# Kähler and non-Kähler Shafarevich–Tate twists

Anna Abasheva

# Shafarevich–Tate groups of holomorphic Lagrangian fibrations II

Anna Abasheva

###### Abstract

Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration
$\pi\colon X\to B$. A Shafarevich–Tate twist of $X$ is a holomorphic
symplectic manifold with a Lagrangian fibration $\pi^{\varphi}\colon
X^{\varphi}\to B$ which is isomorphic to $\pi$ locally over the base. In
particular, $\pi^{\varphi}$ has the same fibers as $\pi$. A twist
$X^{\varphi}$ corresponds to an element $\varphi$ in the Shafarevich–Tate
group $\Sha$ of $X$. We show that $X^{\varphi}$ is Kähler when a multiple of
$\varphi$ lies in the connected component of unity of $\Sha$ and give a
necessary condition for $X^{\varphi}$ to be bimeromorphic to a Kähler
manifold.

##  1 Introduction

###  1.1 Definitions

######  Definition 1.1.1.

An irreducible holomorphic symplectic manifold $X$ is a compact complex simply
connected manifold admitting a closed holomorphic symplectic form $\sigma$
such that $H^{0}(\Omega^{2}_{X})=\mathbb{C}\cdot\sigma$. If $X$ is Kähler,
then we will call $X$ irreducible hyperkähler.

######  Definition 1.1.2.

A Lagrangian fibration on an irreducible holomorphic symplectic manifold $X$
is a morphism $\pi\colon X\to B$ to a normal variety $B$ such that the
restriction of $\sigma$ to a smooth fiber is zero.

If the base $B$ is smooth, then $B$ is necessarily isomorphic to
$\mathbb{P}^{n}$ [Hwa08]. No examples of Lagrangian fibrations over a base
other than $\mathbb{P}^{n}$ have been discovered. By [SY22, Theorem 0.2] Hodge
numbers of $B$ coincide with those of $\mathbb{P}^{n}$.

In the sequel we will sometimes impose an additional assumption on $\pi\colon
X\to B$, namely we will assume that the fibration has no multiple fibers in
codimension one (Definition 2.1.11).

######  Definition 1.1.3.

Define the sheaf $T_{X/B}$ of vertical vector fields on $X$ as the kernel of
the map $T_{X}\to\pi^{*}T_{B}$, where $T_{B}:=(\Omega_{B})^{\vee}$.

The flow of a vertical vector field $v$ induces a vertical automorphism
$\exp(v)$ of $X$.

######  Definition 1.1.4.

Consider the sheaf $Aut^{0}_{X/B}$ on $B$ consisting of all vertical
automorphisms that are locally of the form $\exp(v)$ for some vertical vector
field $v$. The Shafarevich–Tate group of the fibration $\pi\colon X\to B$ is
defined to be $\Sha=H^{1}(B,Aut^{0}_{X/B})$.

The group $\Sha$ has a nice geometric meaning. Cover $B$ by open disks so that
$B=\bigcup U_{i}$. For each subset $I$ of indices, we denote
$\bigcap\limits_{i\in I}U_{i}$ by $U_{I}$ and $\pi^{-1}(U_{I})$ by $X_{I}$.
Every class $\varphi\in\Sha$ can be represented by a Čech $1$-cocycle with
coefficients in $Aut^{0}_{X/B}$. In other words, we have a vertical
automorphism $\varphi_{ij}$ of $X_{ij}$ for each pair of indices $(i,j)$, and

$\varphi_{ik}=\varphi_{jk}\circ\varphi_{ij}.$ (1)

For each $(i,j)$ glue $X_{i}$ to $X_{j}$ by the automorphism $\varphi_{ij}$ to
get a new variety $X^{\varphi}$. By the cocycle condition (1), the variety
$X^{\varphi}$ is a smooth Hausdorff complex manifold admitting a fibration

$\pi^{\varphi}\colon X^{\varphi}\to B.$

######  Definition 1.1.5.

The manifold $X^{\varphi}$ constructed above is called the Shafarevich–Tate
twist of $X$ with respect to the class $\varphi\in\Sha$.

#### 1.1.6.

Shafarevich–Tate group has a structure of a topological group, possibly non-
Hausdorff [AR21, Subsection 3.1]. Denote its connected component of unity by
$\Sha^{0}$. When the fibration $\pi$ has no multiple fibers in codimension
one, the group $\Sha^{0}$ is a quotient of $\mathbb{C}$ by a finitely
generated subgroup and the discrete part $\Sha/\Sha^{0}$ of $\Sha$ satisfies:

$(\Sha/\Sha^{0})\otimes\mathbb{Q}\simeq H^{2}(R^{1}\pi_{*}\mathbb{Q}).$ (2)

For a class $\varphi\in\Sha$, we will denote by $\overline{\varphi}$ its image
in $\Sha/\Sha^{0}\otimes\mathbb{Q}$.

###  1.2 Statement of the results

######  Theorem A.

Let $\pi\colon X\to B$ be a Lagrangian fibration on a projective irreducible
holomorphic symplectic manifold $X$. Pick a class $\varphi\in\Sha$.

  1. 1.

If $\varphi$ is torsion, then $X^{\varphi}$ is projective.

  2. 2.

If $r\varphi$ lies in $\Sha^{0}$ for some positive integer $r$, i.e.
$\overline{\varphi}=0$, then $X^{\varphi}$ is Kähler.

######  Remark 1.2.1.

The first statement of Theorem A holds when $X$ is only assumed to be Kähler.

We proved a version of this theorem in [AR21, Theorem B] for a general
hyperkähler manifold assuming $\pi$ has no multiple fibers in codimension one
and $\varphi\in\Sha^{0}$. The new proof does not require any of this
assumptions.

######  Remark 1.2.2.

A weaker version of Theorem A appeared in [SV24] a few days ago. However, the
proof presented in this paper is simpler, uses completely different ideas, and
works in a more general case.

#### 1.2.3.

The manifold $X^{\varphi}$ is holomorphic symplectic and the fibration
$\pi^{\varphi}$ is a Lagrangian fibration [AR21, Corollary 3.7]. We can show
more.

######  Theorem B.

Let $\pi\colon X\to\mathbb{P}^{n}$ be a Lagrangian fibration on an irreducible
hyperkähler manifold with no multiple fibers in codimension one. Then
$X^{\varphi}$ is irreducible holomorphic symplectic for any $\varphi\in\Sha$.

#### 1.2.4.

In the next theorem we compute the second Betti number of Shafarevich–Tate
twists. Note that the differential $d_{2}$ on the second page of the Leray
spectral sequence of $\mathbb{Q}_{X}$ maps $H^{0}(B,R^{2}\pi_{*}\mathbb{Q})$
to $H^{2}(B,R^{1}\pi_{*}\mathbb{Q})\simeq(\Sha/\Sha^{0})\otimes\mathbb{Q}$.

######  Theorem C.

Let $\pi\colon X\to\mathbb{P}^{n}$ be a Lagrangian fibration with no multiple
fibers in codimension one on an irreducible hyperkähler manifold $X$, and
$\varphi\in\Sha$. Then exactly one of the following two cases occurs.

  1. 1.

If the image $\overline{\varphi}$ of $\varphi$ in
$H^{2}(R^{1}\pi_{*}\mathbb{Q})$ lies in the image of $d_{2}$, then
$b_{2}(X^{\varphi})=b_{2}(X)$. Moreover, there is a cohomology class $h\in
H^{2}(X^{\varphi})$ which restricts to an ample class on a smooth fiber.

  2. 2.

If $\overline{\varphi}$ is not in the image of $d_{2}$, then
$b_{2}(X^{\varphi})=b_{2}(X)-1$. In this case all cohomology classes $h\in
H^{2}(X^{\varphi})$ restrict trivially to a smooth fiber.

######  Definition 1.2.5.

A complex manifold is said to be in Fujiki class $\mathcal{C}$ if it is
bimeromorphic to a Kähler variety.

We will derive the following criterion for non-Kählerness of Shafarevich–Tate
twists as an easy corollary of Theorem A.

######  Theorem D.

Let $\pi\colon X\to\mathbb{P}^{n}$ be a Lagrangian fibration with no multiple
fibers in codimension one on a hyperkähler manifold $X$. Pick $\varphi\in\Sha$
such that $\overline{\varphi}$ is not in the image of $d_{2}$. Then
$X^{\varphi}$ is not in Fujiki class $\mathcal{C}$, in particular, not Kähler.

#### 1.2.6. Outline of the paper.

We start by recalling basic facts abour Lagrangian fibrations and their
Shafarevich–Tate twists in Section 2. Section 3 is devoted to the proof of
Theorem A. Theorems B,C, and D will be proven in Section 4.

#### 1.2.7. Acknowledgements.

I thank my advisor Giulia Saccà, as well as (in alphabetical order) Rodion
Déev, Daniel Huybrechts, Yoon-Joo Kim, Evgeny Shinder, Morena Porzio, Sasha
Viktorova, and Claire Voisin for their interest and helpful conversations.
Sasha Viktorova proofread this paper in record time, and my gratitude to her
is as hot as New York in summer. This paper was completed during my stay in
Oberwolfach, Germany at the workshop “Algebraic Geometry: Wall Crossing and
Moduli Spaces, Varieties and Derived Categories”. I am deeply grateful to the
organizers of the workshop for the invitation and the opportunity to present
the results of this paper. The period when I actively worked on this project
was very hard for me for multiple reasons. I thank everyone who supported me
during this time, especially my friends Masha, Morena, Sasha, and Zoe.

##  2 Preliminaries

###  2.1 Lagrangian fibrations

#### 2.1.1. Beauville-Bogomolov-Fujiki form.

One of the key properties of irreducible hyperkähler manifolds is existence of
the Beauville-Bogomolov-Fujiki form (BBF form) on their second cohomology
group. We can define the BBF form via the following theorem.

######  Theorem 2.1.2.

Let $X$ be an irreducible hyperkähler manifold of dimension $2n$. Then there
exists an integral symmetric non-degenerate form $q$ on $H^{2}(X,\mathbb{Z})$
such that $\forall\alpha\in H^{2}(X,\mathbb{Z})$,

$c_{X}q(\alpha)^{n}=\int_{X}\alpha^{2n}.$

The constant $c_{X}$ is positive and depends only on the deformation type of
$X$.

#### 2.1.3. Fibers of Lagrangian fibrations are abelian varieties.

Consider a Lagrangian fibration $\pi\colon X\to B$ on an irreducible
holomorphic symplectic manifold $X$. A general fiber of $\pi$ is a complex
torus and even an abelian variety [Cam06]. The projectivity of smooth fibers
follows easily from the theorem below.

######  Theorem 2.1.4.

[Voi86] Let $\pi\colon X\to B$ be a Lagrangian fibration on a hyperkähler
manifold, $F$ its smooth fiber. Then the restriction map

$H^{2}(X,\mathbb{Q})\to H^{2}(F,\mathbb{Q})$ (3)

has rank one.

For any Kähler class $h\in H^{2}(X,\mathbb{R})$, some multiple of $h$
restricts to an integral class on $F$. The class $h|_{F}$ is Kähler and
integral, hence ample.

#### 2.1.5. Higher pushforwards of $\mathcal{O}_{X}$.

Matsushita showed that the higher pushforward sheaves
$R^{i}\pi_{*}\mathcal{O}_{X}$ along a Lagrangian fibration are locally free
for all $i\geq 0$ [Mat05]. Let $[\omega]$ be the class of a Kähler form on
$X$. Consider the composition of the maps

$\Omega_{B}\xrightarrow{\sigma}\pi_{*}T_{X/B}\xrightarrow{f_{\omega}}R^{1}\pi_{*}\mathcal{O}_{X}.$
(4)

The map $f_{\omega}$ sends a vector field $v$ to the class of the form
$\iota_{v}\omega$ under the $\bar{\partial}$-differential.

######  Theorem 2.1.6.

[Mat05] Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration. Then
the map $\Omega_{B}\to R^{1}\pi_{*}\mathcal{O}_{X}$ and its wedge powers
$\Omega^{i}_{B}\to R^{i}\pi_{*}\mathcal{O}_{X}$ are isomorphisms.

The base change theorem [Mum12, Chapter 5, Corollary 2&3] implies that the
dimension of $H^{i}(F,\mathcal{O}_{F})$ does not depend on a fiber $F$. In
particular, $h^{0}(\mathcal{O}_{F})=1$ for every fiber.

#### 2.1.7. Discriminant.

The image in $B$ of singular fibers of $\pi$ is called the discriminant of the
Lagrangian fibration and will be denoted by $\Delta$. It is known to be a
divisor [HO09, Proposition 3.1]. We define $B^{0}$ to be the complement of
$\Delta$ and $X^{0}:=\pi^{-1}(B^{0})$.

#### 2.1.8. Vertical vector fields.

The holomorphic symplectic form $\sigma$ on $X$ enables us to construct many
vertical vector fields on $X$. Start with a holomorphic $1$-form $\varphi$ on
some open subset $U\subset B$. Since $\sigma$ is non-degenerate, there exists
a unique vector field $v$ on $\pi^{-1}(U)$ such that

$\iota_{v}\sigma=\pi^{*}\varphi.$

This vector field can be easily shown to be vertical. We have just constructed
a map

$\Omega_{B}\to\pi_{*}T_{X/B},$ (5)

where $T_{X/B}:=\operatorname{ker}(\pi_{*}\colon T_{X}\to\pi^{*}T_{B})$ is the
sheaf of vertical vector fields.

#### 2.1.9. Singular fibers.

While singular fibers of a Lagrangian fibration can be wild, there is a nice
description of singular fibers in codimension one [HO09, Theorem 1.3, 1.4]. We
will not need an explicit description, but instead will state an immediate
corollary of Hwang and Oguiso results.

######  Theorem 2.1.10.

([HO09, Theorem 1.4]) Let $D$ be an irreducible component of the discriminant
$\Delta$. Decompose $\pi^{*}D$ into the sum of its irreducible components with
multiplicities

$\pi^{*}D=\sum\limits_{i=1}^{r}m_{i}D_{i}.$

Let $\mathrm{mul}(D)$ denote the number $\operatorname{gcd}\limits_{1\leq
g\leq r}{(m_{i})}$. Then $m_{i}=\mathrm{mul}(D)$ for some $i$.

In particular, if $\mathrm{mul}(D)=1$, there exists a component of $\pi^{*}D$
of multiplicity one. It follows that for a general point $b$ of $D$, the
fibration $\pi$ has a local section in a neighborhood of $b$.

######  Definition 2.1.11.

A Lagrangian fibration $\pi$ is said to have no multiple fibers in codimension
one if for every irreducible component $D$ of $\Delta$ $\mathrm{mul}(D)=1$.
Equivalently, a Lagrangian fibration with no multiple fibers in codimension
one has a local section in a neighborhood of every point of $B$ outside a
closed subset of codimension at least two.

######  Theorem 2.1.12.

([AR21, Theorem 2.6]) Assume that a Lagrangian fibration $\pi\colon X\to
B=\mathbb{P}^{n}$ has no multiple components in codimension one. Then the map
(5) $\Omega_{B}\to\pi_{*}T_{X/B}$ is an isomorphism. In particular, the sheaf
$\pi_{*}T_{X/B}$ is locally free.

###  2.2 Shafarevich–Tate group

#### 2.2.1. Structure of the Shafarevich–Tate group.

By the definition of $Aut^{0}_{X/B}$, the exponential map $\pi_{*}T_{X/B}\to
Aut^{0}_{X/B}$ is surjective. Define $\Gamma$ to be the kernel of this map.
The short exact sequence

$0\to\Gamma\to\pi_{*}T_{X/B}\to Aut^{0}_{X/B}\to 0$ (6)

induces the long exact sequence of cohomology groups:

$H^{1}(\Gamma)\to H^{1}(\pi_{*}T_{X/B})\to\Sha\to H^{2}(\Gamma).$ (7)

We will call the image of $H^{1}(\pi_{*}T_{X/B})$ in $\Sha$ the connected
component of unity of $\Sha$ and will denote it by $\Sha^{0}$. The quotient
$\Sha/\Sha^{0}$ is the discrete part of $\Sha$.

When $\pi$ has no multiple fibers in codimension one, the sequence is exact on
the right. Indeed, the cohomology group $H^{2}(\pi_{*}T_{X/B})\simeq
H^{1,2}(B)$ vanishes. The vector space $H^{1}(\pi_{*}T_{X/B})$ is isomorphic
to $H^{1,1}(B)\simeq\mathbb{C}$ in this case.

#### 2.2.2. Degenerate twistor deformations.

There is a useful differential geometric point of view on Shafarevich–Tate
deformations [AR21, Subsection 2.3]. Let $\omega$ denote a Kähler form on $B$.
The form $\Omega+t\pi^{*}\omega$ is obviously not holomorphic, but it turns
out that there exists a different complex structure $I_{t}$ on $X$ making
$\Omega+t\pi^{*}\omega$ holomorphic symplectic. Moreover, such a complex
structure is unique.

######  Definition 2.2.3.

Denote by $X_{t}$ the manifold $X$ with the new complex structure $I_{t}$. It
is called a degenerate twistor deformation of $X$.

It is not hard to see that the fibration $\pi\colon X_{t}\to B$ is holomorphic
and Lagrangian with respect to the new complex structure.

######  Theorem 2.2.4.

([AR21, Theorem A]) Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian
fibration with no multiple fibers in codimension one. For every $t\in
H^{1}(\pi_{*}T_{X/B})\simeq\mathbb{C}$ consider its image $\varphi_{t}\in\Sha$
by the map (7). Then the degenerate twistor deformation $X_{t}$ is isomorphic
to the Shafarevich–Tate twist $X^{\varphi_{t}}$ of $X$ by $\varphi_{t}$. This
isomorphism preserves the Lagrangian fibrations.

#### 2.2.5.

Assume that a Lagrangian fibration $\pi\colon X\to B=\mathbb{P}^{n}$ has no
multiple fibers in codimension one. The map $f_{\omega}\colon\pi_{*}T_{X/B}\to
R^{1}\pi_{*}\mathcal{O}_{X}$ from 2.1.5 sends the subsheaf
$\Gamma_{\mathbb{Q}}:=\Gamma\otimes\mathbb{Q}$ isomorphically to
$R^{1}\pi_{*}\mathbb{Q}$ [AR21, Proposition 4.4]. By the exact sequence (7),
the discrete part $\Sha/\Sha^{0}$ satisfies

$(\Sha/\Sha^{0})\otimes\mathbb{Q}\simeq H^{2}(\Gamma_{\mathbb{Q}})\simeq
H^{2}(R^{1}\pi_{*}\mathbb{Q}).$

##  3 Kähler twists

#### 3.0.1.

The goal of this section is to prove Theorem A. First, note that the second
statement of the theorem immediately follows from the first. Indeed, the set
of Kähler deformations of any Kähler manifold is open in any space of
deformations. Torsion elements in $\Sha$ form a dense subset of all elements
$\varphi\in\Sha$ such that $r\varphi\in\Sha^{0}$ for some
$r\in\mathbb{Z}_{>0}$, hence the claim.

######  Remark 3.0.2.

Every Lagrangian fibration on a hyperkähler manifold admits a projective
Shafarevich–Tate deformation. Indeed, by [AR21] for every class $h\in
H^{2}(X,\mathbb{Q})$ such that $h$ restricts non-trivially to a general fiber,
there exists a degenerate twistor deformation $X_{t}$ such that $h$ is of type
(1,1) on $X_{t}$. The value of $t$ is equal to
$-q(\sigma,h)/q(h,\pi^{*}\omega)$, where $q$ is the Beauville-Bogomolov-Fujiki
form and $\omega$ is an ample class on $B$. We can find $h\in
H^{2}(X,\mathbb{Q})$ such that $q(h)>0$ and $t$ is arbitrarily close to zero.
Arbitrarily small deformations of $X$ are Kähler. By [Huy99, Theorem 3.11] a
hyperkähler manifold admitting a rational (1,1)-class $h$ with $q(h)>0$ is
projective, hence $X_{t}$ is projective. Every degenerate twistor deformation
induces a Shafarevich–Tate twist by the proof of [AR21, Theorem A]. Therefore
we can rephrase Theorem A as follows. Let $\pi\colon X\to B$ be a Lagrangian
fibration on a hyperkähler manifold. Then $\forall\varphi\in\Sha$ such that
$\overline{\varphi}\in\Sha/\Sha^{0}\otimes\mathbb{Q}$ vanishes, the twist
$X^{\varphi}$ is Kähler.

#### 3.0.3. Torsion elements in $\Sha$ can be represented by a torsion
cocycle.

In the rest of the section we will suppose that $X$ is projective and
$\varphi\in\Sha$ is a torsion element.

######  Lemma 3.0.4.

Let $\varphi$ be an $r$-torsion element in $\Sha$. Cover $B$ by small open
subsets $U_{i}$ and represent $\varphi$ by a Čech cocycle $(\varphi_{ij})$,
$\varphi_{ij}\in Aut^{0}_{X/B}(U_{ij})$. Then we can choose $\varphi_{ij}$ in
such a way that $r\varphi_{ij}$ is the identity automorphism of $X_{ij}$ for
each $(i,j)$.

######  Proof: .

Since the class of $r\varphi$ is trivial in $\Sha$, we can find automorphisms
$\beta_{i}\in Aut^{0}_{X/B}(U_{i})$ such that

$r\varphi_{ij}=\beta_{j}-\beta_{i}.$

There exist automorphisms $\gamma_{i}$ such that $r\gamma_{i}=\beta_{i}$.
Indeed, we can write $\beta_{i}=\exp(v_{i})$ for some vertical vector field
$v_{i}$. The automorphism $\gamma_{i}:=\exp(v_{i}/r)$ will do the job. Replace
$\varphi_{ij}$ with $\varphi_{ij}+\gamma_{i}-\gamma_{j}$. The new set of
automorphisms satisfies the condition of the lemma. ∎

#### 3.0.5. Gluing of an ample line bundle.

The proof of Theorem A relies on the following idea. Pick an ample line bundle
$L$ on $X$ and cover $B$ by open disks $U_{i}$. Let $L_{i}$ denote the
restriction of $L_{i}$ to $X_{i}$. We will see that for some $s$, the line
bundles $L_{i}^{s}$ can be glued into a line bundle on $X^{\varphi}$. This
result will follow from the lemma below.

######  Lemma 3.0.6.

Let $L$ be a line bundle on an abelian variety $A$ and $t$ an $r$-torsion
element of $A$. Then

$t^{*}L^{r}\simeq L^{r}.$

######  Proof: .

Consider the morphism $\varphi_{L}\colon A\to A^{\vee}$ sending $x$ to
$x^{*}L\otimes L^{-1}$. The map $\varphi_{L}$ is a homomorphism because any
morphism of abelian varieties sending zero to zero is a homomorphism.
Therefore $\varphi_{L}(t)$ is an $r$-torsion line bundle, i.e.,

$(\varphi_{L}(t))^{r}=t^{*}L^{r}\otimes L^{-r}\simeq\mathcal{O}_{A}.$

∎

######  Lemma 3.0.7.

Let $\pi\colon Y\to B$ be a proper flat morphism such that $h^{0}(Y_{b})=1$
for all fibers $F_{b}$. Consider a line bundle $M$ on $Y$ with the following
properties:

  1. 1.

the restriction of $M$ to any smooth fiber is trivial;

  2. 2.

the restriction of $M$ to any fiber $Y_{b}$ lies in $Pic^{0}(Y_{b})$. Here
$Pic^{0}(Y_{b})$ is the connected component of unity of $Pic(Y_{b})$.

Then some positive multiple $M^{s}$ of $M$ is isomorphic to a pullback of a
line bundle from $B$.

######  Proof: .

Consider the group $E_{b}\subset Pic(Y_{b})$ of line bundles $L$ on $Y_{b}$
with the following property: there exists a line bundle $M$ on $Y$ which is
trivial on smooth fibers and restricts to $L$ on $Y_{b}$. By Raynaud’s theorem
[Ray70], $E_{b}$ has dimension $h^{0}(\mathcal{O}_{Y_{b}})-1$. Our assumptions
imply that $E_{b}$ is discrete for any fiber of $\pi$. The line bundle
$M_{b}:=M|_{Y_{b}}$ is in $E_{b}$ by the first property. Since $M_{b}\in
Pic^{0}(Y_{b})$ and $Pic^{0}(Y_{b})$ is of finite type [FGI+05, Proposition
9.5.3], we have two options:

  1. 1.

$M_{b}$ is a torsion line bundle;

  2. 2.

the Zariski closure of the subgroup of $Pic^{0}(Y_{b})$ generated by $M_{b}$
has dimension at least one.

Since $E_{b}$ is discrete, the second alternative does not occur, and $M_{b}$
is torsion. Therefore, some power $M^{s}$ of $M$ restricts trivially to all
fibers. Define a line bundle $K:=\pi_{*}M^{s}$. The natural map $\pi^{*}K\to
M^{s}$ is an isomorphism. ∎

#### 3.0.8.

We are now ready to finish the proof of Theorem A.

###### Proof of Theorem A, 2.

Step 1. As before, choose a Čech cocycle $(\varphi_{ij})$ with
$r\varphi_{ij}=0$ representing an $r$-torsion class $\varphi\in\Sha$. Let $L$
be an ample line bundle on $X$. The line bundle $\varphi_{ij}^{*}L^{r}\otimes
L^{-r}$ restricts trivially to smooth fibers by Lemma 3.0.6. Moreover, it
satisfies the second condition of Lemma 3.0.7 because $\varphi_{ij}\in
Aut^{0}$. Hence its multiple is the pullback of a line bundle on $U_{ij}$.
When the subsets $U_{i}$’s are sufficiently small, all line bundles on
$U_{ij}$’s are trivial. Therefore there is an isomorphism

$f_{ij}\colon\varphi_{ij}^{*}L^{s}|_{\pi^{-1}(U_{ij)}}\to
L^{s}|_{\pi^{-1}(U_{ij)}}.$

for some $s\in\mathbb{Z}_{>0}$.

Step 2. The isomorphisms $f_{ij}$ might not a priori satisfy the cocycle
condition. In other words, the following map

$f_{ij}^{-1}\circ\varphi_{ij}^{*}f_{jk}^{-1}\circ f_{ik}$

is some automorphism of $L|_{U_{ijk}}$, which might not be trivial. Denote it
by $\lambda_{ijk}$. The automorphism $\lambda_{ijk}$ is a multiplication by a
non-zero holomorphic function on $X_{ijk}$, which must be the pullback of a
function on the base. Therefore the automorphisms $\lambda_{ijk}$ define a
Čech 2-cocycle on $B$ with coefficients in $\mathcal{O}_{B}^{\times}$. The
group $H^{2}(B,\mathcal{O}_{B}^{\times})$ is trivial [SY22, Theorem 0.2],
hence the cocycle $(\lambda_{ijk})$ is a coboundary. Therefore, we can assume
that the isomorphisms $(f_{ij})$ satisfy the cocycle condition. It follows
that we can glue the line bundles $L^{s}|_{\pi^{-1}(U_{ij})}$ into a global
line bundle on $X^{\varphi}$. This line bundle is relatively ample, hence
$X^{\varphi}$ is projective. ∎

##  4 Topology of Shafarevich–Tate twists

#### 4.0.1. Higher pushforwards of $\mathbb{Q}_{X}$ do not depend on a twist.

Let $\pi\colon X\to B$ be a Lagrangian fibration and $\pi^{\varphi}\colon
X^{\varphi}\to B$ its Shafarevich–Tate twist. Then the sheaves
$R^{k}\pi_{*}\mathbb{Z}$ and $R^{k}\pi^{\varphi}_{*}\mathbb{Z}$ are
canonically identified. Indeed, represent $\varphi$ as a Čech cocycle
$(\varphi_{ij})$, where $\varphi_{ij}\in Aut^{0}_{X/B}(U_{ij})$. The
automorphisms $\varphi_{ij}$ are flows of vector fields, hence they act
trivially on $H^{k}(X_{ij})$.

In particular the vector spaces $H^{0}(R^{2}\pi_{*}\mathbb{Q})$ and
$H^{0}(R^{2}\pi^{\varphi}_{*}\mathbb{Q})$ are canonically identified. However,
the differentals

$d_{2}\colon H^{0}(R^{2}\pi_{*}\mathbb{Q})\to
H^{2}(R^{1}\pi_{*}\mathbb{Q})\>\>\>\>\>\text{and}\>\>\>\>\>d_{2}^{\varphi}\colon
H^{0}(R^{2}\pi_{*}\mathbb{Q})\to H^{2}(R^{1}\pi_{*}\mathbb{Q})$

from the Leray spectral sequence of $X$ and $X^{\varphi}$ respectively may be
different.

###  4.1 Fundamental group of a twist

#### 4.1.1. First cohomology group of a twist vanishes.

Suppose that $X$ is hyperkähler. For any surjective map of compact Kähler
manifolds $f\colon Y\to Z$ the pullback map $H^{i}(Z,\mathbb{Q})\to
H^{i}(Y,\mathbb{Q})$ is injective for all $i$. Therefore, for a Lagrangian
fibration $\pi\colon X\to B=\mathbb{P}^{n}$ the pullback map
$H^{2}(B,\mathbb{Q})\to H^{2}(X,\mathbb{Q})$ is injective. Since
$H^{2}(B,\mathcal{R})\simeq\mathcal{R}$ for any coefficient ring
$\mathcal{R}$, the map $H^{2}(B,\mathcal{R})\to H^{2}(X,\mathcal{R})$ is
injective as well. It follows from Leray spectral sequence that

$0\to H^{1}(B,\mathcal{R})\to H^{1}(X,\mathcal{R})\to
H^{0}(B,R^{1}\pi_{*}\mathcal{R})\to 0$ (8)

is exact. Since $B$ and $X$ are simply connected, the group
$H^{0}(B,R^{1}\pi_{*}\mathbb{Z})$ vanishes. Therefore, for any
Shafarevich–Tate twist $X^{\varphi}$,

$H^{1}(X^{\varphi},\mathcal{R})\simeq H^{1}(B,\mathcal{R})=0.$ (9)

The universal coefficient theorem implies that
$H_{1}(X^{\varphi},\mathbb{Z})=0$ for any twist $X^{\varphi}$ of $X$.

######  Theorem 4.1.2.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration on an
irreducible hyperkähler manifold $X$ with no multiple fibers in codimension
one, and $X^{\varphi}$ its Shafarevich–Tate twist. Then $X^{\varphi}$ is
simply connected.

######  Proof: .

Since $H_{1}(X^{\varphi},\mathbb{Z})=0$, it is enough to show that
$\pi_{1}(X^{\varphi},x)$ is abelian. The open subset
$(X^{\varphi})^{0}=(\pi^{\varphi})^{-1}(B^{0})$ is the complement of a
divisor, hence $\pi_{1}((X^{\varphi})^{0},x)$ surjects onto
$\pi_{1}(X^{\varphi},x)$. The fibration $(X^{\varphi})^{0}\to B^{0}$ is
smooth, hence the following sequence is exact:

$\pi_{1}(F,x)\to\pi_{1}((X^{\varphi})^{0},x)\to\pi_{1}(B^{0},y)\to 0.$ (10)

The fundamental group of $B^{0}$ is generated by loops around irreducible
components of $\Delta$. More precisely, let $b\in\Delta$ be a smooth point and
$U_{b}$ an analytic neighborhood of $b$. Choose a non-trivial loop in
$U_{b}\setminus\Delta$ and connect it with the point $y=\pi(x)\in B^{0}$ to
obtain a loop $\gamma_{b}$. The fundamental group $\pi_{1}(B^{0},y)$ is
generated by classes of $\gamma_{b}$ for some $b\in\Delta$.

The fibration $\pi^{\varphi}\colon X^{\varphi}\to B$ admits a section in a
neighborhood of a general point of $\Delta$. Therefore, we can lift a loop
$\gamma_{b}$ in $B^{0}$ to a loop $\tilde{\gamma}_{b}$ in $X^{0}$ which is
contractible in $X$. It follows from the exact sequence (10) and the
surjectivity of the map
$\pi_{1}((X^{\varphi})^{0},x)\to\pi_{1}(X^{\varphi},x)$ that the map

$\pi_{1}(F,x)\to\pi_{1}(X^{\varphi},x)$

is surjective. The group $\pi_{1}(F,x)$ is abelian, hence so is
$\pi_{1}(X^{\varphi},x)$. ∎

###  4.2 Hodge numbers of twists

Recall that by [AR21, Corollary 3.7] a Shafarevich–Tate twist $X^{\varphi}$
inherits a holomorphic symplectic form $\sigma$ from $X$.

######  Proposition 4.2.1.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration with no
multiple fibers in codimension one on an irreducible hyperkähler manifold $X$.
Then $\forall\varphi\in\Sha$:

$H^{0,k}(X^{\varphi})=H^{k}(\mathcal{O}_{X^{\varphi}})=\begin{cases}0,\text{
if }k\text{ is odd;}\\\ \mathbb{C}\cdot\overline{\sigma}^{k/2},\text{ if
}k\text{ is even.}\end{cases}$ (11)

######  Proof: .

For any $\varphi\in\Sha$, the sheaf
$R^{i}\pi^{\varphi}_{*}\mathcal{O}_{X^{\varphi}}$ is isomorphic to
$\Omega^{i}_{B}$ by Theorem 2.1.6. The Leray spectral sequence for
$\mathcal{O}_{X^{\varphi}}$ has the form

$E_{2}^{p,q}=H^{q}(R^{p}\pi_{*}\mathcal{O}_{X^{\varphi}})\simeq
H^{q}(\Omega^{p}_{B})=\begin{cases}0,\text{ if }p\neq q;\\\ \mathbb{C},\text{
otherwise.}\end{cases}$

This spectral sequence degenerates at $E^{2}$, hence $H^{0,k}(X^{\varphi})=0$
for $k$ odd and $H^{0,k}(X^{\varphi})=H^{k/2,k/2}(B)=\mathbb{C}$ when $k$ is
even.

The form $\overline{\sigma}^{d}$ is $d$-closed and not $\bar{\partial}$-exact.
Indeed, if $\overline{\sigma}^{r}=\bar{\partial}\alpha$, then

$0=\int_{X}d(\alpha\overline{\sigma}^{(n-r)}\sigma^{n})=\int_{X}\overline{\sigma}^{n}\sigma^{n}\neq
0,$

contradiction. Hence the class of $\overline{\sigma}^{r}$ is non-trivial in
$H^{0,2r}(X^{\varphi})$ and thus generates $H^{0,2r}(X^{\varphi})$. ∎

######  Lemma 4.2.2.

Let $\pi\colon Y\to S$ be a Lagrangian fibration with no multiple fibers in
codimension one over a smooth possibly non-compact base. Consider a
holomorphic $k$-form $\alpha$ on $Y$. Suppose that the restriction of $\alpha$
to $Y^{0}=\pi^{-1}(S^{0})$ satisfies

$\alpha|_{Y^{0}}=\pi^{*}\beta^{0}$

for some holomorphic $k$-form $\beta^{0}$ on $S^{0}$. Then
$\alpha=\pi^{*}\beta$ for some form $\beta$ on $S$.

######  Proof: .

Let $b\in\Delta$ be a general point of the discriminant. The fibration $\pi$
admits a local section $s$ in a neighborhood of $b$. The form $s^{*}\alpha$
coincides with $\beta^{0}$ over $S^{0}$, hence $\beta$ extends to a
holomorphic form in codimension one. By Hartogs theorem, $\beta^{0}$ extends
to a holomorphic form $\beta$ on $S$. The forms $\pi^{*}\beta$ and $\alpha$
coincide on an open subset, hence coincide on $Y$. ∎

######  Theorem 4.2.3.

As before, let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration
with no multiple fibers in codimension one on an irreducible hyperkähler
manifold $X$. Pick $\varphi\in\Sha$. Then $H^{0}(\Omega^{2}_{X^{\varphi}})$ is
generated by $\sigma$.

######  Proof: .

Let $\xi$ be a holomorphic $2$-form on $X^{\varphi}$. The restriction of $\xi$
to every smooth fiber is $d$-closed because all holomorphic forms on Kähler
manifolds are closed. Therefore, $\xi$ defines a section of the local system
$R^{2}\pi_{*}\mathbb{C}|_{B^{0}}$. This local system has just one non-trivial
section, which is the class of a form of type (1,1). The class $[\xi|_{F}]$ is
of type (2,0), hence it must be trivial. There are no non-trivial exact
holomorphic forms on $F$, hence $\xi|_{F}=0$ for every smooth fiber.

Consider the map

$\iota_{\xi}\colon\colon T_{X/B}\to\Omega_{X}$

sending a vector field $v$ to $\iota_{v}\xi$. Compose this map with the
natural map $\Omega_{X}\to\Omega_{X/B}=\Omega_{X}\pi^{*}\Omega_{B}$. The map

$T_{X/B}\to\Omega_{X/B}$

vanishes on $X^{0}$, hence the image of $T_{X/B}$ in $\Omega_{X/B}$ is a
torsion subsheaf. It follows that the image of $\iota_{\xi}$ lies in the sheaf
$(\pi^{*}\Omega_{B})^{sat}$ consisting of $1$-forms $\alpha$ such
$f\alpha\in\pi^{*}\Omega_{B}$ for some holomorphic function $f$. By taking
pushforwards, we obtain a map

$\iota_{\xi}\colon\pi_{*}T_{X/B}\to\pi_{*}(\pi^{*}\Omega^{1}_{B})^{sat}.$

Note that $(\pi^{*}\Omega^{1}_{B})^{sat}\simeq\Omega_{B}$. Indeed, this is
definitely true over $B^{0}$. Let $\alpha$ be a local section of
$(\pi^{*}\Omega^{1}_{B})^{sat}$. Then the restriction of $\alpha$ to $X^{0}$
is the pullback of a form on the base. By Lemma 4.2.2 the form $\alpha$ must
be the pullback of a form on the base.

The space of homorphisms between $\pi_{*}T_{X/B}$ and $\Omega^{1}_{B}$ is one-
dimensional because

$\mathrm{Hom}(\pi_{*}T_{X/B},\Omega^{1}_{B})\simeq\mathrm{End}(\Omega^{1}_{B})\simeq\mathbb{C}.$

Hence, $\iota_{\xi}$ is proportional to $\iota_{\sigma}$. We obtain that for
some coefficient $\lambda$, the form $\xi^{\prime}=\xi-\lambda\sigma$
satisfies

$\iota_{v}\xi^{\prime}=0$ (12)

for any vertical vector field $v$.

Consider the restriction of $\xi^{\prime}$ to $X^{0}$. It follows from the
property (12) that $\xi^{\prime}\in H^{0}(X^{0},\pi^{*}\Omega_{B^{0}})\simeq
H^{0}(B^{0},\Omega_{B^{0}})$. Hence there exists a holomorphic $2$-form
$\alpha^{0}$ on $B^{0}$ such that $\xi^{\prime}|_{X^{0}}=\pi^{*}\alpha^{0}$.
By Lemma 4.2.2, $\xi^{\prime}=\pi^{*}\alpha$ for some holomorphic $2$-form
$\alpha$ on $\mathbb{P}^{n}$. There are no holomorphic two forms on $B$,
therefore, $\xi^{\prime}=0$ and $\xi=\lambda\sigma$, as desired. ∎

###  4.3 Second cohomology of a twist

#### 4.3.1. The restriction map $H^{2}(X)\to H^{2}(F)$ has rank at most one.

Suppose that $X$ is hyperkähler and let $F$ be a smooth fiber of $\pi$. By
Theorem 2.1.4, the restriction map $H^{2}(X)\to H^{2}(F)$ has a one-
dimensional image generated by an ample class. Global invariant cycle theorem
implies that

$\operatorname{im}\left(H^{2}(X,\mathbb{Q})\to
H^{2}(F,\mathbb{Q})\right)=H^{2}(F,\mathbb{Q})^{\pi_{1}(B^{0})}=H^{0}(B^{0},R^{2}\pi_{*}\mathbb{Q}|_{B^{0}}).$

Here, $H^{2}(F,\mathbb{Q})^{\pi_{1}(B^{0})}$ denotes the subspace of
$H^{2}(F)$ invariant under the monodromy action of $\pi_{1}(B^{0})$. It
follows that $H^{2}(F,\mathbb{Q})^{\pi_{1}(B^{0})}$ is one-dimensional and
generated by an ample class.

Let $X^{\varphi}$ be a Shafarevich–Tate twist of $X$, not necessarily Kähler.
Then the image of the map $H^{2}(X^{\varphi},\mathbb{Q})\to
H^{2}(F,\mathbb{Q})$ lies in $H^{2}(F,\mathbb{Q})^{\pi_{1}(B^{0})}$. This
subspace is isomorphic to $\simeq
H^{0}(B^{0},R^{2}\pi_{*}\mathbb{Q}|_{B^{0}})$, hence does not depend on a
twist. We obtain the following statement.

######  Proposition 4.3.2.

Let $\pi\colon X\to B$ be a Lagrangian fibration on an irreducible hyperkähler
manifold $X$ and $X^{\varphi}$ be its Shafarevich–Tate twist. Then the
restriction map

$H^{2}(X^{\varphi})\to H^{2}(F)$

is either zero or has a one-dimensional image generated by an ample class of
$F$.

######  Lemma 4.3.3.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration with no
multiple fibers in codimension one on an irreducible hyperkähler manifold $X$.
Define the sheaf $\mathcal{NS}$ on $B$ as the image of the Chern class map
$R^{1}\pi_{*}\mathcal{O}_{X}^{\times}\to R^{2}\pi_{*}\mathbb{Z}$. Then

$H^{0}(B,\mathcal{NS})=H^{0}(R^{2}\pi_{*}\mathbb{Z}),$

i.e., for every section $\xi$ of $R^{2}\pi_{*}\mathbb{Z}$ there are line
bundles $L_{i}$ on $X_{i}$ such that $\xi|_{U_{i}}=c_{1}(L_{i})$.

######  Proof: .

Consider the exponential exact sequence

$0\to\mathbb{Z}_{X}\to\mathcal{O}_{X}\to\mathcal{O}_{X}^{\times}\to 0.$

It induces the long exact sequence of pushforward sheaves:

$R^{1}\pi_{*}\mathcal{O}_{X}^{\times}\to R^{2}\pi_{*}\mathbb{Z}\to
R^{2}\pi_{*}\mathcal{O}_{X}.$

The sheaf
$R^{2}\pi_{*}\mathbb{Z}/\operatorname{im}(R^{1}\pi_{*}\mathcal{O}_{X}^{\times})=R^{2}\pi_{*}\mathbb{Z}/\mathcal{NS}$
is a subsheaf of $R^{2}\pi_{*}\mathcal{O}_{X}\simeq\Omega^{2}_{B}$. Since
$H^{0}(\Omega^{2}_{B})=0$, the sheaf $R^{2}\pi_{*}\mathbb{Z}/\mathcal{NS}$ has
no global sections. Hence the natural inclusion $H^{0}(B,\mathcal{NS})\to
H^{0}(R^{2}\pi_{*}\mathbb{Z})$ is an isomorphism. ∎

#### 4.3.4. Isomorphisms between $T_{X/B}$ and $R^{1}\pi_{*}\mathcal{O}_{X}$.

As before, assume that $\pi$ has no multiple fibers in codimension one and the
base is $\mathbb{P}^{n}$. Let $\xi$ be a global section of
$H^{0}(R^{2}\pi_{*}\mathbb{Q})$. Then it defines a map
$f_{\xi}\colon\pi_{*}T_{X/B}\to R^{1}\pi_{*}\mathcal{O}_{X}$ in a similar way
that a class $\omega\in H^{2}(X,\mathbb{Q})$ defines a map $f_{\omega}$ in
2.1.5. By Lemma 4.3.3 we can represent $\xi|_{U_{i}}$ by closed $(1,1)$-form
$\xi_{i}$’s on $X_{i}$. Consider the map

$f_{\xi_{i}}\colon\pi_{*}T_{X_{i}/U_{i}}\to
R^{1}\pi_{*}\mathcal{O}_{X}|_{U_{i}}$

sending $v$ to the class of $[\iota_{v}\xi_{i}]$ under the
$\bar{\partial}$-differential. Since the sheaves $\pi_{*}T_{X/B}$ and
$R^{1}\pi_{*}\mathcal{O}_{X}$ are locally free, the map $f_{\xi_{i}}$ is
determined completely by its restriction to $B^{0}\cap U_{i}$. Therefore, the
map $f_{\xi_{i}}$ depends only on the class $[\xi|_{F}]$ of the restriction of
$\xi$ to smooth fibers. In particular, maps $f_{\xi_{i}}$ do not depend on the
choice of $\xi_{i}$ representing $\xi\in H^{0}(R^{2}\pi_{*}\mathbb{Q})$ and
glue into a map

$f_{\xi}\colon\pi_{*}T_{X/B}\to R^{1}\pi_{*}\mathcal{O}_{X}.$

#### 4.3.5.

The map
$H^{0}(B,R^{2}\pi_{*}\mathbb{C})\to\operatorname{Hom}(\pi_{*}T_{X/B},R^{1}\pi_{*}\mathcal{O}_{X})$
sending $\xi$ to $f_{\xi}$ factors through the restriction to a smooth fiber:

$H^{0}(B,R^{2}\pi_{*}\mathbb{C})\to
H^{0}(B^{0},R^{2}\pi_{*}\mathbb{C})=H^{2}(F)^{\pi_{1}(B_{0})}\to\operatorname{Hom}(\pi_{*}T_{X/B},R^{1}\pi_{*}\mathcal{O}_{X}).$
(13)

The vector space $H^{2}(F)^{\pi_{1}(B_{0})}$ is one-dimensional by 4.3.1. The
sheaves $\pi_{*}T_{X/B}$ and $R^{1}\pi_{*}\mathcal{O}_{X}$ are both isomorphic
to $\Omega_{B}$, hence

$\operatorname{Hom}(\pi_{*}T_{X/B},R^{1}\pi_{*}\mathcal{O}_{X})\simeq\operatorname{End}(\Omega_{B})=\mathbb{C}.$

Therefore, the map
$H^{2}(F)^{\pi_{1}(B^{0})}\to\operatorname{Hom}(\pi_{*}T_{X/B},R^{1}\pi_{*}\mathcal{O}_{X})$
is an isomorphism. Fix an element $\xi_{0}\in H^{0}(B,R^{2}\pi_{*}\mathbb{Z})$
which restricts non-trivially to $F$ and let $f_{0}:=f_{\xi_{0}}$ be the
induced isomorphism $\pi_{*}T_{X/B}\to R^{1}\pi_{*}\mathcal{O}_{X}$. For every
$\xi\in H^{0}(B,R^{2}\pi_{*}\mathbb{C})$ we define a number $\lambda_{\xi}$ by
the identity

$f_{\xi}=\lambda_{\xi}f_{0}.$ (14)

Recall that the isomorphism $f_{0}\colon\pi_{*}T_{X/B}\to
R^{1}\pi_{*}\mathcal{O}_{X}$ sends
$\Gamma_{\mathbb{Q}}=\Gamma\otimes\mathbb{Q}\subset\pi_{*}T_{X/B}$
isomorphically onto $R^{1}\pi_{*}\mathbb{Q}\subset
R^{1}\pi_{*}\mathcal{O}_{X}$ by 2.2.5. We identify the group
$H^{2}(\Gamma_{\mathbb{Q}})=(\Sha/\Sha^{0})\otimes\mathbb{Q}$ with
$H^{2}(R^{1}\pi_{*}\mathbb{Q})$ using the isomorphism $f_{0}$.

#### 4.3.6. Boundary map $\Sha\to H^{2}(\Gamma)$.

The boundary map $\Sha\to H^{2}(\Gamma)$ can be described in terms of Čech
cocycles as follows. Pick $\varphi\in\Sha$ and represent it by a $1$-cocycle
$\varphi_{ij}\in Aut^{0}_{X/B}(U_{ij})$. We can find a vertical vector field
$v_{ij}$ on $X_{ij}$ such that $\exp(v_{ij})=\varphi_{ij}$. The vector field
$v_{ij}+v_{jk}+v_{ki}$ on $X_{ijk}$ lies in $\Gamma$ and represents the class
$\overline{\varphi}\in H^{2}(\Gamma)$.

#### 4.3.7. Boundary map $H^{0}(R^{2}\pi_{*}\mathbb{Q})\to
H^{2}(R^{1}\pi_{*}\mathbb{Q})$.

We will describe the boundary map $d_{2}\colon
H^{0}(R^{2}\pi_{*}\mathbb{Q})\to H^{2}(R^{1}\pi_{*}\mathbb{Q})\simeq
H^{2}(\Gamma_{\mathbb{Q}})$ from the Leray spectral sequence of $\pi$ in terms
of Čech cocycles. Let $\xi$ be a section of $H^{0}(R^{2}\pi_{*}\mathbb{Q})$.
Represent it locally by $(1,1)$-forms $\xi_{i}$ on $X_{i}$. The difference
$\xi_{j}-\xi_{i}$ is an exact form, hence

$\xi_{j}-\xi_{i}=d\rho_{ij}$

for some $1$-form $\rho_{ij}$ on $X_{ij}$. The form
$\rho_{ij}+\rho_{jk}+\rho_{ki}$ is closed on $X_{ijk}$, hence defines a
cocycle with coefficients in $R^{1}\pi_{*}\mathbb{Q}$. By 2.2.5 there exists a
unique vertical vector field $w_{ijk}$ on $X_{ijk}$ such that $f_{0}(w_{ijk})$
is equal to the class of the (0,1)-form
$\rho^{0,1}_{ij}+\rho^{0,1}_{jk}+\rho^{0,1}_{ki}$ under the
$\bar{\partial}$-differential.

######  Proposition 4.3.8.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration with no
multiple fibers in codimension one. Pick a class $\varphi\in\Sha$. Let

$d_{2}^{\varphi}\colon H^{0}(R^{2}\pi_{*}\mathbb{Q})\to
H^{2}(R^{1}\pi_{*}\mathbb{Q})\simeq H^{1}(\Gamma_{\mathbb{Q}})$

be the differential in the Leray spectral sequence for $\pi^{\varphi}\colon
X^{\varphi}\to B$. Then for any $\xi\in H^{0}(R^{2}\pi_{*}\mathbb{Q})$

$d_{2}^{\varphi}(\xi)=d_{2}(\xi)+\lambda_{\xi}\overline{\varphi}.$

######  Proof: .

Represent $\xi\in H^{0}(R^{2}\pi_{*}\mathbb{Q})$ by a collection of closed
$(1,1)$-forms $\xi_{i}$ on $X_{i}$. When we view $X_{i}$ as an open subset of
$X^{\varphi}$, we will denote the same forms by $\xi_{i}^{\varphi}$. The
difference $\xi_{j}^{\varphi}-\xi_{i}^{\varphi}$ is not the same as
$\xi_{j}-\xi_{i}$ because there is a twist by $\varphi_{ij}\in
Aut^{0}_{X/B}(U_{ij})$ involved. Namely

$\xi_{j}^{\varphi}-\xi_{i}^{\varphi}=\varphi_{ij}^{*}\xi_{j}-\xi_{i}=(\varphi_{ij}^{*}\xi_{j}-\xi_{j})+(\xi_{j}-\xi_{i}).$

As in 4.3.7, write $\xi_{j}-\xi_{i}=d\rho_{ij}$. Find a vector field $v_{ij}$
such that $\varphi_{ij}=\exp(v_{ij})$. Then

$\varphi_{ij}^{*}\xi_{j}-\xi_{j}=\int\limits_{0}^{1}\frac{d}{dt}\exp(tv_{ij})^{*}\xi_{j}dt=\int\limits_{0}^{1}\exp(tv_{ij})^{*}\mathrm{L}_{v_{ij}}\xi_{j}dt=\\\
=\int\limits_{0}^{1}\exp(tv_{ij})^{*}(d\iota_{v_{ij}}+\iota_{v_{ij}}d)\xi_{j}dt=d\int\limits_{0}^{1}\exp(tv_{ij})^{*}\iota_{v_{ij}}\xi_{j}dt.$

Set $\gamma_{ij}=\int\limits_{0}^{1}\exp(tv_{ij})^{*}\iota_{v_{ij}}\xi_{j}dt$.
Then

$\xi_{j}^{\varphi}-\xi_{i}^{\varphi}=d(\rho_{ij}+\gamma_{ij})$

The form $\gamma_{ij}$ is of type $(0,1)$ and $\bar{\partial}$-closed. Its
class under the $\bar{\partial}$-differential is

$\int\limits_{0}^{1}[\exp(tv_{ij})^{*}\iota_{v_{ij}}\xi_{j}]dt=\int\limits_{0}^{1}[\iota_{v_{ij}}\xi_{j}]dt=[\iota_{v_{ij}}\xi_{j}]=f_{\xi}(v_{ij})=\lambda_{\xi}f_{0}(v_{ij}).$

As in 4.3.7, let $w_{ijk}$ be the element in $\Gamma_{\mathbb{Q}}(U_{ijk})$
such that $f_{0}(w_{ijk})=[\rho^{0,1}_{ij}+\rho^{0,1}_{jk}+\rho^{0,1}_{ki}]$.
It represents $d_{2}(\xi)$. Then $d_{2}^{\varphi}(\xi)-d^{2}(\xi)$ can be
represented by the cocycle

$\lambda_{\xi}(v_{ij}+v_{jk}+v_{ki}).$

The cocycle $v_{ij}+v_{jk}+v_{ki}$ represents $\overline{\varphi}$ by 4.3.6,
hence the claim.

∎

######  Corollary 4.3.9.

As before, let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration
with no multiple fibers in codimension one. Consider the restriction maps

$r\colon H^{2}(X)\to H^{0}(R^{2}\pi_{*}\mathbb{Q}),\>\>\>\>\>r^{\varphi}\colon
H^{2}(X^{\varphi})\to H^{0}(R^{2}\pi_{*}\mathbb{Q}).$

Let $H^{2}(X)^{0}$ (resp. $H^{2}(X^{\varphi})^{0})$ denote the subspace of
classes in $H^{2}(X)$ (resp. $H^{2}(X^{\varphi}))$ that restrict trivially to
a smooth fiber. Then

$\operatorname{im}r\cap\operatorname{im}r^{\varphi}=r(H^{2}(X)^{0})=r^{\varphi}(H^{2}(X^{\varphi})^{0}).$
(15)

######  Proof: .

The image of the restriction map $r$ (resp. $r^{\varphi}$) coincides with the
kernel of $d_{2}$ (resp. $d_{2}^{\varphi}$). By Proposition 4.3.8 a class
$\xi$ lies in the kernel of both $d_{2}$ and $d_{2}^{\varphi}$ if and only if
$\lambda_{\xi}=0$, i.e., the restriction of $\xi$ to a smooth fiber is
trivial. The claim follows. ∎

######  Proposition 4.3.10.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration on an
irreducible hyperkähler manifold and $X^{\varphi}$ its Shafarevich–Tate twist.
Then either $b_{2}(X^{\varphi})=b_{2}(X)$ or $b_{2}(X^{\varphi})=b_{2}(X)-1$.
The first case occurs if and only if there is a class $h\in H^{2}(X)$ whose
restriction to a smooth fiber is non-trivial.

######  Proof: .

It follows from the Leray spectral sequence for $\mathbb{Q}_{X^{\varphi}}$
that

$b_{2}(X^{\varphi})=b_{2}(B)+\dim
H^{1}(R^{1}\pi_{*}\mathbb{Q})+\operatorname{rk}r^{\varphi}.$ (16)

Indeed, the vector space $H^{0}(R^{1}\pi_{*}\mathbb{Q})$ vanishes by 4.1.1,
hence the map $H^{2}(B)\to H^{2}(X^{\varphi})$ is injective. Moreover, the
second differential $d_{2}\colon H^{1}(R^{1}\pi_{*}\mathbb{Q})\to
H^{3}(B,\mathbb{Q})$ is zero because $H^{3}(B,\mathbb{Q})=0$. The formula (16)
implies that

$b_{2}(X^{\varphi})-b_{2}(X)=\operatorname{rk}r^{\varphi}-\operatorname{rk}r$

Since $X$ is projective, the subspace $H^{2}(X)^{0}$ has codimension $1$ in
$H^{2}(X)$, hence

$\operatorname{rk}r=\dim r(H^{2}(X)^{0})+1=\dim
r^{\varphi}(H^{2}(X^{\varphi})^{0})+1,$

where the last equality holds by Corollary 4.3.9. If there is a class in
$H^{2}(X^{\varphi})$ restricting non-trivially to a smooth fiber, then
$H^{2}(X^{\varphi})^{0}$ is a hyperplane in $H^{2}(X^{\varphi})$ and

$\operatorname{rk}r^{\varphi}=\dim
r^{\varphi}(H^{2}(X^{\varphi})^{0})+1=\operatorname{rk}r.$

In this case, $b_{2}(X)=b_{2}(X^{\varphi})$. Otherwise,
$H^{2}(X^{\varphi})^{0}=H^{2}(X^{\varphi})$, hence

$\operatorname{rk}r^{\varphi}=\dim
r^{\varphi}(H^{2}(X^{\varphi})^{0})=\operatorname{rk}r-1,$

and

$b_{2}(X)=b_{2}(X^{\varphi})+1.$

∎

######  Proposition 4.3.11.

Let $\pi\colon X\to B=\mathbb{P}^{n}$ be a Lagrangian fibration on with no
multiple fibers in codimension one and $X^{\varphi}$ its Shafarevich–Tate
twist. Then there is a class $h\in H^{2}(X^{\varphi})$ which restricts non-
trivially to a smooth fiber if and only if $\overline{\varphi}\in
H^{2}(\Gamma_{\mathbb{Q}})$ is not in the image of the boundary map
$d_{2}\colon H^{0}(R^{2}\pi_{*}\mathbb{Q})\to
H^{2}(R^{1}\pi_{*}\mathbb{Q})\simeq H^{2}(\Gamma_{\mathbb{Q}})$.

######  Proof: .

Suppose that there is a class $h\in H^{2}(X^{\varphi})$ which restricts non-
trivially to a smooth fiber. Let $\overline{h}$ be its image in
$H^{0}(B,R^{2}\pi_{*}\mathbb{Q})$. By Proposition 4.3.8

$0=d_{2}^{\varphi}(\overline{h})=d_{2}(h)+\lambda_{h}\overline{\varphi}.$

Therefore, $\overline{\varphi}=-d_{2}(h)/\lambda_{h}$ is in the image of
$d_{2}$.

Conversely, suppose that $\overline{\varphi}$ is in the image of $d_{2}$,
i.e., there is a class $\xi\in H^{0}(R^{2}\pi_{*}\mathbb{Q})$ such that
$d_{2}\xi=\overline{\varphi}$. Let $h_{0}$ be an ample class of $X$ and
$\overline{h_{0}}$ its image in $H^{0}(R^{2}\pi_{*}\mathbb{Q})$. Consider the
class

$h:=(1+\lambda_{\xi})\overline{h_{0}}-\xi.$

Then $\lambda_{h}=1$ and

$d_{2}^{\varphi}(h)=d_{2}(h)+\overline{\varphi}=(1+\lambda_{\xi})d_{2}(\overline{h_{0}})-d_{2}(\xi)+\overline{\varphi}=-d_{2}(\xi)+\overline{\varphi}=0.$

Therefore, $h$ lifts to a class in $H^{2}(X^{\varphi})$, which restricts non-
trivially to smooth fibers. ∎

Propositions 4.3.10 and 4.3.11 immediately imply Theorem C.

###### Proof of Theorem D.

Suppose $X^{\varphi}$ is in Fujiki class $\mathcal{C}$, i.e., there is a
rational map $f\colon X^{\varphi}\dashrightarrow Y$ to a Kähler manifold $Y$.
Let $h\in H^{2}(X,\mathbb{R})$ be the pullback of a Kähler form on $Y$. The
restriction of $f$ to a general fiber $F$ of $\pi^{\varphi}$ is birational
onto its image, hence $h|_{F}$ is non-trivial. By Theorem C, the twist
satisfies $\overline{\varphi}\in\operatorname{im}d_{2}$. ∎

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Anna Abasheva
Columbia University,
Department of Mathematics,
2990 Broadway,
New York, NY, USA
aa4643(at)columbia(dot)edu