gluing the holomorphic symplectic man-ifolds π´1pUiq by the automorphisms ϕij. Since the automorphisms ϕij preserve σ, theforms σ|π´1pUiq are glued to a well-defined holomorphic symplectic form σs on Xs. Locallyπs coincides with π, therefore it is Lagrangian with respect to σs.

3.4. Degenerate twistor deformations as Shafarevich–Tate twists. Let π : X Ñ B

be a Lagrangian fibration. Consider the Shafarevich–Tate family XXT Ñ ĂX (Proposition3.3) and the degenerate twistor family Xdeg.tw Ñ C (Definition 2.14) associated to π. Fixa holomorphic symplectic form σ on X. It induces an isomorphism ĂX – H1,1pBq. Letrαs P H1,1pBq denote the class of a hyperplane section of B “ Pn. The class rαs inducesthe natural isomorphism H1,1pBq – C.

Theorem 3.8. There exists an isomorphism of XXT and Xdeg.tw as families of complex

manifolds that lifts the isomorphism ĂX – C. In other words, we have the commutativediagram:

Xdeg.twΦ //

Πdeg.tw

XXT

ΠXT

C» // ĂX.

Proof. Step 1: First, let us recall the relation between Dolbeault and Čech cohomol-ogy groups of the sheaf Ω1

B. The class rαs considered as a Dolbeault cohomology class,can be represented by a closed p1, 1q-form α on B. Take an affine open cover B “ Ť

Ui.The class rαs may be represented by a Čech cocycle taiju where aij P H0pUij ,Ω

1

Bq areholomorphic 1-forms on Uij. Given a closed p1, 1q-form α, one can construct taiju asfollows. There exists a p1, 0q-form ai on Ui such that ω|Ui

“ dai “ Bai for every i. Wedefine the form aij as

aij :“ ai|Uij´ aj|Uij

.

Step 2: In this step we will construct an isomorphism from XXT to Xdeg.tw locally onthe base B. Consider the vertical vector fields wi on π´1pUiq ˆC Ă X ˆC defined by theformula

ιwiσ “ tπ˚ai


Here t is the linear coordinate function on C and σ is the pullback of the holomorphicsymplectic form on X to π´1pUiq ˆ C. For any t P C we have the following identity onπ´1pUiq ˆ ttu:

Lwiσ “ tπ˚dai “ tπ˚α|π´1pUiq.

Let ϕi be the exponential of the vector field wi. This gives the following equality of formson π´1pUiq ˆ ttu:

ϕ˚i σ “ σ ` tπ˚α,

so that for any t P C the map ϕi is a biholomorphism

ϕi|π´1pUiqˆttu : pπ´1pUiq, Itq Ñ π´1pUiq,where It is the complex structure from Lemma 2.12. In fact, since every automorphismϕi commutes with the projection π´1pUiq ˆ C Ñ C, it induces a biholomorphism

ϕi : pπ´1pUiq ˆ C, Itwq Ñ π´1pUiq ˆ C.

Here Itw is the complex structure introduced in Definition 2.14.

Step 3: It remains to prove that the maps ϕi can be glued together to a globalisomorphism of families Xdeg.tw Ñ XXT. Define the vector fields wij on π´1pUijq ˆ C by

ιwijσ “ tπ˚paijq.

Let ϕij be the exponential of wij. Then ϕij is a holomorphic automorphism of π´1pUijqˆC.Since aij “ aj ´ ai one has

ϕij “ ϕj ˝ ϕ´1

i .

By Proposition 3.3 the manifold XXT is the twist ofXˆĂX by the cocycle tϕiju. Thereforethe maps ϕi glue together to give rise to a global biholomorphism Φ: Xdeg.tw Ñ XXT.

3.5. The period map of a Shafarevich–Tate family. Here we will describe the periodmap of a Shafarevich–Tate family. That can be viewed as a generalization of a Markman’sresult [Markman, Thm. 7.11].

The following definition is standard. Let X be a hyperkähler manifold. Its perioddomain D is defined as

D :“ trxs | qpx, xq “ 0 and qpx, xq ą 0u Ă PpH2pX,Cqqwhere q is the BBF form. Let p : X Ñ T be a holomorphic family of hyperkähler manifoldsover a connected simply connected base T . One can associate the period map PT : T Ñ D

to p. This is the holomorphic map defined as

t ÞÑ rH2,0pp´1ptqqs.Its importance is manifested by the Torelli theorem that says the following. Let

TeichpXq be the Teichmüller space of X i.e. the space of complex structures of hy-perkähler type on X modulo isotopies. Then the period map

P : TeichpXq Ñ D

is generically injective on connected components of TeichpXq [Ver13, Huy10].

Proposition 3.9. Let Xdeg.tw Ñ C be the degenerate twistor family of a Lagrangianfibration π : X Ñ B. Then its period map Pdeg.tw : C Ñ D is injective. The image ofPdeg.tw is the entire curve that is obtained as the intersection of D with a projective linelying on the quadric tqpx, xq “ 0u.


Proof: Let α be a Kähler form on B. The period map of the degenerate twistor familyis given by

Pdeg.twptq “ rσ ` tπ˚αsLet η denote the class of π˚α in H2pX,Cq. Consider the projective line L :“ Pptσ, ηuqlying in PpH2pX,Cqq. The map Pdeg.tw sends C isomorphically to L z rηs. All points ofthis line satisfy qpx, xq “ 0. The point corresponding to η does not lie in the perioddomain D since qpη, ηq “ 0. Therefore, the image of Pdeg.tw is L X D.

Remark 3.10. One may think of ĂX as an entire curve in the Teichmüller space of Xand X

0 as its image inside the moduli space M. The moduli space M can be definedas TeichpXq modulo the mapping class group of X. It is known that the action of themapping class group on TeichpXq is very non-discrete. Therefore, M is non-Hausdorff[Ver13]. This phenomenon is often seen already at the level of Shafarevich–Tate groups(Theorem 5.7)

4. Connected component of unity of a Shafarevich–Tate group

4.1. The sheaf Γ. Consider the exponential exact sequence

0 Ñ ZX Ñ OX Ñ OˆX Ñ 0

Apply the higher direct image functor R‚π˚p´q to it. Since the fibers of π are connectedand proper, we have π˚ZX » ZB, π˚OX » OB, and π˚O

ˆX » O

ˆB . The sequence

0 Ñ π˚ZX Ñ π˚OX Ñ π˚OˆX Ñ 0

is therefore exact. Hence we get a long exact sequence of sheaves:

0 Ñ R1π˚ZX Ñ R1π˚OX Ñ R1π˚OˆX Ñ R2π˚ZX Ñ . . .

Proposition 4.1. The isomorphism

ω : R1π˚OX Ñ π˚TXB

from Proposition 2.4 sends R1π˚ZX Ă R1π˚OX into Γ :“ kerpπ˚TXB Ñ Aut0XBq.

Proof: We need to prove that the composition of the maps

R1π˚ZiÝÑ R1π˚OX

rωÝÑ π˚TXBεÝÑ Aut0XB

vanishes. Let b P B˝ and Fb be a smooth fiber. Then pR1π˚OXqb “ H1pFb,OFbq,

pπ˚TXBq|b “ H0pFb, TFbq and rωb is the map

H0,1pFbq Ñ H1,0pFbq˚ – H0pFb, TFbq,given by the polarization rωsFb

on the abelian variety Fb. By Corollary 2.2 we can chooseω in such a way that this polarization is integral, so rωb maps H1pFb,Zq Ă H1pFb,OFb

q toΓb “ H1pFb,Zq Ă H0pFb, TFbq.

It follows that the statement of the lemma holds after the restriction to B˝. Considera local section τ of R1π˚ZX over an open subset U Ă B. Denote U

ŞB˝ by U˝. We

have seen that the restriction of the automorphism pε ˝ rω ˝ iqpτq to π´1pU˝q is trivial.An automorphism that is trivial on a dense open subset is trivial. Hence, the verticalautomorphism pε ˝ rω ˝ iqpτq of π´1pUq is trivial.

We have constructed a morphism of sheaves α : R1π˚ZX Ñ Γ. Note that it is injective,since it is a composition of an isomorphism rω : R1π˚OX Ñ π˚TXB and an embeddingi : R1π˚ZX Ñ R1π˚OX .


There is the following diagram with exact rows:

0 // Γ // π˚TXBε // Aut0XB

// 0

0 // R1π˚ZXi //

α

OO

R1π˚OX

rωOO

// R1π˚OˆX

Recall the local invariant cycle theorem:

Proposition 4.2. ([BBDG, Cor. 6.2.9]) Let f : X Ñ Y be a proper map of complexalgebraic varieties with X non-singular. For a subset U Ă Y , denote by U˝ the set ofnon-critical values of f in U . Then for any b P Y there exists a small ball Ub Ă Y withthe center b such that the following holds.

(1) For any i the following groups are isomorphic: H ipf´1pbq,Qq » H ipf´1pUbq,Qq;(2) For any point b˝ P U˝

b the cohomology group H ipf´1pUbq,Qq surjects onto the localmonodromy invariants H ipf´1pb˝q,Qqπ1pU˝

bq “ Riπ˚QXpU˝

b q.Remark 4.3. Let π : X Ñ B be a Lagrangian fibration on a hyperkähler manifold. Foreach b P B there exists an open subset U Ă B containing b such that π´1pUq is a smoothalgebraic variety [SV, Cor. 3.4]. Therefore, the local invariant cycle theorem can beapplied in this situation.

Let ΓQ denote the sheaf Γ b QB.

Proposition 4.4. The map α b Q : R1π˚Q Ñ ΓQ is an isomorphism.

Proof: Step 1. First, notice that the lemma holds after the restriction to the non-critical set B˝ Ă B. Indeed, if Fb “ π´1pbq is a smooth fiber, then

αb : pR1π˚QXqb “ H1pFb,Qq Ñ pΓQqb “ H1pFb,Qqis the map given by the polarization on the abelian variety F . Hence it is an isomorphism.Note that in general the same might not hold with integral coefficients.

Step 2. Take a point b P B and let Ub Ă B be as in Proposition 4.2. We need to provethat the map

αUb: H1pUb,Qq Ñ ΓQpUbq

is an isomorphism. We already know that it is injective, hence we only need to checksurjectivity. Take a local section γ P ΓQpUbq. Let γ˝ be its restriction to U˝

b . By Step 1

there exists β˝ P H0pU˝b , R

1π˚QXq, such that αU˝bpβ˝q “ γ˝. The section β˝ can be lifted

to a section β of H0pUb, R1π˚QXq “ H1pπ´1pUbq,Qq by the local invariant cycle theorem.

The local sections αUbpβq P ΓQpUbq and γ P ΓQpUbq coincide after restriction to U˝

b bytheir construction. The sheaf ΓQ is a subsheaf of the locally free sheaf R1π˚OX , henceαUb

pβq and γ coincide.

4.2. The connected component of unity of Shafarevich–Tate groups. The goalof this Subsection is to describe the group X

0 i.e. the connected component of unity ofthe group X. This group fits into the short exact sequence

H1pB,Γq Ñ ĂX Ñ X0 Ñ 0.

In the previous Subsection, we constructed an injective map α : R1π˚Z Ñ Γ and provedthat α is an isomorphism after tensoring with Q (Proposition 4.4). Although we aremostly concerned with the sheaf Γ, the sheaf R1π˚Z is often much easier to understand.


Define the subspace W Ă H2pX,Cq as the kernel of the restriction map

H2pX,Cq Ñ H0pB,R2π˚CXqThis is a Hodge substructure in H2pX,Zq that can be described as

(4) WZ “č

bPB

kerpH2pX,Zq Ñ H2pFb,Zqq,

Let us denote WQ :“ WŞH2pX,Qq.

As before, let η P H2pX,Zq be the pullback of the class of a hyperplane section. LetηK be the orthogonal to η with respect to the BBF form.

Proposition 4.5. Let W be as above. Then we have the following chain of inclusions

tη, rσs, rσsu Ă W Ă ηK.

Proof: The class η is the Chern class of the line bundle π˚OPnp1q. This line bundle

restricts trivially to every fiber of π, hence η is contained in W . By Proposition 2.1 thekernel of the restriction map H2pX,Cq Ñ H2pFb,Cq to a smooth fiber Fb is ηK. We obtainthat W Ă ηK.

We are left to prove that rσs is contained in W . Indeed, W is a Hodge substructure,thus it will imply that rσs is in W as well. The fibration π is Lagrangian, hence therestriction of the 2-form σ to every smooth fiber of π vanishes. Therefore the class rσs inH2pX,Cq restricts trivially to a smooth fiber.

Let Fb be a singular fiber. By [Cl, Thm. 6.9] there is a neighborhood Ub Ă B of b anda smooth retraction of π´1pUbq to Fb i.e. a one-parameter family of diffeomorphisms ftsuch that

ft : π´1pUbq Ñ π´1pUbq f0 “ idπ´1pUbq, impf1q “ Fb, ft|Fb

“ idFb@t

One can see that f˚0σ “ σ and f˚

1σ “ 0. Let ξt denote the tangent vector field to the

one-parameter family ft. Compute

σ|π´1pUbq “ ´pf˚1σ ´ f˚

0σq “ ´

1ż

0

d

dtf˚t σ “ ´

1ż

0

f˚t Lξσ “ ´d

ˆż1

0

f˚t ιξσ

˙.

We conclude that the form σ becomes exact after the restriction to π´1pUbq. Therefore,the class of σ is indeed contained in W .

Remark 4.6. If all fibers of π are irreducible, then ker pH2pX,Cq Ñ H2pFb,Cqq does notdepend on b. In particular, W “ ηK (Proposition 2.1).

Proposition 4.7. Consider the map

H1pB,R1π˚Zq Ñ H1pR1π˚OXq » ĂXinduced by the embedding R1π˚Z Ñ R1π˚OX . There are canonical isomorphisms

ĂX » H0,2pXqand

H1pB,R1π˚Zq » WZηthat fit into the commutative diagram

H1pB,R1π˚Zq //

»

ĂX»

WZη p // H0,2pXq


Here p is induced by the Hodge projection WZ Ă H2pX,Zq Ñ H0,2pXq.Proof: Consider the Leray spectral sequence for the sheaf OX . The terms of the secondpage of this spectral sequence can be computed using Proposition 2.4 as

Ep,q2

“ HppB,Rqπ˚OXq “ HppB,ΩqBq “

#C if p “ q,

0 otherwise.

This spectral sequence degenerates on the second page. It converges to the cohomologygroups of the sheaf OX . Therefore,

H2pX,OXq – E1,12

“ H1pR1π˚OXq – ĂX.

Let us now consider the second page of the Leray spectral sequence for the sheaf ZX .The differential d2 : E

1,12

Ñ E3,02

vanishes because E3,02

“ H3pPn,Zq “ 0. All the otherdifferentials starting in E1,1 vanish because their targets are in negative grading. There-fore,

E1,18 “ E

1,12

“ H1pB,R1π˚Zq.The filtration on the Leray spectral sequence induces a filtration F ‚H2pX,Zq on thecohomology groups of X. We obtain the following short exact sequences:

0 Ñ Z¨ηÝÑ F 1H2pX,Zq Ñ H1pB,R1π˚Zq Ñ 0

0 Ñ F 1H2pX,Zq Ñ H2pX,Zq Ñ H0pB,R2π˚Zq.

The second exact sequence implies that F 1H2pX,Zq “ WZ. We see from the firstexact sequence that H1pB,R1π˚ZXq » WZZη. The commutativity of the diagram inthe statement of the proposition follows from functoriality properties of Leray spectralsequences.

Corollary 4.8. Let π : X Ñ B be a Lagrangian fibration. Define the group xX0 to be thecokernel of the map

WZη pÝÑ H0,2pXq.Then the group X

0 is a quotient of xX0 by a finite group.

Proof: The map α : R1π˚Z Ñ Γ induces the map H1pαq : H1pR1π˚Zq Ñ H1pB,Γq. Themap H1pαq has a finite cokernel by Proposition 4.4. By Proposition 4.7 there is thefollowing commutative diagram

WZη //

H1pαq

H0,2pXq //

»

xX0 //

0

H1pB,Γq // ĂX // X0 // 0

The kernel of the map xX0 Ñ X0 is isomorphic to cokerpH1pαqq by the Snake lemma.

Hence it is finite.

5. Kählerness and projectivity properties

In this Section we study Kählerness and algebraicity properties of degenerate twistordeformations. Unfortunately we are not able to prove that a denegerate twistor deforma-tion of a hyperkähler manifold X is always Kähler. We need to impose some additionalassumption on X. We call manifolds that do not satisfy this assumption M-special.


5.1. M-special Hodge structures. In this Subsection, we summarize several resultsabout Hodge structures of K3 type. In particular, we introduce and discuss M-specialHodge structures. A pure Hodge structure W is said to be of K3 type if it is of weight 2

and dimW 2,0 “ 1.Let W be a Z-Hodge structure of weight 2 and K Ă C a subring. We will write

W1,1K :“ WK

ŞW 1,1 and W

2,0`0,2K :“ WK

Ş pW 2,0 ‘ W 0,2q. We denote the Hodge projec-tion by p : W Ñ W 0,2. Let q be a polarisation on W . If W is a Hodge structure of K3type, then the projection p : W Ñ W 0,2 is given by the map

v Ñ qpv, σqσfor an appropriate choice of σ P W 2,0.

We define the transcendental Hodge substructure T Ď W to be the orthogonal comple-ment to W 1,1

Z . The lattice T has the rank at least two because

W2,0`0,2R Ď TR.

Note that the restriction of the Hodge projection p to T is an isomorphism to its image.

Definition 5.1. Let W be a Z-Hodge structure of K3 type. It is said to be M-special ifW

2,0`0,2Z ‰ 0.

Remark 5.2. Let V Ă W be an embedding of Hodge structures of K3 type. Then V isM-special if and only if so is W .

The following definition is a straightforward generalization of [Markman, Def. 1].

Lemma 5.3. Let W be a polarized Z-Hodge structure of K3 type. The following areequivalent.

(1) The subspace W 1,1 is rational ("Picard rank is maximal").(2) The subspace W 2,0`0,2 is rational.(3) The rank of the transcendental Hodge substructure T is two.(4) The image of WZ under the Hodge projection p : WZ Ñ W 0,2 is a discrete subgroup

of W 0,2.

The proof of this lemma is straightforward and is left to the reader. If one of theconditions of the lemma holds, then W is M-special. However, the converse is not true.

Lemma 5.4. ([Markman, Lemma 5.5]) Let W be a polarized Z-Hodge structure of K3type that is not M-special. Assume that rkT ě 3. Then for every sublattice T 1 Ă T ofcorank one, the group ppT 1q generates W 0,2 as an R-vector space.

Proof: Let T 1 Ă T be a sublattice of corank one. Suppose that ppT 1q does not generateW 0,2 over R. In that case there exist real numbers a, b such that

a ¨ qpRepσq, tq ` b ¨ qpImpσq, tq “ 0 @t P T 1

Hence the vector l “ a ¨ Repσq ` b ¨ Impσq P W2,0`0,2R is orthogonal to T 1. Observe that

lK “ pW 1,1Z ` T 1q b R since pW 1,1

Z ` T 1q b R Ď lK and the dimensions of the two spacescoincide. The subspace lK is therefore rational. Hence, a multiple of l is rational. Thiscontradicts the assumption that W is not M-special.

Lemma 5.5. ([Markman, Lemma 5.5]) Let T Ă R2 be a finitely generated subgroup ofR2 of rank at least three. Assume that every subgroup T 1 Ă T of corank one generates R2

over R. Then T is dense in R2.

Theorem 5.6. ([Markman, Lemma 5.4, 5.5]) Let W be a polarized Z-Hodge structure ofK3 type. Then the following are equivalent


(1) The Hodge structure W is M-special.(2) The image of WZ under the Hodge projection p : W Ñ W 0,2 is not dense in W 0,2.

Proof: p1q ñ p2q Suppose that W is M-special. Choose a non-zero element l P W 2,0`0,2Z .

There exists a unique element σ P W 2,0 such that Reσ “ l. The projection p : W Ñ W 0,2

is identified with the map v P W ÞÑ qpσ, vq up to scaling. For every v P WZ the num-ber Repqpσ, vqq “ qpl, vq is an integer. Hence, there exists a constant λ P R such thatRepzq P Zλ for every vector z in ppWZq. Therefore, ppWZq cannot be dense in W 0,2.

p2q ñ p1q Suppose W is not M-special. In that case the rank of T is at least three(Lemma 5.3). Lemma 5.4 implies that for every sublattice T 1 Ă T of corank one, thegroup ppT 1q generates W 0,2 over R. It follows from Lemma 5.5 that ppTZq “ ppWZq isdense in W 0,2.

5.2. Kählerness of degenerate twistor deformations. Let π : X Ñ B be a La-grangian fibration. Recall that we defined the Hodge substructure W Ă H2pX,Zq as

W :“ ker`H2pX,Zq Ñ H0pB,R2π˚ZXq

˘.

The Hodge structure W is of K3 type by Proposition 4.5. Let us denote the Hodgeprojection by p : H2pXq Ñ H0,2pXq.

Theorem 5.7. Let π : X Ñ B be a Lagrangian fibration, X0 the connected componentof unity of its Shafarevich–Tate group X. Then X

0 is Hausdorff if and only if X is ofmaximal Picard rank. In this case X

0 is isomorphic to an elliptic curve.

Proof: The group X0 is Hausdorff if and only if the topological group xX0 defined as

H0,2pXqppWZq is Hausdorff. Indeed, the latter group is a finite unramified cover of X0

(Corollary 4.8). We see that X0 is Hausdorff if and only if ppWZq Ă H0,2pXq is a discretesubgroup. Apply Lemma 5.3 to the Hodge structure W η. We obtain that ppWZq isdiscrete in H0,2pXq if and only if W 2,0`0,2 “ H2,0pXq ‘ H0,2pXq is a rational subspace.This is equivalent to saying that H1,1pXq is rational i.e. X has the maximal Picard rank(Lemma 5.3) If ppWZq is discrete, then ppWZq is a lattice in H0,2pXq of rank two and X

0

is an elliptic curve.

Definition 5.8. Let X be a hyperkähler manifold. It is called M-special if

pH2,0pXq ‘ H0,2pXqq X H2pX,Zq ‰ t0u

In other words, a hyperkähler manifold is M-special if the Hodge structure on its secondcohomology is M-special (Definition 5.1).

Lemma 5.9. Let π : X Ñ B be a Lagrangian fibration on a projective hyperkähler mani-fold X. Then X is M-special if and only if the Hodge structure WZη is M-special.

Proof: Let l P H1,1pXqZ be an ample class. Since qpl, ηq ą 0 the pairing with l inducesa non-zero functional on W . Thus, lK X W is a hyperplane in W not containing η. Weobtain the direct sum decomposition W “ η ‘ plK X W q. The subspace lK X W is apolarized Hodge substructure of H2pX,Zq isomorphic to W η. It follows from Remark5.2 that lK X W is M-special if and only if so is H2pX,Zq.

Remark 5.10. The statement of Lemma 5.9 in general does not hold for non-projectivemanifolds. However, it is still true that if X is M-special then so is W η.

We are now ready to prove the main theorem of this Section.


Theorem 5.11. Let π : X Ñ B be a Lagrangian fibration and XXT Ñ ĂX the Shafarevich–Tate family (Subsection 3.2). Assume that X is projective and not M-special. Then the

members of the family Xs Ă XXT are Kähler for each s P ĂX.

Proof: The zero fiber X0 is Kähler by assumption. Kählerness is an open condition.Thus there exists an open subset U Ă ĂX such that for every s P U the twist Xs is Kähler[Vois, Thm. 9.23]. The manifolds Xs and Xs`λ are isomorphic for each λ in the subgroupΛ :“ impH1pB,Γqq Ă ĂX. The vector spaces ppWZq b Q and Λ b Q coincide (Proposition4.4). By Lemma 5.9 the Hodge structure W η is not M-special. It follows from Theorem5.6 that ppWZq is dense in ĂX » H0,2pXq. Hence Λ is dense in ĂX. We obtain that forevery s P ĂX there exists s1 P U such that Xs » Xs1. Consequently, Xs is Kähler for everys.

Remark 5.12. The statement of Theorem 5.11 remains true for non-projective manifoldsif we assume that the Hodge structure H2pX,Zqη is not M-special.

Remark 5.13. We do not know if the statement of Theorem 5.11 holds for M-specialmanifolds. Another open question is the following. Consider an element s P X zX0. Isthe twist Xs a Kähler manifold? Note that from results of Subsection 3.3 we know thatXs possesses a holomorphic symplectic form. It would be interesting to know if one canobtain examples of non-Kähler holomorphic symplectic manifolds, such as in [Gu, Bog96].

Remark 5.14. Being M-special is a restrictive condition. Hence the statement of Theo-rem 5.11 holds for most Lagrangian fibrations. Consider the set Dη of Hodge structureson H2pX,Zq such that η P H1,1pXq. It can be identified with a complex manifold ofcomplex dimension b2 ´ 3. The set of Hodge structures in Dη such that H2pX,Zqη isM-special is a countable union of real analytic subvarieties of real dimension b2 ´ 3.

5.3. Algebraic points in Shafarevich–Tate families. In this subsection we will de-scribe the set of projective deformations in the Shafarevich–Tate family.

Lemma 5.15. Consider the degenerate twistor family Xdeg.tw Ñ C and let X “ X0 be thefiber over 0 P C. Let α be a class in H2pX,Rq. Then exactly one of the following holds:

(1) α P ηK;(2) there exists a unique s P C such that α P H1,1pXsq.

Moreover,Ş

sPCH1,1pXsq “ pηKq1,1.

Proof: For each s P C the complex structure on Xs is defined by a c-symplectic formwith cohomology class rσss “ rσ0s ` sη. A real class α lies in H1,1pXsq if and only if it isorthogonal to σs i.e.

qpα, σsq “ qpα, σ0q ` sqpα, ηq “ 0.

Suppose that α P H2pX,Rq z ηK. Then sα :“ ´qpα, σ0qqpα, ηq is the unique numbersatisfying qpα, σsαq “ 0. If α is contained in ηK, then it is of type p1, 1q for every degeneratetwistor deformation.

Corollary 5.16. The set R :“ ts P ĂX | Xs is algebraic u is at most countable. Inparticular, a very general member of the Shafarevich–Tate family is non-algebraic.

Proof: If Xs is algebraic, it carries an ample divisor L and c1pLq P H1,1Z pXq. Moreover,

we have the inequality qpc1pLq, ηq ą 0 in this case. For every α P H2pX,Zq z ηK there isonly one point sα such that α P H1,1pXsαq (Lemma 5.15). Thus, R is contained in theset

tsα | α P H2pX,Zq z ηKu.It follows that R is at most countable.


See Theorem 5.20 below for a refinement of Corollary 5.16. A similar theorem holdsfor twistor families Xtw Ñ P1 ([Ver96, Fu82]).

Lemma 5.17. Let π : X Ñ B be a Lagrangian fibration on a hyperkähler manifold overB “ Pn. The following are equivalent:

(1) X is projective;(2) π admits a multisection i.e. there exists a subvariety Z Ă X such that π|Z : Z Ñ B

is a finite morphism;(3) there exists a class α P H1,1

Q pXq such that qpα, ηq ‰ 0;

(4) there exists a class ω P H1,1Q pXq such that qpω, ωq ą 0.

Proof: p1q ñ p2q Choose a holomorphic embedding i : X ãÑ PN . Pick a point x P X

outside of the singular locus of π. Consider a general linear subspace L Ă PN of codi-mension n passing through x and transversal to the fibers of π. Let Z be a component ofZ 1 :“ L

ŞX Ă X which is transversal to the fibers of π. Then Z is a multisection of π.

p2q ñ p3q Let Z Ă X be a multisection. Let C0 Ă B be a smooth curve and letC :“ π´1pC0q Ş

Z be its preimage in Z. Consider the homology class rCs P H2pX,Zq ofthe curve C. The BBF form defines an isomorphism

H2pX,Qq Ñ H2pX,Qq˚ » H2pX,Qq.Let α P H2pX,Qq be the class corresponding to rCs under this isomorphism. Thenqpα, ηq “

şCη ą 0.

p3q ñ p4q Let α be a rational p1, 1q-class such that qpα, ηq ‰ 0. We may assume thatqpα, ηq ą 0. Let us find a number t such that ω :“ α ` tη satisfies qpω, ωq ą 0. Wecompute that

qpα` tη, α ` tηq “ qpα, αq ` 2tqpα, ηq,Therefore, the class ω is rational and has a positive square for any rational t ą ´qpα,αq

2qpα,ηq.

p4q ñ p1q See [Huy01, Thm. 3.11]

Remark 5.18. If a Kähler manifold is algebraic3, it is necessarily projective by theMoishezon theorem ([Moish, Thm. 11]). Therefore, we can replace the first condition inthe theorem above with the one that X is algebraic.

Denote by p : H2pX,Cq Ñ H0,2pXq the Hodge projection. Take σ to be the holomor-phic symplectic form such that ppσq is identified with the class of the Fubini-Study formunder the isomorphism H0,2pXq » ĂX » H1,1pBq. In the following lemma we will give acharacterization of torsion in X

0 in terms of the BBF form.

Lemma 5.19. Let π : X Ñ B be a Lagrangian fibration. Consider a class tσ P H0,2pXq,t P C. Let rtσs denote its image in X

0. Then rtσs is torsion if and only if there exists arational class l P WQ Ă H2pX,Qq such that qpl, σq “ t (see formula (4) for the definitionof WQ).

Proof: We know from the short exact sequence (1) that X0 “ XΛ where Λ denotes

the group ImpH1pB,Γq Ñ ĂXq. Hence, the class rtσs is torsion if and only if tσ lies in theimage of the group H1pB,Γq b Q. This is equivalent to saying that tσ lies in the imageof WQ Ă H2pX,Qq under the Hodge projection H2pX,Qq Ñ H0,2pXq (see Proposition

3i.e., the analytification of a smooth proper algebraic variety over C


4.4, Proposition 4.7). Since H1,1pXq is orthogonal to H2,0pXq ‘ H0,2pXq, the projectionH2pX,Cq Ñ H0,2pXq is given by the formula

v Ñ qpv, σqσ.Hence, tσ lies in the image of WQ under the Hodge projection if and only if there existsa class l P WQ such that t “ qpl, σq.

Theorem 5.20. Let π : X Ñ B be a Lagrangian fibration. Suppose that X is projective.Let s be an element of X0. The following are equivalent.

(1) The Shafarevich–Tate twist Xs of X is projective.(2) The manifold Xs is Kähler and s is a torsion element of X0.

Proof: p2q ñ p1q Suppose that the element s is torsion. Let s “ tσ denote a preimageof s in H0,2pXq » ĂX. There exists a class l P WQ such that qpl, σq “ t (Lemma 5.19).As X is projective, there exists a class α P H1,1

Q pXq such that qpα, ηq “ 1 (Lemma 5.17).The class αs :“ α ´ l satisfies qpαs, ηq “ qpα, ηq “ 1 as l is contained in WQ Ă ηK. Weclaim that αs lies in H1,1

Q pX sq. Indeed, αs is orthogonal to H2,0pX sq because

qpαs, σ ` tηq “ qpα ´ l, σ ` tηq “ qpα, σq ´ tqpl, ηq “ t´ t “ 0.

Lemma 5.17 implies that Xs is projective.

p1q ñ p2q Suppose that X and Xs are both projective. Let s “ tσ denote a preimageof s in H0,2pXq as before. There exist two rational classes α P H1,1

Q pXq and αs P H1,1Q pX sq

such that qpα, ηq “ qpαs, ηq “ 1 (Lemma 5.17). Hence, the class l :“ α ´ αs is a rationalclass orthogonal to η. This class satisfies

qpl, σq “ qpl, σ ` tηq “ qpα, σ ` tηq “ aqpα, ηq “ t.

We can not conclude directly that rtσs is torsion because l might not lie in WQ. Therefore,we need to adjust l.

Consider the subspace WK Ă H2pX,Qq. It follows from Proposition 4.5 that

η P WK Ă`ηK X H1,1pXq

˘.

The BBF form q has signature p1, h1,1pXq ´ 1q on H1,1pXq and η is isotropic with respectto this form. Therefore, the restriction of q to WK is semi-negative definite with kernelgenerated by η.

Let U Ă WK be a rational hyperplane in WK not containing η. The form q|U is negativedefinite, in particular, it is non-degenerate. Therefore there exists a unique rational vectoru P U such that for every v P U the following holds

qpl, vq “ qpu, vq.The vector l ´ u is orthogonal to every vector in WK. Hence l ´ u is contained in WQ.Since u P H1,1pXq, we have qpl´u, σq “ qpl, σq “ t. Lemma 5.17 concludes the proof.

Remark 5.21. By [SV, Cor. 3.4] the set of s P X0 such that Xs is projective is non-

empty.

6. Sections of Lagrangian fibrations

6.1. Obstruction for existence of a section. We move on to study obstructions toexistence of sections of Lagrangian fibrations. In this Section, we will always assume thatπ : X Ñ B is a Lagrangian fibration with reduced irreducible fibers.


In this case, the fibration π : X Ñ B admits a local section in a neighborhood of everypoint b P B. Consider an open cover B “ Ť

Ui and choose a collection of local sectionssi : Ui Ñ π´1pUiq.

By [Markush96, Prop. 2.1 (iii)] for every pair i, j there exists a unique automorphismϕij P Aut0XBpUijq such that ϕijpsi|Uij

q “ sj |Uij. The collection of automorphisms tϕiju

satisfies the cocycle condition. Therefore, tϕiju defines a class

αpX, πq P H1pB,Aut0XBq “ X.

We denote this class by αpXq when the structure of the Lagrangian fibration on X isclear.

Lemma 6.1. (1) The class αpXq does not depend on the choice of local sectionssi : Ui Ñ π´1pUiq.

(2) For any element s P X we have αpXsq “ αpXq ` s.(3) The class αpXq vanishes if and only if the fibration π : X Ñ B admits a section.

Proof: We will prove only the third statement. The proof of the first two ones follows thesame lines. Suppose that αpXq “ 0. Then there exist automorphisms ϕi P Aut0XBpUiqsuch that ϕij “ ψ´1

j ψi. The sections ϕipsiq coincide on intersections, so they define aglobal section of π. The converse implication is straightforward.

Let apXq to be the image of αpXq in XX0 “ H2pB,Γq.Corollary 6.2. Let π : X Ñ B be a Lagrangian fibration with reduced irreducible fibers.Then the class apXq vanishes if and only if there exists a deformation Xs of X in theShafarevich–Tate family that admits a holomorphic section. Moreover, in this case, theclass of s in X

0 is uniquely defined.

Proof: The class apXq P H2pB,Γq vanishes if and only if α :“ αpXq lies in X0. The

class αpX´αq vanishes by Lemma 6.1 (2). Therefore

π´α : X´α Ñ B

admits a holomorphic section. Conversely, if α R X0, then for every deformation Xs in

the Shafarevich–Tate family we have apXsq ‰ 0.

It was proved in [BDV, Thm. 3.5] that a Lagrangian fibration admits a smooth sectionif and only if some of its degenerate twistor deformations admits a holomorphic section.Combined with Corollary 6.2 we get that apXq is indeed a complete topological obstructionfor existence of a section on a Lagrangian fibration with reduced irreducible fibers.

For the rest of the paper we will be proving the following theorem.

Theorem 6.3. Let π : X Ñ B be a Lagrangian fibration on a compact hyperkähler man-ifold over a smooth base. Let Γ be the kernel of the exponential map π˚TXB Ñ Aut0XB.Assume that the following holds:

‚ the fibers of π are reduced and irreducible;‚ H3pX,Qq “ 0;‚ H2pB,Γq is torsion-free.

Then there exists a unique deformation pXs, πsq of pX, πq in the Shafarevich–Tate familysuch that πs : Xs Ñ B admits a holomorphic section.

Note that the condition H3pX,Qq “ 0 holds if X is deformation equivalent to theHilbert scheme of points on a K3 surface or to one of the exceptional O’Grady examples.

Unfortunately, we are not able to get rid of the condition on H2pB,Γq. However, westrongly believe, that the theorem should be true without this assumption. Therefore wepose the following conjecture.


Conjecture 1. Let π : X Ñ B be a Lagrangian fibration. Assume that b3pXq “ 0. Thenthere exists a degenerate twistor deformation of π admitting a holomorphic section.

We believe that the assumption that fibers are reduced and irreducible is not too re-strictive.

Conjecture 2. ([Bog]) Let π : X Ñ B be a Lagrangian fibration on a compact hyper-kähler manifold. Then it has no multiple fibres.

Conjecture 3. Let π : X Ñ B be a Lagrangian fibration. Consider the space Mη ofdeformations of X such that η remains of type p1, 1q. Then a very general deformation ofX in Mη is a Lagrangian fibration with irreducible fibers.

Both conjectures hold for K3 surfaces (see [Huy16, Prop. 1.6 (ii)] for Conjecture 2 and[FM, I.1.Thrm. 4.8] for Conjecture 3).

6.2. Hard Lefschetz type theorems for higher direct images of QX . Let π : X Ñ B

be a Lagrangian fibration. In this Subsection we prove a version of the Hard Lefschetz the-orem for the sheaf R1π˚QX which will be used in the proof of the Theorem 6.3. Through-out this Subsection we assume that X is projective. Let l P H1,1

Q pXq be an ample class.Abusing notation, we will denote by the same letter the induced section of R2π˚QX .

For each p ě 0, multiplication by l induces a map

L : Rpπ˚QX Ñ Rp`2π˚QX .

We will refer to L as the Lefschetz map.

Lemma 6.4. (Lefschetz decomposition for R2π˚QX) Let π : X Ñ B be a Lagrangianfibration with X projective. Assume that all fibers of π are reduced and irreducible. Thenthe sheaf R2π˚QX decomposes as

R2π˚QX “ QB ¨ l ‘ pR2π˚QXqprimwhere pR2π˚QXqprim is the kernel of the map

Ln´1 : R2π˚QX Ñ R2nπ˚QX .

Proof: Since the fibres are irreducible we have R2nπ˚QX » QB . The restriction ofLn´1 : R2π˚QX Ñ R2nπ˚QX on the subsheaf generated by l is an isomorphism. Hence theclaim.

We move on to study the pn ´ 1q-th power of the Lefschetz map on R1π˚QX , that is

(5) Ln´1 : R1π˚QX Ñ R2n´1π˚QX .

First, note that multiplication by l P H1,1Q pXq induces maps on Rpπ˚Ω

qX as well:

L : Rpπ˚ΩqX Ñ Rp`1π˚Ω

q`1

X .

for any p, q “ 0, . . . , n. Abusing notation, we will denote these maps also by L. Lefschetzmaps commute with the natural morphisms R‚π˚QX Ñ R‚π˚OX , the relative Hodgeprojections. In particular, the following diagram is commutative.

R1π˚QX//

Ln´1

R1π˚OX

Ln´1

R2n´1π˚QX// Rnπ˚Ω

n´1

X


The horizontal arrows in this diagram are Hodge projections. The holomorphic symplecticform σ on X induces the isomorphism Ω1

X » TX . The composition of this isomorphismwith the natural map TX Ñ π˚TB gives us the map

θ : Ω1

X Ñ π˚TB.

For every pair of integers p, q one can apply the functor Rqπ˚Λpp´q and get the following

map of sheaves on B:

Rqπ˚pΛpθq : Rqπ˚ΩpX Ñ Rqπ˚pπ˚ΛpTBq.

The sheaf on the right is:

Rqπ˚pπ˚ΛpTBq » Rqπ˚OX b ΛpTB » ΩqB b ΛpTB » Rqπ˚OX b pRpπ˚OXq˚

.

The isomorphisms follow from the projection formula and Matsushita’s theorem (Propo-sition 2.4).

For each p, q we have constructed a map

fp,q : Rqπ˚Ω

pX Ñ Rqπ˚OX b pRpπ˚OXq˚

.

In particular, there are the following maps:

f0,1 : R1π˚OX Ñ R1π˚OX

f1,1 : R1π˚Ω

1

X Ñ EndpR1π˚OXqfn´1,n : R

nπ˚Ωn´1

X Ñ Rnπ˚OX b`Rn´1π˚OX

˘˚ » R1π˚OX .

The map f0,1 is the identity map by construction. It is easy to see that the image of lunder f1,1 is the identity operator.

Lemma 6.5. fn´1,n ˝ Ln´1 “ idR1π˚OX.

Proof: The maps fp,q commute with the multiplication of forms in the sense that thefollowing diagram is commutative for any p1, q1, p2, q2 “ 0, . . . , n.

Rq1π˚Ωp1X b Rq2π˚Ω

p2X

//

fp1,q1bfp2,q2

Rq1`q2π˚Ωp1`p2X

fp1`p2,q1`q2

Rq1π˚OX b pRp1π˚OXq˚ b Rq2π˚OX b pRp2π˚OXq˚ // Rq1`q2π˚OX b pRp1`p2π˚OXq˚

It follows that the following diagram is commutative

R1π˚OX b pR1π˚Ω1

Xqb n´1 //

id bpf1,1qb n´1

Rnπ˚Ωn´1

X

fn´1,n

R1π˚OX b pEndpR1π˚OXqqb n´1 // R1π˚OX

Since f1,1plq “ idR1π˚OX, we obtain that for every local section α of R1π˚OX

fn´1,npα ¨ ln´1q “ fn´1,n ˝ Ln´1pαq “ α.

Corollary 6.6. Let π : X Ñ B be a Lagrangian fibration on a projective hyperkählermanifold X. Then there exists a sheaf N on B such that the sheaf R2n´1π˚QX decomposesinto the direct sum

R2n´1π˚QX » R1π˚QX ‘ N.

The embedding of the first summand is given by the map (5).


Proof: The map (5) is an isomorphism after the restriction to B˝ Ă B by the HardLefschetz theorem. Together with Lemma 6.5 this implies that the map fn´1,n|B˝ sendsR2n´1π˚Q|B˝ isomorphically to R1π˚Q|B˝ . A local section of R1π˚OX whose restriction toB˝ lies in R1π˚Q is necessarily a section of R1π˚Q by the proof of Proposition 4.4. Hencethe map fn´1,n descends to a map

fn´1,n|R2n´1π˚QX: R2n´1π˚Q Ñ R1π˚Q.

This map satisfies the following property (Lemma 6.5):

fn´1,n|R2n´1π˚QX˝ Ln´1 “ idR1π˚QX

.

The claim now follows.

Remark 6.7. It can be proven that the sheaf N from Proposition 6.6 is supported ona codimension two subset. The result follows from the description of a general singularfiber of a Lagrangian fibration given in [HO]. We do not know whether the sheaf N canbe non-trivial.

6.3. The discrete part of Shafarevich–Tate groups. Let X be the Shafarevich–Tategroup of a Lagrangian fibration π : X Ñ B. Recall that the group XX0 of connectedcomponents of X is isomorphic to H2pB,Γq (see the exact sequence (3)). We sometimesrefer to XX0 as the discrete part of X.

By Proposition 4.4, there is an isomorphism

H2pB,Γq bZ Q » H2pB,R1π˚QXq.

The natural map Ep,02

“ HppB,Qq Ñ HppX,Qq is given by the pullback map π˚. Thepullback map on cohomology is injective for every surjective map of compact Kählermanifolds (see e.g. [Vois, Lem. 7.28]). This implies that Ep,0

2“ E

p,08 . Therefore, the

differential d2 : E2,12

Ñ E4,02

vanishes. All the higher differentials with the source in E2,1

vanish because their targets are trivial groups. For the same reason, all the higher differ-entials dn, n ą 2 with the source in E0,2 vanish too. Since E3,0

2“ H3pB,Qq is trivial, there

is an embedding of E2,18 “ E

2,12

impd2q into H3pX,Qq. Moreover, we have the followingexact sequence of Q-vector spaces

(6) H2pX,Qq rÝÑ H0pB,R2π˚QXq d2ÝÑ H2pB,R1π˚QXq eÝÑ H3pX,Qq.

Our goal is to prove the following theorem:

Theorem 6.8. Let π : X Ñ B be a Lagrangian fibration with reduced irreducible fibers.Then the differential

d2 : H0pB,R2π˚QXq Ñ H2pB,R1π˚QXq

in the Leray spectral sequence of π vanishes.

Proof: Assume that X is projective. Lemma 6.4 implies that

H0pB,R2π˚QXq “ H0pB,Qq ‘ H0pB, pR2π˚QXqprimqwhere pR2π˚QXqprim is the kernel of the map Ln´1 : R2π˚QX Ñ R2nπ˚QX . The summandH0pB,Qq is generated by the image of the ample class l of X in H0pR2π˚QXq. Henced2|H0pB,Qq vanishes.

We are left to prove that d2|H0ppR2π˚QXqprimq vanishes. The differentials in the Lerayspectral sequence commute with the Lefschetz maps. In particular, the following diagram


is commutative.H0ppR2π˚QXqprimq d2 //

Ln´1

H2pR1π˚QXqLn´1

H0pR2nπ˚QXq d2 // H2pR2n´1π˚QXqThe vertical arrow on the left-hand side vanishes by the definition of the primitive partof R2π˚QX . The vertical map on the right-hand side is injective. Indeed, by Corollary6.6 the map Ln´1 embeds H2pR1π˚QXq into H2pR2n´1π˚QXq as a direct summand. Itfollows that d2|H0ppR2π˚QXqprimq must vanish.

Step 2: In the case when X is only assumed to be Kähler there exists a degeneratetwistor deformation π1 : X 1 Ñ B of X such that X 1 is projective ([SV, Cor. 3.4]). Thestatement of the theorem holds for X 1. Since topologically the maps π and π1 coincide,the statement of the theorem holds for X as well.

Corollary 6.9. In the setting of Theorem 6.8, the following holds:

(1) The restriction map r : H2pX,Qq Ñ H0pB,R2π˚QXq is surjective.(2) The map e : H2pR1π˚QXq Ñ H3pX,Qq from the exact sequence (6) is injective.

Proof: Follows from Theorem 6.8 and the exact sequence (6).

Proof of Theorem 6.3: It follows from Corollary 6.9 (2) and Proposition 4.4 that thereexists an embedding

pXX0q bZ Q ãÑ H3pX,Qq.In particular, if b3pXq vanishes, XX0 is finite. Since XX0 » H2pB,Γq this finishesthe proof.

Remark 6.10. If π : X Ñ B is an elliptic fibration on a K3 surface, then XX0 is trivial([FM, I.1, Lemma 5.1]).

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Anna Abasheva

Columbia University

Department of Mathematics,

2990 Broadway, New York, NY, USA

also:Independent University of Moscow

Bolshoy Vlasievskiy per., 11,

Moscow, Russia

aa4643(at)columbia.edu

Vasily Rogov

Humboldt Universität zum Berlin

Institut für Mathematik, Rudower

Chaussee 25, 12489 Berlin, Germany

vasirog (at) gmail.com

arXiv:2112.10921v2 [math.AG] 26 Dec 2021

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