# Isotrivial Lagrangian fibrations of compact hyper-kähler manifolds

Yoon-Joo Kim  Department of Mathematics, Columbia University, New York, NY
10027<EMAIL_ADDRESS>,  Radu Laza  Department of Mathematics, Stony
Brook University, New York, NY 11794<EMAIL_ADDRESS>and  Olivier
Martin  Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320
<EMAIL_ADDRESS>
###### Abstract.

This article initiates the study of isotrivial Lagrangian fibrations of
compact hyper-Kähler manifolds. We present four foundational results that
extend well-known facts about isotrivial elliptic fibrations of K3 surfaces.
First, we prove that smooth fibers of an isotrivial Lagrangian fibration are
isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy
between two types of isotrivial Lagrangian fibrations, which we call A and B.
Third, we give a classification result for type A isotrivial Lagrangian
fibrations assuming a well-known conjecture on the base of Lagrangian
fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a
rational section, then it is birational to one of two straightforward examples
of isotrivial fibrations of hyper-Kähler manifolds of $\text{K3}^{[n]}$-type
and $\text{Kum}_{n}$-type. Finally, we prove that a genericity assumption on
the smooth fiber of an isotrivial Lagrangian fibration ensures that the
fibration is of type A.

##  1\. Introduction

Hyper-Kähler manifolds are one of the three building blocks of compact Kähler
manifolds with trivial canonical bundles. Due to their rich geometry and
relation to other objects of interest, they have become the subject of intense
study over the past decades. However, only few examples of such manifolds are
known, namely two deformation types, $\text{K3}^{[n]}$ and $\textup{Kum}_{n}$,
in each even dimension $2n\geq 4$, and two sporadic examples, OG6 and OG10, in
dimensions 6 and 10. A central question in the field is whether there are
finitely many deformation types in each dimension.

One possible approach to this problem is via Lagrangian fibrations as in
[Saw16]. Specifically, it is expected that all deformations types of hyper-
Kähler manifolds admit specializations to Lagrangian-fibered hyper-Kähler
manifolds, thereby possibly reducing this boundedness problem to the study of
families of abelian varieties. Bakker [Bak22], improving earlier results of
van Geemen–Voisin [vGV16], showed that Lagrangian fibrations are either
isotrivial or have maximal variation. Since isotrivial Lagrangian fibrations
are relatively tractable they are particularly interesting objects of study in
the search for new deformation types.

In this article, we establish some foundational results on isotrivial
Lagrangian fibrations of projective hyper-Kähler manifolds, which we call
hyper-Kähler manifolds for simplicity111The degenerate twistor deformation of
[Ver15, BDV22] can be used to deform an isotrivial Lagrangian fibration into a
projective isotrivial Lagrangian fibration. We may thus assume that $X$ is
projective without loss of generality.. We start by recalling the
classification of isotrivial elliptic fibrations of K3 surfaces
$\pi:S\longrightarrow\mathbb{P}^{1}$ in terms of the monodromy group $G$ of
the local system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$, where
$\pi_{0}:S_{0}\longrightarrow\mathbb{P}^{1}\setminus\Sigma$ is the smooth
isotrivial fibration obtained by restricting $\pi$ to the complement of the
singular fibers. The following theorem (see Appendix A) is well-known to
experts but we were unable to find a suitable reference (see [FM94, §1.4.2],
[Saw14], and [Moo18] for related statements):

######  Theorem 1.1.

Let $\pi:S\longrightarrow\mathbb{P}^{1}$ be an isotrivial fibration of a
projective K3 surface with smooth fiber $F$ and let
$G:=\operatorname{im}\left(\pi_{1}(\mathbb{P}^{1}\setminus\Sigma)\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z})\right))$
be its global monodromy group. Then $G$ is a subgroup of the automorphism
group of the elliptic curve $F$. In particular, $G$ is either
$\mu_{2},\mu_{3},\mu_{4},\text{ or }\mu_{6}$. Furthermore:

  1. (1)

If $G=\mu_{2}$, there is an abelian surface $A$ and a fibration
$p:A\longrightarrow E$ to an elliptic curve $E$ with fiber $F$, such that
$\pi$ is birational to the quotient of $p$ by the involution $[-1]$. In
particular, $S$ is a Kummer surface.

  2. (2)

If $\pi$ admits a section, there is a curve $C$ with a $G$-action such that
$\pi$ is birational to the quotient of the projection $F\times
C\longrightarrow C$ by a diagonal $G$-action. The possible curves $C$ for
$G=\mu_{3},\mu_{4}$, and $\mu_{6}$ are classified in tables 2, 3, and 4
respectively.

Note that every elliptic fibration of a K3 surface is a Tate–Shafarevich twist
of an elliptic fibration with a section (see, e.g., [Huy16, §11]). Theorem 1.1
thus provides a complete answer to the classification of isotrivial fibrations
for K3 surfaces up to Tate–Shafarevich twist.

######  Remark 1.2.

Note that if the elliptic curve $F$ has $j$-invariant different from $0$ or
$1728$ then we are automatically in case (1). The case (2) has the property
that the K3 surface $S/\mathbb{P}^{1}$ admits an automorphism of order $m\geq
3$ fixing pointwise the base $\mathbb{P}^{1}$ of the fibration (thus
$m\in\\{3,4,6\\}$). All examples of fibrations arising in (2) are obtained by
considering degree $d$ cyclic covers $C$ of $\mathbb{P}^{1}$ branched in
appropriate configurations of points (e.g. generically $12$ points for $m=6$)
and performing a birational modification of the diagonal quotient $(E\times
C)/\mu_{m}\longrightarrow C/\mu_{m}=\mathbb{P}^{1}$, where $E$ is the elliptic
curve with $\mu_{m}$ automorphisms. This construction is closely related to
the work of Deligne–Mostow [DM86] (see also [Moo18]) and, from the perspective
of flat metrics on $S^{2}$ with conifold singularities, to the work of
Thurston [Thu98]. The Hodge theoretic underpinning for such examples is
provided by van Geemen’s theory of half-twists [vG01].

Moving to higher dimension, the goal of this article is to provide an
analogous classification for isotrivial Lagrangian fibrations of hyper-Kähler
manifolds. Along the way we will discover several analogies with the K3 case.
The first of these is the surprising fact that smooth fibers of an isotrivial
Lagrangian fibration $\pi:X\longrightarrow B$ are isogenous to the power of an
elliptic curve. We note that throughout the article we assume that Lagrangian
fibrations have a smooth base $B$ (and thus that $B$ is isomorphic to
$\mathbb{P}^{n}$ by [Hwa08]).

######  Theorem 1.3.

Let $\pi:X\longrightarrow B$ be an isotrivial Lagrangian fibration of a hyper-
Kähler $2n$-fold with smooth fiber $F$. Then there is an elliptic curve $E$
such that $F$ is isogenous to $E^{n}$.

Consider an isotrivial Lagrangian fibration $\pi:X\longrightarrow B$ with
general fiber $F$. Let $B_{0}=B\setminus\Sigma$ be the complement of the
discriminant locus $\Sigma$ and $X_{0}$ be the base change of $X$ to $B_{0}$.
We write $\pi_{0}:X_{0}\longrightarrow B_{0}$ for the the restriction of $\pi$
to $X_{0}$, which is a (smooth) isotrivial fibration. As in the K3 case, we
consider the monodromy group
$G=\operatorname{im}\left(\pi_{1}(B_{0})\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z})\right)$
and the minimal Galois covering $U\longrightarrow B_{0}$ that trivializes the
local system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$. The base change
$Z\longrightarrow U$ of $\pi_{0}$ by $U\longrightarrow B_{0}$ may not be a
trivial family but it plays an important role. For this reason, we refer to
$Z\longrightarrow U$ as the intermediate trivialization. We will argue that
the geometry of the isotrivial fibration $X/B$ is controlled to a large extent
by the Kodaira dimension (of a smooth compactification) of the intermediate
base $U$, which is either $0$ or maximal. We call $\pi$ of _type A_ or _type
B_ accordingly.

######  Theorem 1.4.

Let $\pi:X\longrightarrow B$ be an isotrivial Lagrangian fibration of a hyper-
Kähler $2n$-fold and $U$ be the base of the intermediate trivialization. Then
either

  1. (A)

$U$ compactifies to an abelian variety isogenous to a power of an elliptic
curve, or

  2. (B)

$U$ compactifies to a smooth projective variety of general type.

For any $n>1$ there are two straightforward examples of isotrivial Lagrangian
fibrations, which we call $\textup{K3}^{[n]}$-fibration and
$\textup{Kum}_{n}$-fibration (see Section 2.4 for a detailed discussion).

######  Definition 1.5 ($\text{K3}^{[n]}$ and $\text{Kum}_{n}$ fibrations).

An isotrivial Lagrangian fibration of a hyper-Kähler $2n$-fold
$\pi:X\longrightarrow B$ is a

  * _i_)

_$\text{K3}^{[n]}$ -fibration_ if there is an isotrivial elliptic K3 surface
$f:S\longrightarrow\mathbb{P}^{1}$ and a commutative diagram

${X}$${S^{[n]}}$${B}$${\mathbb{P}^{n}.}$$\scriptstyle{\cong}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$$\scriptstyle{f^{[n]}}$$\scriptstyle{\cong}$

  * _ii_)

_$\text{Kum}_{n}$ -fibration_ if there is an abelian surface $A$, an elliptic
curve $E^{\prime}$, a surjective homomorphism $f:A\longrightarrow E^{\prime}$,
and a commutative diagram

${X}$${\textup{Kum}_{n}(A)}$${B}$${\mathbb{P}^{n},}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{\pi}$$\scriptstyle{g}$$\scriptstyle{\cong}$

where the morphism $g$ is induced from $f$ (see Section 2.4 for a discussion).

We say that $\pi$ is _birational to a $\text{K3}^{[n]}$-fibration_ (resp.
_birational to a $\text{Kum}_{n}$-fibration_) if the isomorphism $\phi$
appearing in the above definitions is replaced by a birational map.

Our next result states that, assuming the following well-known conjecture, a
type A fibration with a rational section is birational to either a
$\textup{K3}^{[n]}$-fibration or a $\textup{Kum}_{n}$-fibration.

######  Conjecture 1.6.

Let $\pi:X\longrightarrow B$ be a nontrivial and non-birational fibration of a
hyper-Kähler manifold $X$ of dimension $2n$. Then $B$ is isomorphic to
$\mathbb{P}^{n}$.

######  Remark 1.7.

Conjecture 1.6 was verified for hyper-Kähler fourfolds by Ou [Ou19] and
Huybrechts–Xu [HX22], and for all known deformation types of hyper-Kähler
manifolds by the cumulative work of many people in [BM14], [Mar14], [Mat17],
[Wie18], [MR21], and [MO22].

######  Theorem 1.8.

Let $\pi:X\longrightarrow B$ be an isotrivial Lagrangian fibration of type A
of a hyper-Kähler $2n$-fold. Assume that

  * (1)

Conjecture 1.6 holds for hyper-Kähler manifolds birational to $X$ (for
instance if $\dim X=4$ or if $X$ belongs to one of the known deformation
types), and

  * (2)

$\pi$ has a rational section.

Then $\pi$ is birational to either a $\textup{K3}^{[n]}$-fibration or a
$\textup{Kum}_{n}$-fibration.

In analogy with Remark 1.2 in the K3 case, a certain genericity assumption on
the fiber $F$ ensures that $\pi$ is of type A. A fiber of $\pi$ is called
_non-multiple_ if its cycle class is of the form $\sum_{j}a_{j}[Y_{j}]$ with
$\gcd\\{a_{j}\\}=1$, where $Y_{j}$ denotes the irreducible components of the
fiber.

######  Theorem 1.9.

Assume the general singular fibers of $\pi$ are non-multiple. If the
endomorphism field of the elliptic factor $E$ of $F$ is different from
$\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$ then $\pi$ is of type A.
Moreover, the degree of the reduced discriminant divisor
$\Sigma_{\textup{red}}$ is $2(n+1)$.

We do not know examples of Lagrangian fibrations of hyper-Kähler manifolds
with a multiple general singular fiber. For example, if $\pi$ has a rational
section then the general singular fibers are not multiple. Note also that the
degree of the reduced discriminant divisor of any Lagrangian fibration is at
least $n+2$ by [Par22, Rmk 1.11].

######  Remark 1.10.

One can consider the endomorphism algebra
$\textup{End}_{\text{HS}}(H^{2}(X,\mathbb{Q})_{\mathrm{tr}})$ of the
transcendental cohomology, which is either a totally real field or a CM field
by [Zar83]. One says that $X$ has weak CM by a CM field $K$ (e.g.
$\mathbb{Q}(\sqrt{-d})$) if
$K\subseteq\operatorname{End}_{\textup{HS}}(H^{2}_{\operatorname{tr}}(X,\mathbb{Q}))$.
Theorem 1.9 (as well as Remark 1.2) essentially says that if $X$ does not have
weak CM by $\mathbb{Q}(\sqrt{-1})$ or $\mathbb{Q}(\sqrt{-3})$, then
$\pi:X\longrightarrow B$ is of type A. Note that K3-type Hodge structures with
weak CM by a CM field $K$ are parameterized by Shimura subvarieties of ball
quotient type (of dimension at most half the dimension of the period domain).

An interesting question is whether every deformation class of hyper-Kähler
manifolds contains a member with an isotrivial Lagrangian fibration. We are
not able to answer this question for the exceptional OG6 and OG10 types.
However, our results above that if such isotrivial Lagrangian fibrations exist
they have to be somewhat exotic. We discuss the OG10 case here. Specifically,
recall that Laza–Saccà–Voisin [LSV17] constructed a locally complete family of
Lagrangian fibrations for the OG10 type starting from generic cubic fourfolds;
we call members of this family LSV fibrations for OG10. By construction, the
LSV fibration admits a rational section. The construction of the LSV fibration
can be extended to all smooth cubics [Sac23], singular cubics with ADE
singularities [HN08] (corresponding to points on the $\mathcal{C}_{6}$ Hassett
divisor), and to degenerations to the secant to the Veronese surface [OR],
[dCRS21] (corresponding to points on the $\mathcal{C}_{2}$ Hassett
divisor)222Some of these constructions only describe the degenerate LSV
fibration $X_{0}/B_{0}$ over an open set $B_{0}\subset B\cong\mathbb{P}^{5}$
of the base. The existence of a suitable compactification $X/B$ follows from
[Laz10], [KLSV18], and arguments similar to those of [Sac23]. In any case, for
the purpose of identifying isotrivial fibrations the fibration $X_{0}/B_{0}$
suffices.. In conclusion, one gets a good control of all Lagrangian fibrations
deformation equivalent to the LSV fibration on OG10. In view of Theorem 1.8
and Theorem 1.9, we conclude:

######  Corollary 1.11.

Let $\pi:X\longrightarrow B$ be deformation equivalent to the LSV fibration
for OG10. If $\pi$ is isotrivial, its general fiber is isogenous to $E^{5}$,
where $E$ is an elliptic curve with $j(E)\in\\{0,1728\\}$. Additionally,
$H^{2}_{\operatorname{tr}}(X)$ has weak CM by $\mathbb{Q}(\sqrt{-1})$ or
$\mathbb{Q}(\sqrt{-3})$.∎

######  Remark 1.12.

We believe that it is possible to further analyze the structure of $X/B$ as in
the corollary above, leading to either an isotrivial example in the LSV
deformation class, or to the exclusion of such isotrivial fibrations. We have
some partial results in this direction and plan to revisit the question in the
future.

We close by mentioning two additional results regarding isotrivial Lagrangian
fibrations. First, isotriviality is a closed condition in moduli for
Lagrangian fibered hyper-Kähler manifolds (Proposition 6.1). Second, a general
deformation of a Lagrangian fibration of a hyper-Kähler manifold $X$ with
$b_{2}(X)\geq 7$ is of maximal variation. (Note that all known examples of
hyper-Kähler manifolds satisfy $b_{2}\geq 7$.)

######  Theorem 1.13.

Let $\pi:X\longrightarrow B$ be a Lagrangian fibration of a hyper-Kähler
manifold with $b_{2}(X)\geq 7$. There exists a deformation of the Lagrangian
fibration $\pi$ which is not isotrivial.

Let us briefly review the content of the paper and sketch some key points. Let
$G$ be the monodromy group of the local system
$R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$. To tackle Theorem 1.3, we exploit
the compatibility of two structures on the cohomology of the fiber $F$: the
Hodge structure and the $G$-module structure. The $G$-equivariant Hodge
structure $H^{1}(F,\mathbb{Q})$ is significantly constrained due to the
simplicity of the variation of Hodge structures
$R^{1}(\pi_{0})_{*}\underline{\mathbb{Q}}$, a result that goes back to
[Voi92], [Ogu09], [Mat16], and [SY22]. Representation theory of Mumford–Tate
and finite groups shows that $F$ is necessarily isogenous to a power of a
simple abelian variety (Corollary 2.12). To show this simple abelian variety
is an elliptic curve, we exploit a second rational fibration
$\pi^{\prime}:X\dashrightarrow B^{\prime}$, where $B^{\prime}=\bar{F}/G$ is a
finite quotient of some abelian variety $\bar{F}$ isogenous to $F$ (see
Theorem 3.1). The variety $B^{\prime}$ is $\mathbb{Q}$-Fano as it is dominated
by a hyper-Kähler manifold, so the quotient map $\bar{F}\longrightarrow
B^{\prime}$ must be ramified in codimension $1$. Its ramification locus
consists of abelian subvarieties of codimension $1$, forcing $\bar{F}$ to have
an elliptic curve factor.

For Theorem 1.4, we study the equivariant Hodge structure on the cohomology of
the intermediate base $U$. The Künneth formula together with the fact that
$h^{2,0}(X)=1$ provides a relationship between $H^{1}(F,\mathbb{Q})$ and
$H^{1}(U,\mathbb{Q})$, which allows us translate the information we have about
the former into information about the latter. The resulting properties of
$H^{1}(U,\mathbb{Q})$ have geometric consequences for $U$ and its Albanese
morphism. Finally, an equivariant version of the theorem of Ueno [Uen75] and
Kawamata [Kaw81] reviewed in Appendix C allow us to conclude that $\kappa(U)$
is equal to $0$ or $n$. When $\kappa(U)=0$, the normalization of $B$ in the
function field $K(U)$ is an abelian variety (Proposition 4.4) and
compactification of $U$.

For Theorem 1.9, we need to understand the behavior of isotrivial fibrations
in codimension $1$. The two principal tools that we use for this purpose (see
Section 4) are the classification of codimension $1$ fibers by Hwang–Oguiso
[HO09, HO11], and the canonical bundle formula for $K$-trivial fibrations
(e.g. [Kol07]). Both of them take a simpler form in the case of isotrivial
fibrations. Druel–Bianco [DB22] provide a version of the canonical bundle
formula for isotrivial fibrations. For the codimension $1$ singular fibers
arising in isotrivial fibrations, we will refine Hwang–Oguiso’s result and
show that only the type II, III, IV, $\text{I}_{0}^{*}$, $\text{II}^{*}$,
$\text{III}^{*}$, and $\text{IV}^{*}$ fibers may occur (Proposition 4.16). The
simplest situation is when the elliptic curve factor $E$ in $F$ doesn’t have
CM by $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$: the codimension $1$
fibers are necessarily of type $\text{I}_{0}^{*}$ and the local monodromy
around the discriminant locus is $\mu_{2}$, allowing us to conclude the
theorem.

Theorem 1.4 and its more precise formulation Theorem 3.2 describe quite
explicitly the birational behavior of the isotrivial fibration
$\pi:X\longrightarrow B$. To obtain Theorem 1.8, we use again the second
rational fibration $\pi^{\prime}:X\dashrightarrow B^{\prime}$. When $\pi$ is
of type A, then $X$ admits a birational hyper-Kähler model $X^{\prime}$
obtained by a symplectic resolution of a quotient of an abelian $2n$-fold.
Moreover, the rational map $\pi^{\prime}:X^{\prime}\dashrightarrow B^{\prime}$
is a morphism and an isotrivial Lagrangian fibration. Assuming Conjecture 1.6,
$B^{\prime}$ is then isomorphic to $\mathbb{P}^{n}$. This forces the abelian
varieties together with their $G$-actions to belong to a short list of
possibilities by the recent classification of smooth quotients of abelian
varieties by Auffarth, Lucchini Arteche, and Quezada [ALA20, ALAQ22, ALA22].

Additionally, we include three appendices. In Appendix A, we discuss the proof
of the classification of isotrivial fibrations for K3 surfaces (Theorem 1.1)
in detail. While fairly straightforward, our treatment highlights some of the
issues arising in higher dimensions. The reader may use this discussion as an
introduction to the general case. The remaining two appendices contain
technical statements needed in the main text.

### Notation and Conventions

An abelian variety $A$ may be considered as an abelian torsor by forgetting
its origin. For clarity, we will sometimes write $A^{\operatorname{tor}}$ to
emphasize that we consider $A$ as an abelian torsor. Given an abelian variety
$A$, we write $\operatorname{Aut}A^{\operatorname{tor}}$ for the automorphism
group of the torsor $A^{\operatorname{tor}}$ and $\operatorname{Aut}_{0}A$ for
the subgroup of automorphisms fixing the origin. The two groups are related by
an isomorphism $\operatorname{Aut}A^{\operatorname{tor}}\cong
A\rtimes\operatorname{Aut}_{0}A$.

Given a finite group $G$, we use the term _$G$ -variety_ for an algebraic
variety with a $G$-action. A _$G$ -abelian variety_ is an abelian variety $A$
together with a $G$-action given by a homomorphism
$G\longrightarrow\operatorname{Aut}_{0}A$. A _$G$ -abelian torsor_ is an
abelian torsor $A^{\operatorname{tor}}$ together with a $G$-action given by
$G\longrightarrow\operatorname{Aut}A^{\operatorname{tor}}$. Note that the
group action for a $G$-abelian variety fixes the origin whereas the action for
a $G$-abelian torsor may not.

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$X$ an $A$-torsor, $Y$ a $B$-torsor, and $f:X\longrightarrow Y$ a morphism. We
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to express the compatibility

$f(a\cdot x)=g(b)\cdot f(x).$

We will use similar notation for more involved diagrams as well.

Finally, given a hyper-Kähler manifold $X$, a _rational Lagrangian fibration_
of $X$ is a rational fibration $X\dashrightarrow B^{\prime}$ whose general
fiber is a Lagrangian subvariety of $X$. In the literature, this name is
sometimes reserved for a composition $X\dashrightarrow
X^{\prime}\longrightarrow B^{\prime}$ of a birational map from $X$ to another
hyper-Kähler manifold $X^{\prime}$ and a Lagrangian fibration of $X^{\prime}$.
The former definition is strictly weaker than the latter.

### Acknowledgments

The first author was supported by the ERC Synergy Grant HyperK (ID 854361).
The second author was partially supported by NSF (DMS-2101640). The third
author was partially supported by the AMS-Simons Travel Grant and would like
to thank the Hausdorff Research Institute for Mathematics for its hospitality
and support during the Junior Trimester Program titled _Algebraic geometry:
derived categories, Hodge theory, and Chow groups_ (Fall of 2023).
We thank Benjamin Bakker, Robert Friedman, Philip Engel, and Lenny Taelman for
many useful comments.

##  2\. Trivializations of isotrivial fibrations and Galois action on
cohomology

Throughout the paper we will consider an isotrivial Lagrangian fibration
$\pi:X\longrightarrow B=\mathbb{P}^{n}$, with generic fiber $F$, which we view
as a (fixed) polarized abelian variety. We will denote by $\Delta\subset B$
the discriminant locus, which is a subvariety of pure codimension $1$, and by
$B_{0}=B\setminus\Delta$ its complement. The variety $X_{0}=\pi^{-1}(B_{0})$
then admits a smooth isotrivial fibration $\pi_{0}=\pi_{\mid
X_{0}}:X_{0}\longrightarrow B_{0}$. After a finite étale base change, such a
smooth isotrivial family of abelian varieties can be trivialized. It will be
useful to perform this trivialization in two steps. The first étale base
change trivializes the global monodromy on the first cohomology of the fibers.
The second, which may be trivial, provides us with a rational section.

To be precise, let

$G:=\operatorname{im}\left(\pi_{1}(B_{0})\to\operatorname{Aut}\big{(}H^{1}(F,\mathbb{Z})\big{)}\right)$

be the monodromy group of the local system
$R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$. Since the moduli space of
polarized abelian variety with a sufficiently high level structure is a fine,
$G$ is a finite group. The finite index kernel of the monodromy representation
then determines a finite étale cover $U\longrightarrow B_{0}$ trivializing the
local system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$ (see §2.4 for examples
of this base change for well-known isotrivial Lagrangian fibrations). We
denote by $Z\longrightarrow U$ the base change in the following diagram and
refer to it as the _intermediate trivialization_

${Z}$${X_{0}}$${U}$${B_{0}.}$$\scriptstyle{p}$$\scriptstyle{\pi_{0}}$ (2.1)

If $\pi$ admits a rational section, the intermediate trivialization $Z/U$ is a
trivialization of the fibration $\pi_{0}:X_{0}\longrightarrow B_{0}$;
equivalently, $X$ is birational to a diagonal quotient $(F\times U)/G$ (Lemma
2.4). In general, a further base change is needed to trivialize $X_{0}/B_{0}$
but in any case there is a $G$-action on the cohomology of the fiber $F$ and
the base $U$, which is compatible with the Hodge structure. The assumption
that $X$ is a hyper-Kähler manifold imposes strong restrictions on the
equivariant cohomology of $F$ and $U$ (e.g. Proposition 2.11, Lemma 2.23). In
particular, we will obtain two important consequences to be used towards our
main results:

  * •

$F$ is isogenous to a power of a simple abelian variety (Corollary 2.12). This
falls short of Theorem 1.3 but plays a central role in its proof.

  * •

$U$ admits a map to a simple $G$-abelian torsor which is generically finite on
its image. This will be applied in conjunction with Theorem C.2, an
equivariant version of a theorem of Kawamata, to deduce Theorem 1.4.

###  2.1. Galois groups associated to isotrivial Lagrangian fibrations

As discussed above, the cover $U\longrightarrow B_{0}$ trivializes the
polarizable VHS $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$. There is an
abelian scheme $P_{0}\longrightarrow B_{0}$ associated to this VHS, which is
also trivialized by $U\longrightarrow B_{0}$, and the smooth fibration
$\pi_{0}:X_{0}\longrightarrow B_{0}$ is a torsor for $P_{0}/B_{0}$333This
picture holds more generally and without isotriviality assumptions and we
refer to [AF16] and [Kim21] for further discussion..

After, a further Galois base change $\tilde{U}\longrightarrow U\longrightarrow
B_{0}$, which we assume minimal (but which may not be unique), we can fully
trivialize the family $\pi:X_{0}\longrightarrow B_{0}$. This is because the
intermediate trivialization $p$, as a _projective_ $F$-torsor, determines a
torsion element $\alpha\in H^{1}_{\text{\'{e}t}}(U,F)$ (see [Ray70, Prop
XIII.2.3]). If $\alpha$ is $d$-torsion, it can be lifted to an element
$\tilde{\alpha}\in H^{1}_{\text{\'{e}t}}(U,F[d])$, where $F[d]$ is the kernel
of the multiplication homomorphism $[d]:F\longrightarrow F$. The finite étale
covering $\tilde{U}\longrightarrow U$ corresponding to $\tilde{\alpha}$
trivializes the family. Passing to a connected component or Galois closure if
needed, we may assume that $\tilde{U}$ is connected and that
$\tilde{U}\longrightarrow U\longrightarrow B_{0}$ is Galois. We get a
cartesian diagram as follows and call the map
$\text{pr}_{2}:F\times\tilde{U}\longrightarrow U$ the _full trivialization_ of
$\pi$.

${F\times\tilde{U}}$${Z}$${X_{0}}$${\tilde{U}}$${U}$${B_{0}.}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{p}$$\scriptstyle{\pi}$
(2.2)

We denote the three Galois groups arising from the bottom row of this diagram
by

$G=\operatorname{Gal}(U/B_{0}),\qquad\tilde{G}=\operatorname{Gal}(\tilde{U}/B_{0}),\qquad\tilde{G}_{\operatorname{tr}}=\operatorname{Gal}(\tilde{U}/U),$
(2.3)

and call $G$ (resp. $\tilde{G}$) the intermediate (resp. full) Galois group.
These groups fit in a short exact sequence

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Since a finite étale morphism being Galois is a property which is stable under
base change, the morphism $F\times\tilde{U}\longrightarrow X_{0}$ (resp.
$Z\longrightarrow X_{0}$) is also a Galois cover with Galois group $\tilde{G}$
(resp. $G$). In other words, $F\times\tilde{U}$ (resp. $Z$) admits a free
$\tilde{G}$-action (resp. $G$-action) with quotient $X_{0}$. We emphasize that
the $\tilde{G}$-action on $F\times\tilde{U}$ may not be diagonal when $\pi$
does not have a rational section. In contrast, when $\pi$ has a rational
section, we have $G=\tilde{G}$ (thus, $\tilde{G}_{\operatorname{tr}}=1$ and
$Z=F\times U$) and the $G$-action on $Z$ is diagonal:

######  Lemma 2.4.

The following are equivalent.

  1. (1)

$\pi:X\longrightarrow B$ has a rational section.

  2. (2)

The intermediate trivialization $p:Z\longrightarrow U$ is $G$-equivariantly
isomorphic to $\operatorname{pr}_{2}:F\times U\longrightarrow U$ equipped with
a diagonal $G$-action

$g\cdot(x,y)=(f_{g}(x),\ g\cdot y)\qquad\mbox{for}\quad
f_{g}\in\operatorname{Aut}_{0}F.$ (2.5)

###### Proof.

(2) $\Longrightarrow$ (1): The ($G$-equivariant) zero section
$U\longrightarrow F\times U$ given by $y\mapsto(0,y)$ descends to a rational
section of $\pi:X\longrightarrow B$.

(1) $\Longrightarrow$ (2): Recall that $p:Z\longrightarrow U$ is a an
$F$-torsor, or a torsor under the constant abelian scheme
$\operatorname{pr}_{2}:F\times U\longrightarrow U$. In Proposition 2.8, we
will construct a homomorphism $f:G\longrightarrow\operatorname{Aut}_{0}F$, but
let us take it momentarily for granted. We can use (2.5) to endow the constant
abelian scheme with a $G$-action. With respect to this action, the group
scheme action morphism $\rho:(F\times U)\times_{U}Z\longrightarrow Z$ is
equivariant. Now, any rational section $s:B\dashrightarrow X$ is necessarily
defined over $B_{0}$ by [Kim21, §3.3], so we can lift it to an equivariant
section $s:U\longrightarrow Z$ of the intermediate fibration $p$. We have
constructed a sequence of equivariant morphisms

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and the theory of torsors implies that the composition
$\rho\circ(\operatorname{id},s)$ is an isomorphism. Hence $F\times U$ and $Z$
are equivariantly isomorphic. ∎

As the following example illustrates, isotrivial Lagrangian fibrations with
$\tilde{G}_{\operatorname{tr}}\neq 0$ exist already in dimension $2$. In the
same fashion, it is not hard to write down examples of Hilbert schemes or
generalized Kummer varieties with isotrivial Lagrangian fibrations with
$\tilde{G}_{\operatorname{tr}}\neq 0$.

######  Example 2.6.

Consider an abelian surface $A$ with a fibration $f:A\longrightarrow E$, where
$E$ is an elliptic curve. Let $F$ be the fiber of this fibration, $X$ the
Kummer surface associated to $A$, and $\pi:X\longrightarrow
E/[-1]\cong\mathbb{P}^{1}$ the elliptic fibration arising from $f$. In this
situation, we have:

  * •

The global monodromy group is $G=\mu_{2}$.

  * •

The discriminant divisor $\Delta\subset E/[-1]$ is the image of the
$2$-torsion $E[2]\subset E$.

  * •

$U=E\setminus E[2]$.

  * •

$Z=A\setminus f^{-1}(E[2]).$

It follows that the family $p:Z\longrightarrow U$ has a section if and only if
$f$ has a section i.e., if and only if $A\cong F\times E$ and $f$ is the
projection onto the second factor. In fact, given isogenies $\eta:F\times
E^{\prime}\longrightarrow A$ and $\eta^{\prime}:E^{\prime}\longrightarrow E$
such that $f\circ\eta=\eta^{\prime}\circ\text{pr}_{2}$, the full
trivialization diagram takes the form

${F\times
E^{\prime}}$${Z}$${X_{0}}$${E^{\prime}\setminus\eta^{\prime-1}(E[2])}$${E\setminus
E[2]}$${(E\setminus
E[2])/[-1].}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{p}$$\scriptstyle{\pi_{0}}$
(2.7)

Given such a diagram the group $\tilde{G}_{\operatorname{tr}}$ is
$\ker(\eta)=\text{Gal}(F\times E^{\prime}/A)$. Requiring that $\eta$ is
minimal, namely that it does not factor through a similar isogeny $F\times
E^{\prime\prime}\longrightarrow A$ is insufficient to ensure uniqueness.

###  2.2. Cohomology of the fiber $F$

In this subsection, we study the cohomology of the fiber $F$ as a Hodge
structure and a $G$-representation. Recall that the intermediate Galois group
$G$ is the image of the monodromy representation
$\pi_{1}(B_{0})\longrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z}))$
associated to the local system $R^{1}(\pi_{0})_{*}\underline{\mathbb{Z}}$ on
$B_{0}$. A feature of isotrivial Lagrangian fibrations is that $G$ must be a
subgroup of $\text{Aut}_{0}(F)$, and therefore is relatively small. On the
other hand, a result of Shen-Yin-Voisin [SY22] states that the $G$-invariant
cohomology of $F$ is small, so that $G$ is relatively large. It is this
tension that we exploit in order to prove Theorem 1.3. We will first present
the proof of Theorem 1.3 under the assumption that $\pi$ has a rational
section (see Theorem 2.14), while the general case will be treated in Section
3.

As announced, the monodromy group for isotrivial fibrations is very special.

######  Proposition 2.8.

The $G$-action on $H^{*}(F,\mathbb{Z})$ respects the Hodge structure, the cup
product, and the polarization induced from $X$. In particular, $G$ isomorphic
to a subgroup of $\operatorname{Aut}_{0}(F)$.

###### Proof.

Consider the full trivialization
$\operatorname{pr}_{2}:F\times\tilde{U}\longrightarrow\tilde{U}$. Given
$y\in\tilde{U}$, write $F_{y}:=F\times\\{y\\}\subset F\times\tilde{U}$ for the
fiber over $y$. An element $g\in\tilde{G}$ acts on $F\times\tilde{U}$ by
taking $F_{y}$ to $F_{g.y}$. Identifying both fibers with $F$, we obtain an
automorphism of $F$. Hence, for each $g\in\tilde{G}$ there is a morphism

$f_{g}:\tilde{U}\longrightarrow\operatorname{Aut}F^{\operatorname{tor}}.$

Since $\tilde{U}$ is connected, its image lies in a single connected component
of $\operatorname{Aut}F^{\operatorname{tor}}$ and defines an element
$f_{g}\in\operatorname{Aut}_{0}F=\pi_{0}(\operatorname{Aut}F^{\operatorname{tor}})$.
One easily checks that this gives a group homomorphism

$\displaystyle f:\tilde{G}$
$\displaystyle\longrightarrow\operatorname{Aut}_{0}F$ $\displaystyle g$
$\displaystyle\longmapsto\;\;\;f_{g}\;\;\;\;.\vskip 6.0pt plus 2.0pt minus
2.0pt$

The full trivialization trivializes the local system
$R^{k}(\pi_{0})_{*}\underline{\mathbb{Z}}$ for all $k$ and the composition of
$f$ with the map

$\operatorname{Aut}_{0}F\hookrightarrow\operatorname{GL}(H^{1}(F,\mathbb{Z}))\longrightarrow\operatorname{GL}(H^{*}(F,\mathbb{Z}))$

is the monodromy representation
$\tilde{G}\longrightarrow\operatorname{GL}(H^{*}(F,\mathbb{Z}))$. This endows
$H^{*}(F,\mathbb{Z})$ with a $\tilde{G}$-action. It respects the Hodge
structure and cup product because the action of $\operatorname{Aut}_{0}F$ on
cohomology does. By definition, the image of the homomorphism $f$ is $G$ so
$\tilde{G}_{\operatorname{tr}}$ acts trivially on $H^{*}(F,\mathbb{Z})$ and
$G=\tilde{G}/\tilde{G}_{\operatorname{tr}}$ acts faithfully on cohomology.
Finally, the fact that $G$ respects the polarization coming from $X$ (and
indeed the very fact that such a polarization is unique up to scaling) is a
consequence of Proposition 2.10 below. ∎

######  Remark 2.9.

We caution the reader that if $\pi$ does not admit a rational section there is
no preferred action of $G$ on $F$. In particular the injective map
$G\longrightarrow\text{Aut}_{0}(F)$ requires the choice of an identity for
$F$.

On the other hand, the monodromy group of a Lagrangian fibration on a hyper-
Kähler manifold has to be large ([SY22, Thm. 0.4]).

######  Proposition 2.10.

There is an ample class $\theta\in\textup{Hdg}^{2}_{\mathbb{Q}}(F)$ such that
the $G$-invariant subspaces of the cohomology of $F$ are given by

$H^{k}(F,\mathbb{Q})^{G}=\begin{cases}0&\mbox{if}\quad k\mbox{ is odd}\\\
\mathbb{Q}\cdot\theta^{k/2}\quad&\mbox{if}\quad k\mbox{ is
even}\quad.\end{cases}$

###### Proof.

This is a reformulation of the result of Shen–Yin–Voisin [SY22, Thm 0.4].
Given a smooth fiber $F_{b}=\pi^{-1}(b)$, we write $i:F_{b}\hookrightarrow X$
for the closed immersion. The image of the restriction map
$i^{*}:H^{k}(X,\mathbb{Q})\longrightarrow H^{k}(F_{b},\mathbb{Q})$ is zero if
$k$ is odd and $\mathbb{Q}\cdot\theta^{k/2}$ if $k$ is even. The image of
$i^{*}$ coincides with the monodromy invariant cohomology
$H^{k}(F,\mathbb{Q})^{\pi_{1}(B_{0})}=H^{k}(F,\mathbb{Q})^{G}$ by the global
invariant cycle theorem. ∎

By Proposition 2.8, the $G$-equivariant Hodge structure $H^{k}(F,\mathbb{Q})$
is the $k^{\textup{th}}$ exterior power of the $G$-equivariant Hodge structure
$H^{1}(F,\mathbb{Q})$. The following is a consequence of Proposition 2.10.

######  Proposition 2.11.

$H^{1}(F,\mathbb{Q})$ is simple as a $G$-equivariant Hodge structure.

###### Proof.

Suppose that $H^{1}(F,\mathbb{Q})\cong V_{1}\oplus V_{2}$, for nontrivial
$G$-equivariant Hodge structures $V_{1}$ and $V_{2}$, which are necessarily
polarizable. Choose a polarizations $\theta_{i}\in\wedge^{2}V_{i}$ for
$i=1,2$. Given any $g\in G$, the element $g\cdot\theta_{i}\in\wedge^{2}V_{i}$
is also a polarization of $V_{i}$. Since the sum of polarizations is a
polarization and therefore nonzero, averaging $\theta_{i}$ over $G$ gives a
nonzero $G$-invariant element of $\wedge^{2}V_{i}$. This proves that
$\dim(\wedge^{2}V_{i})^{G}\geq 1$ for $i=1,2$, thereby violating Proposition
2.10. ∎

######  Corollary 2.12.

$H^{1}(F,\mathbb{Q})$ is an isotypic Hodge structure. In particular $F$ is
isogenous to a power of a simple abelian variety.

###### Proof.

Letting $\operatorname{MT}$ be the Mumford–Tate group of
$H^{1}(F,\mathbb{Q})$, the $G$-equivariant Hodge structure on
$H^{1}(F,\mathbb{Q})$ is nothing but a $(G,\operatorname{MT})$-bimodule
structure. The claim is now a formal consequence of Lemma 2.13 below, which we
present without proof for conciseness. Notice that the same argument proves
that $H^{1}(F,\mathbb{Q})$ is an isotypic $G$-module. ∎

######  Lemma 2.13.

Let $G$ and $H$ be reductive algebraic groups. Then any finite dimensional
isotypic $(G,H)$-bimodule is both $G$-isotypic and $H$-isotypic. ∎

In light of Corollary 2.12, to prove Theorem 1.3 it suffices to show that $F$
contains a subtorus of codimension $1$. We will discuss the proof of Theorem
1.3 in the next section, but to motivate our argument we first present the
following special case:

######  Theorem 2.14.

If $\pi:X\longrightarrow B$ has at least one rational section then its smooth
fiber $F$ is isogenous to a power of an elliptic curve.

###### Proof.

By Lemma 2.4, the intermediate trivialization is isomorphic to the second
projection $F\times U\longrightarrow U$, where the $G$-action on $F\times U$
is diagonal. The first projection map $F\times U\longrightarrow F$ is also a
$G$-equivariant morphism. Taking its $G$-quotient, we obtain a morphism
$X_{0}\longrightarrow F/G$, or a dominant rational map $X\dashrightarrow F/G$.
The following lemma states that $B^{\prime}:=F/G$ is $\mathbb{Q}$-Fano.

######  Lemma 2.15.

Let $B^{\prime}$ be a $\mathbb{Q}$-factorial variety which is dominated by a
hyper-Kähler variety. Then $B^{\prime}$ is $\mathbb{Q}$-Fano.

###### Proof.

Observe that

  * _i_)

$H^{1}(B^{\prime},\mathbb{Q})=0$.

  * _ii_)

$B^{\prime}$ is simply connected and rationally connected.

  * _iii_)

$\operatorname{Pic}B^{\prime}\cong\mathbb{Z}$.

The first item follows from the fact that $B^{\prime}$ admits a surjective
morphism from a variety with trivial first cohomology, namely an appropriate
blowup of the hyper-Kähler dominating $B^{\prime}$. Moreover, $B^{\prime}$ is
simply connected since a normal variety dominated by a simply connected
variety with irreducible general fibers is also simply connected [Kol95, Prop
2.10.2]. It is rationally connected because it is dominated by a hyper-Kähler
manifold [Lin20, Thm 1.4]. To prove that
$\operatorname{Pic}B^{\prime}\cong\mathbb{Z}$, notice from the first item that
$\operatorname{Pic}B^{\prime}\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}$.
If $B^{\prime}$ had a torsion line bundle, the associated cyclic covering
would give a connected finite étale covering of $B^{\prime}$, violating the
fact that $B^{\prime}$ is simply connected. Because $B^{\prime}$ is
$\mathbb{Q}$-factorial, $\operatorname{Pic}B^{\prime}\cong\mathbb{Z}$ and
since $B^{\prime}$ is rationally connected the canonical divisor
$K_{B^{\prime}}$ must be anti-ample. ∎

Returning to the proof of the theorem, note that $K_{F}$ is trivial and
$K_{F/G}$ is anti-ample, so the quotient morphism $F\longrightarrow F/G$ must
be ramified in codimension $1$. In other words, there is a prime divisor
$D\subset F$ with non-trivial generic stabilizer. Using the fact that the
$G$-action on $H^{1}(F,\mathbb{Q})$ is faithful, it is easy to see $D$ is an
abelian torsor. Hence $F$ contains a subtorus of codimension $1$ and the
simple factor of $F$ from Corollary 2.12 is an elliptic curve. ∎

Turning our discussion to the $G$-module structure on the $(1,0)$-cohomology
of $F$, we have:

######  Lemma 2.16.

Denote by $V$ the $G$-module structure on $H^{1,0}(F)$. Then $\wedge^{k}V$ for
$k=0,1,\cdots,n$ are mutually non-isomorphic simple $G$-modules. In
particular, $V$ is a non-symplectic simple $G$-module (i.e.,
$(\wedge^{2}V)^{G}=0$).

###### Proof.

It is a standard fact from representation theory that given a representation
$W$ of a finite group $G$, the conjugate and dual representations of $W$ are
isomorphic. Hence $H^{0,1}(F)$ is isomorphic to $V^{\vee}$. We thus have
$G$-module isomorphisms

$H^{k,k}(F)\cong\wedge^{k}H^{1,0}(F)\otimes\wedge^{k}H^{0,1}(F)\cong\wedge^{k}V\otimes\wedge^{k}V^{\vee}\cong\operatorname{End}_{\mathbb{Q}}(\wedge^{k}V).$

Taking $G$-invariants, we obtain
$H^{k,k}(F)^{G}\cong\operatorname{End}_{G}(\wedge^{k}V)$, which is
$1$-dimensional by Proposition 2.10. Schur’s lemma implies that $\wedge^{k}V$
is a simple $G$-module. Similarly, the fact that $H^{p,q}(F)^{G}=0$ for $p\neq
q$ implies that $\wedge^{p}V$ and $\wedge^{q}V$ are not isomorphic. In
particular, $\wedge^{2}V$ is a simple $G$-module that is not isomorphic to the
trivial $G$-module, so $(\wedge^{2}V)^{G}=0$. ∎

######  Corollary 2.17.

If $n\geq 2$, the intermediate Galois group $G$ is not abelian.

###### Proof.

If $G$ is abelian, then every simple $G$-module over $\mathbb{C}$ is
$1$-dimensional. Since $H^{1,0}(F)$ is an $n$-dimensional simple $G$-module by
Lemma 2.16, we get $n=1$. ∎

###  2.3. Cohomology of the intermediate base $U$

In this subsection, we study the cohomology of the intermediate base $U$ as a
$G$-representation and a mixed Hodge structure. We exploit numerical
constraints on the cohomology of the hyper-Kähler manifold $X$ and the open
subset $X_{0}$ (Lemma 2.21) to translate information about the cohomology of
$F$ obtained in the last section into information about the cohomology of $U$
(Proposition 2.19 and Lemma 2.23).

Before getting into our main discussion, let us observe that the cohomology of
$Z$ is determined by that of $F$ and $U$:

######  Proposition 2.18.

There exists a $G$-equivariant mixed Hodge structure isomorphism

$H^{*}(Z,\mathbb{Q})\cong H^{*}(F,\mathbb{Q})\otimes H^{*}(U,\mathbb{Q}),$

which also respects the cup product. In particular

$W_{k}H^{k}(Z,\mathbb{Q})\cong\bigoplus_{i=0}^{k}H^{i}(F,\mathbb{Q})\otimes
W_{k-i}H^{k-i}(U,\mathbb{Q}),\qquad 0\leq k\leq 2n.$

###### Proof.

Recall that there is a Galois covering $F\times\tilde{U}\longrightarrow Z$ and
that the Galois group $\tilde{G}_{\operatorname{tr}}$ acts trivially on the
cohomology of $F$. We have a sequence of isomorphisms of $G$-mixed Hodge
structures

$\displaystyle H^{*}(Z,\mathbb{Q})\cong
H^{*}(F\times\tilde{U},\mathbb{Q})^{\tilde{G}_{\operatorname{tr}}}$
$\displaystyle\cong\Big{(}H^{*}(F,\mathbb{Q})\otimes
H^{*}(\tilde{U},\mathbb{Q})\Big{)}^{\tilde{G}_{\operatorname{tr}}}$
$\displaystyle\cong H^{*}(F,\mathbb{Q})\otimes
H^{*}(\tilde{U},\mathbb{Q})^{\tilde{G}_{\operatorname{tr}}}\cong
H^{*}(F,\mathbb{Q})\otimes H^{*}(U,\mathbb{Q}).$

All these isomorphisms are obviously isomorphism of mixed Hodge structures. ∎

The following is the main result of this subsection, illustrating again the
rigid cohomological behavior of the intermediate base $U$. It will play an
important role in both Theorem 1.4 and Theorem 1.8.

######  Proposition 2.19.

There exists a generically finite $G$-equivariant morphism $U\longrightarrow
A$ to a simple $G$-abelian torsor $A$ whose dimension is a positive multiple
of $n$. Moreover, this morphism is unique up to composition with a
$G$-equivariant isogeny on the left.

In particular, $U$ has maximal Albanese dimension. The $G$-abelian torsor $A$
may have dimension strictly greater than $n$. We refer the reader to the cases
$\mu=\mu_{3},\mu_{4},$ or $\mu_{6}$ of §2.4.1 for examples with $\dim A>n$.
However, when $\dim A=n$ the cohomology of $A$ resembles that of $F$.

######  Corollary 2.20.

If $\dim A=n$, the $G$-invariant cohomology of $A$ is

$H^{k}(A,\mathbb{Q})^{G}\cong\begin{cases}0&\mbox{if}\quad k\mbox{ is odd}\\\
\mathbb{Q}\quad&\mbox{if}\quad k\mbox{ is even}\quad.\end{cases}$

###### Proof.

If $\dim A=n$, the construction of $A$ gives a $G$-module isomorphism
$H^{1,0}(A)\cong V^{\vee}$. Hence
$H^{p,q}(A)^{G}\cong\big{(}\wedge^{p}V^{\vee}\otimes\wedge^{q}V\big{)}^{G}\cong\operatorname{Hom}_{G}(\wedge^{p}V,\wedge^{q}V)$,
and the result follows from Lemma 2.16. ∎

To prove Proposition 2.19 we need to understand the mixed Hodge structure on
$H^{1}(U,\mathbb{Q})$, and in particular its weight one associated graded
piece. Since $U$ is a smooth variety, the mixed Hodge structure on
$H^{1}(U,\mathbb{Q})$ has weights $\geq 1$. For notational simplicity, we will
write $H^{1,0}(U)$ to denote the $(1,0)$-part of the weight $1$ associated
graded $W_{1}H^{1}(U,\mathbb{C})=\text{Gr}^{W}_{1}H^{1}(U,\mathbb{C})$, i.e.

$H^{1,0}(U):=\big{(}W_{1}H^{1}(U,\mathbb{C})\big{)}^{1,0}\ \ \subset\ \
H^{1}(U,\mathbb{C}).$

Similarly, we use the notation
$H^{k,0}(X_{0}):=\big{(}W_{k}H^{k}(X_{0},\mathbb{C})\big{)}^{k,0}$ and
$H^{k,0}(Z):=\big{(}W_{k}H^{k}(Z,\mathbb{C})\big{)}^{k,0}$.

######  Lemma 2.21.

Denote by $\sigma_{X}\in H^{2,0}(X)$ the holomorphic symplectic form on $X$
and by $\sigma_{X_{0}}\in H^{2,0}(X_{0})$ its restriction to $X_{0}$.

  1. (1)

We have an isomorphism of mixed Hodge structures

$H^{*}(X_{0},\mathbb{Q})\cong\Big{(}H^{*}(F,\mathbb{Q})\otimes
H^{*}(U,\mathbb{Q})\Big{)}^{G}.$

  2. (2)

$H^{k,0}(X_{0})=\begin{cases}0\quad&\mbox{if }k\text{ is even,}\\\
\mathbb{C}\cdot\sigma_{X_{0}}^{k/2}\quad&\mbox{if }k\text{ is
odd.}\end{cases}$

###### Proof.

The first item follows from the equality $X_{0}=Z/G$ and Proposition 2.18. The
second item is the standard fact that the restriction homomorphism to a
Zariski open subset $H^{*}(X,\mathbb{Q})\longrightarrow
H^{*}(X_{0},\mathbb{Q})$ gives an isomorphism on the $(k,0)$-piece of the
weight $k$ associated graded. ∎

Let $\sigma$ be the holomorphic symplectic form on $Z$ pulled back from the
holomorphic symplectic form on $X$. It is a $G$-invariant symplectic form on
$Z$ which is unique up to scaling.

######  Lemma 2.22.

$\Big{(}H^{1,0}(F)\otimes H^{1,0}(U)\Big{)}^{G}=\mathbb{C}\cdot\sigma$ and
$\Big{(}H^{n,0}(F)\otimes H^{n,0}(U)\Big{)}^{G}=\mathbb{C}\cdot\sigma^{n}$.

###### Proof.

From Lemma 2.21 we obtain

$H^{2,0}(F)^{G}\oplus\Big{(}H^{1,0}(F)\otimes H^{1,0}(U)\Big{)}^{G}\oplus
H^{2,0}(U)^{G}=\mathbb{C}\cdot\sigma.$

The first summand $H^{2,0}(F)^{G}$ vanishes by Proposition 2.10. The third
summand $H^{2,0}(U)^{G}$ also vanishes, because $U/G=B_{0}$ and $B_{0}$ is a
Zariski open subset of $B=\mathbb{P}^{n}$. The first claim follows. The second
claim is proven analogously. ∎

The $G$-module $V:=H^{1,0}(F)$ is simple by Lemma 2.16. We have

$H^{1,0}(F)=V,\qquad H^{n,0}(F)=\wedge^{n}V.$

Note that $\wedge^{n}V$ is a non-trivial one-dimensional simple $G$-module.

######  Lemma 2.23.

The $V^{\vee}$-isotypic component of $H^{1,0}(U)$ has multiplicity one and the
$\wedge^{n}V^{\vee}$-isotypic component of $H^{n,0}(U)$ has multiplicity one.
In particular, $h^{1,0}(U)\geq n$.

###### Proof.

Let $m$ be the multiplicity of the $V^{\vee}$-isotypic component of
$H^{1,0}(U)$. Lemma 2.22 and Schur’s lemma force $m=1$. The same argument
shows that the multiplicity of the $\wedge^{n}V^{\vee}$-isotypic component of
$H^{n,0}(U)$ is one. ∎

###### Proof of Proposition 2.19.

We have seen in Lemma 2.23 that the $V^{\vee}$-isotypic component of
$H^{1,0}(U)$ has multiplicity one. Consider the smallest $G$-equivariant Hodge
structure $W\subset W_{1}H^{1}(U,\mathbb{Q})$ containing $V^{\vee}\subset
H^{1,0}(U)$. It is a simple $G$-equivariant Hodge substructure and determines
a $G$-equivariant homomorphism $\operatorname{Alb}_{U}\longrightarrow A$,
where $A$ is a simple $G$-abelian variety. This morphism is unique up to
composition with a $G$-equivariant isogeny on the left. Its kernel
$K=\ker(\operatorname{Alb}_{U}\longrightarrow A)$ acts on
$\operatorname{Alb}_{U}^{\operatorname{tor}}$ freely, so we may take a
quotient $\operatorname{Alb}_{U}^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}=\operatorname{Alb}_{U}^{\operatorname{tor}}/K$. The
composition
$U\longrightarrow\operatorname{Alb}_{U}^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}$ is the desired morphism.

The symplectic form $\sigma\in H^{1,0}(F)\otimes H^{1,0}(U)\subset H^{2,0}(Z)$
has a nontrivial $n$-th power $\sigma^{n}\in H^{2n,0}(Z)=H^{n,0}(F)\otimes
H^{n,0}(U)$. This implies that the cup product homomorphism restricts to a
nonzero homomorphism

$\Big{(}H^{1,0}(F)\otimes H^{1,0}(U)\Big{)}^{\otimes
n}\longrightarrow\wedge^{n}H^{1,0}(F)\otimes\wedge^{n}H^{1,0}(U)\longrightarrow
H^{n,0}(F)\otimes H^{n,0}(U).$

The conclusion is that the cup product homomorphism $\wedge^{n}H^{1,0}(U)\to
H^{n,0}(U)$ is nonzero and therefore that the pullback
$H^{n,0}(\operatorname{Alb}_{U}^{\operatorname{tor}})\longrightarrow
H^{n,0}(U)$ obtained from the Albanese morphism
$U\longrightarrow\operatorname{Alb}_{U}^{\operatorname{tor}}$ is nonzero. In
particular, $U$ has maximal Albanese dimension. The same argument applies to
the subspace $V^{\vee}\subset H^{1,0}(U)$ because the symplectic form $\sigma$
is in fact contained in $H^{1,0}(F)\otimes V^{\vee}\subset H^{1,0}(F)\otimes
H^{1,0}(U)$. We deduce that the composition
$U\longrightarrow\operatorname{Alb}_{U}^{\operatorname{tor}}\longrightarrow
A^{\operatorname{tor}}$ is generically finite on its image.

It remains to prove that $\dim A$ is divisible by $n$. Recall that we have
constructed $A$ (up to $G$-equivariant isogeny) from a simple $G$-equivariant
Hodge structure $W\subset W_{1}H^{1}(U,\mathbb{Q})$. We claim that there is an
isomorphism of $G$-modules $W^{1,0}\cong V^{\vee}\oplus mV$ for some
$m\in\mathbb{Z}_{\geq 0}$. Since $W$ is a simple $G$-equivariant Hodge
structure, it is an isotypic $G$-module by Lemma 2.13. Because
$H^{1}(F,\mathbb{C})=H^{1,0}(F)\oplus H^{0,1}(F)=V\oplus V^{\vee}$, the
$G$-module $V\oplus V^{\vee}$ is defined over $\mathbb{Q}$. This implies that
$W_{\mathbb{C}}$ is a multiple of $V\oplus V^{\vee}$. Finally, $W^{1,0}$ must
be of the form $V^{\vee}\oplus mV$ since it contains $V^{\vee}$ with
multiplicity one by Lemma 2.23. ∎

###  2.4. Trivializations of the standard examples

In this section we discuss examples of isotrivial Lagrangian fibrations. In
particular, we revisit $\textup{K3}^{[n]}$ and $\textup{Kum}_{n}$-fibrations
introduced in Definition 1.5 and discuss their trivializations and associated
Galois groups. At the moment, all isotrivial Lagrangian fibrations which the
authors known about are $\textup{K3}^{[n]}$ or $\textup{Kum}_{n}$-fibrations.
As discussed in the introduction, we are not able to produce (or exclude) an
isotrivial fibration for the deformation type of LSV fibrations [LSV17] of
manifolds of OG10 type (see however Corollary 1.11).

Furthermore, if $(n-1)$ is not square free, hyper-Kähler manifolds of
$\textup{K3}^{[n]}$ type admit additional deformation types of Lagrangian
fibrations, one for each $k$ such that $k^{2}\mid(n-1)$ (see Markman [Mar14];
similarly, for $\text{Kum}_{n}$, see Wieneck [Wie18]). Note that, in contrast
to $K3^{[n]}$-fibrations, these Lagrangian fibrations do not admit a rational
section (even topologically). The first occurrence of such an additional
fibration on $\text{K3}^{[n]}$ is the case $n=5$ (i.e. $n-1=k^{2}$ with
$k=2$). Interestingly, this Lagrangian fibration $\pi:X\longrightarrow B$ on
$K3^{[5]}$, admits an involution $\iota$ compatible with the Lagrangian
fibration, such that the quotient $X/\iota\longrightarrow B$ is deformation
equivalent to the LSV fibration on the OG10 manifold (see [dCRS21]). More
precisely, $X/\iota$ admits a symplectic resolution which is a hyper-Kähler
manifold of OG10 type, and the Lagrangian fibration $(X/\iota)\longrightarrow
B$ deforms to the LSV fibration associated to the intermediate Jacobian
fibration for cubic fourfolds444$X/\iota\longrightarrow B$ can be interpreted
as the LSV fibration associated to a period point belonging to the
$\mathcal{C}_{2}$ Hassett divisor in the moduli of cubic fourfolds (see
[Laz10]). The second author learned about this connection from K. O’Grady a
long time ago.. It is clear that $X/B$ is an isotrivial fibration iff
$(X/\iota)/B$ is isotrivial. In fact, using the connection between LSV
fibrations and cubic fourfolds, it is possible to prove that if an LSV
fibration admits an isotrivial member, then it has to be of the form
$(X/\iota)/B$ as above. In other words, the deformation class of Markman
fibrations on $K3^{[5]}$ with $k=2$ admits an isotrivial member iff the
deformation class of LSV fibrations on OG10 admits an isotrivial member.

####  2.4.1. Intermediate trivialization of $\textup{K3}^{[n]}$-fibrations

Let $f:S\longrightarrow\mathbb{P}^{1}$ be an isotrivial elliptic fibration of
a K3 surface and $\pi:X=S^{[n]}\longrightarrow\mathbb{P}^{n}$ be its
associated $\text{K3}^{[n]}$-fibration. The intermediate Galois group of the
isotrivial fibration $f$ is $\mu=\mu_{2},\mu_{3},\mu_{4}$, or $\mu_{6}$ (see
Appendix A). Consider its intermediate trivialization

${Z}$${S}$${C}$${\mathbb{P}^{1}.}$$\scriptstyle{f}$

We can construct from it another commutative diagram

${Z^{n}}$${S^{[n]}}$${C^{n}}$${(\mathbb{P}^{1})^{(n)},}$$\scriptstyle{p}$$\scriptstyle{\pi}$

whose restriction to the complement of the discriminant divisor $B_{0}\subset
B$ gives the intermediate trivialization diagram of
$\pi:S^{[n]}\longrightarrow B=(\mathbb{P}^{1})^{(n)}$. The intermediate Galois
group is $G=\mu^{\times n}\rtimes\mathfrak{S}_{n}$.

Notice from the results in Appendix A (or Theorem 3.2) that if $C$ is an
elliptic curve, then $Z$ is an abelian surface. In this case, the intermediate
trivialization $p:Z^{n}\longrightarrow C^{n}$ is a morphism between abelian
torsors (of dimensions $2n$ and $n$, respectively). If $C$ has genus at least
$2$, then $Z$ has Kodaira dimension $1$ and the Kodaira dimensions of the
varieties arising in the intermediate trivialization is
$\kappa(Z^{n})=\kappa(C^{n})=n$. In conclusion, the Kodaira dimensions of
$Z^{n}$ and $C^{n}$ is either $0$ or $n$.

We can also describe the discriminant divisor $\Delta\subset B$. It consists
of the union of hyperplanes

$H_{t}=\\{x_{1}+\cdots+x_{n-1}+t\in\text{Sym}^{n}\mathbb{P}^{1}\\}\subset\text{Sym}^{n}\mathbb{P}^{1}\cong(\mathbb{P}^{1})^{[n]}\cong
B,$

one for each $t$ in the discriminant locus of $f$, together with the big
diagonal

$\delta=\\{x_{1}+\cdots+x_{n}\in\text{Sym}^{n}\mathbb{P}^{1}:x_{i}=x_{j}\text{
for some
}i,j\\}\subset\text{Sym}^{n}\mathbb{P}^{1}\cong(\mathbb{P}^{1})^{[n]}.$

It is interesting to consider the behavior of the discriminant divisor as a
generic elliptic K3 surface $S/\mathbb{P}^{1}$ specializes to an isotrivial
elliptic K3. When $S/\mathbb{P}^{1}$ is generic, the reduced discriminant
divisor is the union of $24$ hyperplanes $H_{t}$ (coming from the $24$ special
fibers $S_{t}$ of $S/\mathbb{P}^{1}$), and the big diagonal $\delta$ as above,
which is a hypersurface of degree $2(n-1)$ (see [Saw22, §4.4]). Note that the
local monodromy around the hyperplanes $H_{t}$ is infinite and the fiber over
the generic point of $H_{t}$ is of Kodaira type $\text{I}_{1}$ in the Hwang-
Oguiso terminology [HO09]. On the other hand, the local monodromy around the
big diagonal is of order $2$, and the generic fiber is of type
$\text{I}_{0}^{*}$. In this situation, the canonical bundle formula for the
fibration $\pi:S^{[n]}\longrightarrow\mathbb{P}^{n}$(see Section 4) reads

$\displaystyle-\deg K_{B}$ $\displaystyle=$ $\displaystyle\frac{1}{12}\deg\sum
H_{t}+\frac{1}{2}\deg\delta,\qquad\text{or}$ $\displaystyle n+1$
$\displaystyle=$ $\displaystyle\frac{1}{12}\cdot 24+\frac{2(n-1)}{2}.$

When $S/\mathbb{P}^{1}$ specializes to an isotrivial elliptic fibration, the
big diagonal $\delta$ remains a component of the discriminant divisor (and its
contribution to the canonical bundle formula is unchanged), but the
hyperplanes $H_{t}$ collide and should be counted with a multiplicity
corresponding to the Euler characteristic of the singular fiber $S_{t}$
(compare with Appendix A). The local monodromy around $H_{t}$ becomes finite,
reflecting the monodromy around $t$ for the elliptic fibration
$S/\mathbb{P}^{1}$. Similarly, the Hwang–Oguiso fiber type at the generic
point of $H_{t}$ corresponds to the Kodaira fiber type at $t$. This discussion
will be generalized in Section 4.

####  2.4.2. Intermediate trivialization of $\textup{Kum}_{n}$-fibrations

We will begin by describing the $\textup{Kum}_{n}$-fibration introduced in
Definition 1.5. Consider a short exact sequence

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where $A$ is an abelian surface and $E,E^{\prime}$ are elliptic curves. Let
$(E^{\prime})^{(n+1)}$ and $(E^{\prime})^{(n+1)}\longrightarrow E^{\prime}$ be
the $(n+1)^{\text{st}}$ symmetric power of $E^{\prime}$ and the summation map.
Define a fiber product $P$ by

${P}$${A}$${(E^{\prime})^{(n+1)}}$${E^{\prime}.}$$\scriptstyle{f}$

The morphism $P\longrightarrow A$ is a $\mathbb{P}^{n}$-bundle and the
summation map $A^{[n+1]}\longrightarrow A$ admits a factorization
$A^{[n+1]}\longrightarrow P\longrightarrow A$ by the universal property of
fiber products. Taking the fiber over $0\in A$, we obtain the isotrivial
Lagrangian fibration $\pi:\text{Kum}_{n}(A)\longrightarrow\mathbb{P}^{n}$,
which was called $g$ in Definition 1.5. Its smooth fiber is isomorphic to
$\ker\big{(}E^{n+1}\longrightarrow E\big{)}\cong E^{n}$.

Next, we will construct the intermediate trivialization and describe the
Galois group $G$. Let $(E^{\prime})^{n+1}\longrightarrow(E^{\prime})^{(n+1)}$
be a quotient map and define a fiber product $Q$ by

${Q}$${P}$${(E^{\prime})^{n+1}}$${(E^{\prime})^{(n+1)}.}$

Then $Q$ is an abelian variety of dimension $n+2$ and $Q\longrightarrow P$ is
the quotient map for the $\mathfrak{S}_{n+1}$-action on $Q$. We have a
commutative diagram

${A^{n+1}}$${A^{[n+1]}}$${Q}$${P}$${A,}$$\scriptstyle{p}$

where morphism $p:A^{n+1}\longrightarrow Q$ is again constructed by the
universal property. Taking the fiber of the whole diagram over $0\in A$, we
obtain a commutative diagram

${\ker(A^{n+1}\to A)}$${X}$${\ker\big{(}(E^{\prime})^{n+1}\to
E^{\prime}\big{)}}$${\mathbb{P}^{n},}$$\scriptstyle{p}$$\scriptstyle{\pi}$

which, after restriction to $B_{0}\subset\mathbb{P}^{n}$, the complement of
the discriminant locus of $\pi$, is an intermediate trivialization diagram.
The intermediate Galois group is $G=\mathfrak{S}_{n+1}$. The intermediate base
$\ker\big{(}(E^{\prime})^{n+1}\longrightarrow E^{\prime}\big{)}$ is isomorphic
to $(E^{\prime})^{n}$ and the total space of the intermediate trivialization
$\ker(A^{n+1}\longrightarrow A)$ is isomorphic to $A^{n}$.

##  3\. Fibrations of type A and B

This section is devoted to the proof of Theorem 1.3 and 1.4. Recall that we
already presented a proof of Theorem 1.3 subject to the additional assumption
that the isotrivial fibration $\pi$ has a rational section (Theorem 2.14). A
key input was the existence of a second isotrivial rational Lagrangian
fibration $X\dashrightarrow F/G$ obtained as a $G$-quotient of the projection
map to the first factor $Z\cong F\times U\longrightarrow F$. We will show in
Proposition 3.12 that an analogous second isotrivial rational Lagrangian
fibration always exists, which will conclude the proof of Theorem 1.3 in its
full generality. The projection map that was used in the presence of a
rational section is replaced by a fibration onto an abelian variety $\bar{F}$
isogenous to $F$.

######  Theorem 3.1.

There exists a $G$-equivariant isotrivial fibration
$p^{\prime}:Z\longrightarrow\bar{F}$ such that:

  1. (1)

$\bar{F}$ is a $G$-abelian torsor isogenous to $F$.

  2. (2)

The product morphism $(p^{\prime},p):Z\longrightarrow\bar{F}\times U$ is
$G$-equivariant, finite étale, and Galois.

Moreover, this morphism $p^{\prime}$ is unique up to $G$-equivariant
isomorphism of $\bar{F}$.

Some care is needed in using the notion of a fibration since $p^{\prime}$ is
not proper and we refer the reader to Appendix B for definitions. If $\pi$
admits a rational section, then $p^{\prime}$ is merely the projection map
$F\times U\longrightarrow F$. One can think of Theorem 3.1 as approximating
$Z$ by a product $\bar{F}\times U$ equipped with a diagonal $G$-action.

Theorem 3.1 also allows us to prove a slightly stronger version of Theorem
1.4, which describes the Kodaira dimensions555By the Kodaira dimension of a
smooth quasi-projective variety we mean the Kodaira dimension of a smooth
projective compactification. of both $U$ and $Z$.

######  Theorem 3.2.

Let $p:Z\longrightarrow U$ be the intermediate trivialization of the
isotrivial fibration $\pi:X\longrightarrow B$. Then one of the following
holds.

  1. (A)

$U$ compactifies to an abelian torsor isogenous to the $n$-th power of an
elliptic curve, and either

    1. (A.1)

$Z$ is birational to an abelian torsor of dimension $2n$, or

    2. (A.2)

$\kappa(Z)=n$ and $p:Z\longrightarrow U$ is the Iitaka fibration of $Z$.

  2. (B)

$U$ compactifies to a variety of general type. Moreover, $\kappa(Z)=n$ and
$p:Z\longrightarrow U$ is the Iitaka fibration of $Z$.

######  Example 3.3.

In view of Theorem 1.1 and Section 2.4.1, it is easy to produce examples of
type A.1 and type B respectively. Namely, if $S\to\mathbb{P}^{1}$ is an
isotrivial elliptic fibration of Kummer type (i.e. Theorem 1.1(1)), then the
associated isotrivial fibration $S^{[n]}\to\mathbb{P}^{n}$ (as in §2.4.1) will
be of type A.1. On the other hand, if $S\to\mathbb{P}^{1}$ is of Deligne-
Mostow type (i.e. Theorem 1.1(2)) and the base $C$ of its associated
intermediate trivialization is not elliptic (in some sense, the generic case;
compare Remark 1.2), then the associated isotrivial fibration
$S^{[n]}\longrightarrow\mathbb{P}^{n}$ in of type B.

######  Remark 3.4.

Again if $\pi$ admits a rational section, then $Z$ is isomorphic to $F\times
U$ by Lemma 2.4 so the Kodaira dimensions of $U$ and $Z$ are the same. Hence
type A.2 may only arise when $\pi$ does not have a rational section. At the
moment we do not have examples of isotrivial Lagrangian fibrations of type
A.2. This type seems to be related to the presence of multiple fibers in
codimension $1$ (conjecturally such Lagrangian fibrations do not exist, but
there are no local obstruction to the existence of multiple fibers, e.g.
[HO11, Ex. 6.3]).

Our approach is to apply an equivariant version of a theorem of Kawamata (see
Appendix C) to the morphism $U\longrightarrow A$ from Proposition 2.19. This
will give information on the Iitaka fibration of $U$ and in particular on the
Kodaira dimension of $U$. The same strategy can be applied to $Z$ by using the
$G$-equivariant composition $Z\longrightarrow\bar{F}\times
U\longrightarrow\bar{F}\times A$.

###  3.1. Diagonal approximation

We will begin with the proof of Theorem 3.1. Recall the full trivialization
diagram

${F\times\tilde{U}}$${Z}$${X_{0}}$${\tilde{U}}$${U}$${B_{0},}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{p}$$\scriptstyle{\pi}$
(2.2 restated)

where the horizontal arrows are finite étale and Galois. We denoted the three
Galois groups by

$G=\operatorname{Gal}(U/B_{0}),\qquad\tilde{G}=\operatorname{Gal}(\tilde{U}/B_{0}),\qquad\text{and}\qquad\tilde{G}_{\operatorname{tr}}=\operatorname{Gal}(\tilde{U}/U),$

which satisfy $G\cong\tilde{G}/\tilde{G}_{\operatorname{tr}}$. To prove
Theorem 3.1 we first establish an analogous statement for the full
trivialization. It can be interpreted as approximating the $\tilde{G}$-action
on $F\times\tilde{U}$ by a diagonal action. We recall that the
$\tilde{G}$-action on $F\times\tilde{U}$ may not be diagonal.

######  Proposition 3.5.

There exists a $\tilde{G}$-equivariant isotrivial fibration
$p_{1}:F\times\tilde{U}\longrightarrow F^{\flat}$ such that:

  1. (1)

$F^{\flat}$ is a $\tilde{G}$-abelian torsor isogenous to $F$.

  2. (2)

The product morphism
$(p_{1},\operatorname{pr}_{2}):F\times\tilde{U}\longrightarrow
F^{\flat}\times\tilde{U}$ is $\tilde{G}$-equivariant, finite étale, and
Galois. Here $F^{\flat}\times\tilde{U}$ is equipped with a diagonal
$\tilde{G}$-action.

Moreover, the morphism $p_{1}$ is unique up to composition with a
$\tilde{G}$-equivariant isomorphism on the left.

Theorem 3.1 follows from Proposition 3.5:

###### Proof of Theorem 3.1.

Consider the $\tilde{G}$-equivariant fibration
$F\times\tilde{U}\longrightarrow F^{\flat}$ from Proposition 3.5. Taking its
$\tilde{G}_{\operatorname{tr}}$-quotient, we obtain the desired
$G$-equivariant fibration
$Z\longrightarrow\bar{F}:=F^{\flat}/\tilde{G}_{\operatorname{tr}}$. The group
$\tilde{G}_{\operatorname{tr}}$ acts by translations on $F^{\flat}$ because it
acts trivially on $H^{*}(F^{\flat},\mathbb{Q})=H^{*}(F,\mathbb{Q})$. Hence
$\bar{F}=F^{\flat}/\tilde{G}_{\operatorname{tr}}$ is again an abelian torsor.
Any such $G$-equivariant isotrivial fibration $Z\longrightarrow\bar{F}$ lifts
to a $\tilde{G}$-equivariant fibration $F\times\tilde{U}\longrightarrow
F^{\flat}$, so uniqueness follows. ∎

The rest of this section will be devoted to the proof of Proposition 3.5. A
$\tilde{G}$-action on $F\times\tilde{U}$ is a collection of automorphisms

$g:F\times\tilde{U}\longrightarrow F\times\tilde{U},\qquad g\in\tilde{G},$

satisfying the condition

$g\cdot(h\cdot(x,y))=(gh)\cdot(x,y).$ (3.6)

Moreover, since in our setting the morphism
$\operatorname{pr}_{2}:F\times\tilde{U}\longrightarrow\tilde{U}$ is
$\tilde{G}$-equivariant, the action is of the form (Proposition B.5)

$g\cdot(x,y)=(f_{g}(x)+e_{g}(g\cdot y),\ g\cdot y),\qquad
f_{g}\in\operatorname{Aut}_{0}F,\quad e_{g}:\tilde{U}\longrightarrow F.$ (3.7)

The group
$\operatorname{Mor}(\tilde{U},\operatorname{Aut}F^{\operatorname{tor}})$
admits a $\tilde{G}$-action induced from that on $\tilde{U}$. Condition (3.6)
boils down to a cocyle condition on the map

$\displaystyle\tilde{G}$
$\displaystyle\longrightarrow\operatorname{Mor}(\tilde{U},F)\rtimes\operatorname{Aut}_{0}F\cong\operatorname{Mor}(\tilde{U},\operatorname{Aut}F^{\operatorname{tor}})$
$\displaystyle g$
$\displaystyle\longmapsto\;\;\;\;\;\;\;\;\;\;(e_{g},f_{g})\;\;\;,$

so we obtain an element in the nonabelian cohomology group

$H^{1}(\tilde{G},\
\operatorname{Mor}(\tilde{U},\operatorname{Aut}F^{\operatorname{tor}})).$

Conversely, one checks that this nonabelian cohomology classifies all possible
$\tilde{G}$-actions on $F\times\tilde{U}$, making
$\operatorname{pr}_{2}:F\times\tilde{U}\longrightarrow\tilde{U}$ equivariant,
up to isomorphism.

The $\tilde{G}$-action on $F\times\tilde{U}$ is diagonal if and only if the
morphisms $e_{g}:\tilde{U}\longrightarrow F$ in (3.7) are constant for all
$g\in\tilde{G}$. We claim that this can also be seen from the
$\tilde{G}$-action on the Albanese variety or Albanese torsor of
$F\times\tilde{U}$. As we did for abelian varieties, we will need to
distinguish between the notion of Albanese variety and Albanese torsor (see
Appendix B for details). The reason for these technicalities is that when
defining the Albanese map without choosing a basepoint one obtains a morphism
to an abelian torsor, not an abelian variety. The Albanese diagram in our
$\tilde{G}$-equivariant setting is

${F\times\tilde{U}}$${\operatorname{Alb}_{F\times\tilde{U}}^{\operatorname{tor}}}$${\operatorname{Alb}_{F\times\tilde{U}}}$${\tilde{U}}$${\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}}$${\operatorname{Alb}_{\tilde{U}},}$$\scriptstyle{\operatorname{pr}_{2}}$

where the diagram on the left is a commutative diagram of
$\tilde{G}$-varieties and the one on the right is a homomorphism of
$\tilde{G}$-abelian varieties.

######  Lemma 3.8.

The following are equivalent:

  1. (1)

The $\tilde{G}$-action on $F\times\tilde{U}$ is diagonal.

  2. (2)

The induced $\tilde{G}$-action on the Albanese variety
$\operatorname{Alb}_{F\times\tilde{U}}=F\times\operatorname{Alb}_{\tilde{U}}$
is diagonal.

  3. (3)

The induced $\tilde{G}$-action on the Albanese torsor
$\operatorname{Alb}_{F\times\tilde{U}}^{\operatorname{tor}}=F^{\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$
is diagonal.

###### Proof.

Note that the Albanese variety of $\tilde{U}$ is determined by the weight $1$
pure Hodge structure
$W_{1}H^{1}(\tilde{U},\mathbb{Z})=\text{Gr}_{W}^{1}H^{1}(\tilde{U},\mathbb{Z})$
together with its polarization. We claim that the $\tilde{G}$-action on
$F\times\tilde{U}$ is diagonal if and only if the Künneth decomposition

$W_{1}H^{1}(F\times\tilde{U},\mathbb{Q})\cong H^{1}(F,\mathbb{Q})\oplus
W_{1}H^{1}(\tilde{U},\mathbb{Q})$

is $\tilde{G}$-equivariant. This will prove the desired equivalence.

The implication $(\Longrightarrow)$ is clear, so we concentrate on the
implication $(\Longleftarrow)$ and assume that the $\tilde{G}$-action is
diagonal on the weight one associated graded. We will use the notation from
(3.7) for the action of $g\in\tilde{G}$ on $F\times\tilde{U}$. With respect to
the Künneth decomposition of the first cohomology

$W_{1}H^{1}(F\times\tilde{U},\mathbb{Q})=H^{1}(F,\mathbb{Q})\oplus
W_{1}H^{1}(\tilde{U},\mathbb{Q}),$

the automorphism $g^{*}$ on $W_{1}H^{1}(F\times\tilde{U},\mathbb{Q})$ takes
the form

$g^{*}=\begin{pmatrix}f_{g}^{*}&0\\\ g^{*}\circ e_{g}^{*}&g^{*}\end{pmatrix}.$

It follows that $g^{*}\circ e_{g}^{*}:H^{1}(F,\mathbb{Q})\to
W_{1}H^{1}(\tilde{U},\mathbb{Q})$ vanishes, or equivalently that
$e_{g}^{*}:H^{1}(F,\mathbb{Q})\to W_{1}H^{1}(\tilde{U},\mathbb{Q})$ vanishes.
By the universal property of the Albanese morphism
$\tilde{U}\to\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$, we have a
factorization:

${\tilde{U}}$${F.}$${\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}}$$\scriptstyle{e_{g}}$$\scriptstyle{e_{g}^{\prime}}$

The homomorphisms $(e_{g}^{\prime})^{*}:H^{1}(F,\mathbb{Q})\to
H^{1}(\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}},\mathbb{Q})=W_{1}H^{1}(\tilde{U},\mathbb{Q})$
vanishes since $(e_{g}^{\prime})^{*}=e_{g}^{*}=0$. But $e_{g}^{\prime}$ is a
morphism between abelian torsors, so $e_{g}^{\prime}$ must be a constant map.
As a result, $e_{g}$ is constant as well. ∎

We can now start the proof of Proposition 3.5. For clarity, we first prove the
following intermediate claim without the uniqueness assertion (if we do not
require $p_{1}:F\times\tilde{U}\longrightarrow F^{\flat}$ to be a fibration
then we lose uniqueness since we can compose on the left with a
$\tilde{G}$-equivariant isogeny).

######  Lemma 3.9.

There exists a $\tilde{G}$-equivariant isotrivial morphism
$p_{1}:F\times\tilde{U}\longrightarrow F^{\flat}$ satisfying (1) and (2) in
Proposition 3.5.

###### Proof.

To minimize potential confusions, we distinguish between the notions of
abelian variety and abelian torsor. We shall divide the proof in three steps.

  * •

Step 1: construction of $F^{\flat}$,

  * •

Step 2: construction of the morphism
$F^{\operatorname{tor}}\times\tilde{U}\longrightarrow
F^{\flat,\operatorname{tor}}$,

  * •

Step 3: proof that $F^{\operatorname{tor}}\times\tilde{U}\longrightarrow
F^{\flat,\operatorname{tor}}\times\tilde{U}$ satisfies the desired properties.

(Step 1) The projection
$\operatorname{pr}_{2}:F^{\operatorname{tor}}\times\tilde{U}\longrightarrow\tilde{U}$
induces a homomorphism
$f:\operatorname{Alb}_{F\times\tilde{U}}\to\operatorname{Alb}_{\tilde{U}}$
between $\tilde{G}$-Albanese varieties and a short exact sequence of
$\tilde{G}$-equivariant Hodge structures

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The reader should recall the subtle fact that $H^{*}(F,\mathbb{Z})$ admits a
canonical $\tilde{G}$-module structure even though $F$ does not. Since a
$\tilde{G}$-equivariant polarizable Hodge structure over $\mathbb{Q}$ is
completely reducible this short exact sequence splits over $\mathbb{Q}$, i.e.,
there exists an injective $\tilde{G}$-equivariant Hodge structure homomorphism
$H^{1}(F,\mathbb{Q})\longrightarrow
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Q})$ splitting the
sequence. Define a $\tilde{G}$-equivariant Hodge substructure $W\subset
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Z})$ by

$W:=\operatorname{im}\Big{[}H^{1}(F,\mathbb{Q})\longrightarrow
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Q})\Big{]}\cap
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Z})\quad\subset\ \
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Z}).$

Using the contravariant equivalence between the category of polarizable
$\tilde{G}$-Hodge structures of weight one and that of $\tilde{G}$-abelian
varieties, we can translate this geometrically to a surjective homomorphism of
$\tilde{G}$-abelian varieties

$\operatorname{Alb}_{F\times\tilde{U}}\longrightarrow F^{\flat},$

where $H^{1}(F^{\flat},\mathbb{Z})=W$. The composition $W\hookrightarrow
H^{1}(\operatorname{Alb}_{F\times\tilde{U}},\mathbb{Z})\twoheadrightarrow
H^{1}(F,\mathbb{Z})$ has finite kernel so the following composition is an
isogeny of abelian varieties inducing a $\tilde{G}$-equivariant map on
cohomology

${F}$${\operatorname{Alb}_{F\times\tilde{U}}}$${F^{\flat}}$${x}$${x^{\flat}.}$

(Step 2) The morphism $\operatorname{Alb}_{F\times\tilde{U}}\to
F^{\flat}\times\operatorname{Alb}_{\tilde{U}}$ is a $\tilde{G}$-equivariant
isogeny of $\tilde{G}$-abelian varieties, where the $\tilde{G}$-action on the
right hand side is diagonal. Let $K$ be its kernel. It is a finite
$\tilde{G}$-abelian group acting freely on the Albanese torsor
$\operatorname{Alb}_{F\times\tilde{U}}^{\operatorname{tor}}=F^{\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$
and the quotient
$(F^{\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}})/K$
is an $F^{\flat}\times\operatorname{Alb}_{\tilde{U}}$-torsor, so it is
isomorphic to
$F^{\flat,\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$.
The quotient morphism is necessarily of the form (Proposition B.5)

$\displaystyle
q^{\prime}:F^{\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$
$\displaystyle\longrightarrow
F^{\flat,\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}$
$\displaystyle(x,y)\;\;\;\;\;\;\;$
$\displaystyle\longmapsto(x^{\flat}+e^{\prime}(y),y)$

for a morphism
$e^{\prime}:\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}\longrightarrow
F^{\flat,\operatorname{tor}}$. Defining a new morphism $q$ by

$\displaystyle q:F^{\operatorname{tor}}\times\tilde{U}$
$\displaystyle\longrightarrow\;F^{\flat,\operatorname{tor}}\times\tilde{U}$
$\displaystyle(x,y)\;\;\;$ $\displaystyle\longmapsto(x^{\flat}+e(y),y),$

where $e$ denotes the composition
$\tilde{U}\to\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}\xrightarrow{e^{\prime}}F^{\flat,\operatorname{tor}}$,
we obtain a _cartesian_ diagram

${F^{\operatorname{tor}}\times\tilde{U}}$${F^{\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}}$${F^{\flat,\operatorname{tor}}\times\tilde{U}}$${F^{\flat,\operatorname{tor}}\times\operatorname{Alb}_{\tilde{U}}^{\operatorname{tor}}.}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$
(3.10)

The desired morphism $p_{1}:F^{\operatorname{tor}}\times\tilde{U}\to
F^{\flat,\operatorname{tor}}$ is the composition of $q$ with the projection to
the first factor i.e., $p_{1}(x,y)=x^{\flat}+e(y)$.

(Step 3) It remains to prove that the morphism $q$ is a Galois covering and
that the $\tilde{G}$-action on $F^{\flat,\operatorname{tor}}\times\tilde{U}$
is diagonal. First, the $\tilde{G}$-action on
$F^{\flat,\operatorname{tor}}\times\tilde{U}$ is diagonal because the induced
action on its Albanese variety $F^{\flat}\times\operatorname{Alb}_{\tilde{U}}$
is diagonal by construction (Lemma 3.8). The map $q$ is a Galois covering
since $q^{\prime}$ is and diagram (3.10) is cartesian. ∎

The Stein factorization of a morphism $F\times\tilde{U}\longrightarrow
F^{\flat}$ as in Lemma 3.9 will be the desired (unique) fibration of
Proposition 3.5.

###### Proof of Proposition 3.5.

We again divide the proof into two steps: the existence and uniqueness of the
fibration.

(Existence) Let $p_{1}:F\times\tilde{U}\longrightarrow F^{\flat}$ be a
morphism as in Lemma 3.9, which takes the form $p_{1}(x,y)=x^{\flat}+e(y)$,
for a morphism $e:\tilde{U}\longrightarrow F^{\flat}$. Consider a smooth
projective compactification $Y\supset\tilde{U}$. Then $e$ uniquely extends to
$e:Y\longrightarrow F^{\flat}$, so that we can extend $p_{1}$ canonically to

$\displaystyle p_{1}:F\times Y$
$\displaystyle\longrightarrow\;\;\;\;F^{\flat}$ $\displaystyle(x,y)\;$
$\displaystyle\longmapsto x^{\flat}+e(y).$

It is a smooth and projective fiber bundle whose fiber is isomorphic to a
(possibly disconnected) finite étale covering of $Y$. Let $F^{\natural}$ be
the variety appearing in the Stein factorization of $p_{1}$:

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The morphism $F\times Y\longrightarrow F^{\natural}$ is an isotrivial
fibration whose fiber $T$ is a finite étale connected covering of $Y$. Its
restriction $F\times\tilde{U}\longrightarrow F^{\natural}$ is also isotrivial
with a smooth and connected fiber $M\subset T$. The morphism $F^{\natural}\to
F^{\flat}$ is finite étale because the number of connected components of the
fibers of $p_{1}$ is constant. This proves that $F^{\natural}$ is an abelian
torsor. Finally, from the definition one easily checks that $F\times
Y\longrightarrow F^{\natural}\times Y$ is a Galois covering.

It remains to prove that $F^{\natural}$ admits a $\tilde{G}$-action making
diagram (3.11) $\tilde{G}$-equivariant. By definition, the function field
$K(F^{\natural})$ of the Stein factorization is the algebraic closure of
$K(F^{\flat})$ in $K(F\times\tilde{U})$. The embedding $K(F^{\flat})\subset
K(F\times\tilde{U})$ is $\tilde{G}$-equivariant, so its algebraic closure is
closed under the $\tilde{G}$-action. This endows $F^{\natural}$ with a
birational $\tilde{G}$-action. However, since $F^{\natural}$ is an abelian
torsor, a birational automorphism of $F^{\natural}$ is an automorphism. It
follows that $F^{\natural}$ is a $\tilde{G}$-abelian torsor and that diagram
(3.11) is $\tilde{G}$-equivariant.

(Uniqueness) Suppose $F_{i}^{\flat}$, $i=1,2$ are two $\tilde{G}$-abelian
varieties isogenous to $F$ and that $p_{i}:F\times\tilde{U}\longrightarrow
F_{i}^{\flat}$, $i=1,2$ are two fibrations satisfying the conditions of the
proposition. Fix a smooth projective compactification $Y\supset\tilde{U}$. The
morphism $p_{i}:F\times\tilde{U}\longrightarrow F^{\flat}_{i}$ is necessarily
of the form $p_{1}(x,y)=x^{\flat}+e(y)$ by Proposition B.5, so it canonically
extends to a fibration $F\times Y\longrightarrow F^{\flat}_{i}$. The abelian
varieties $F_{1}^{\flat}$ and $F_{2}^{\flat}$ are isogenous to $F$ and the
$\tilde{G}$-representations $H^{1}(F_{i}^{\flat},\mathbb{Q})$ for $i=1,2,3$
are isomorphic to each other. We deduce that $F_{1}^{\flat}$ and
$F_{2}^{\flat}$ are $\tilde{G}$-equivariantly isogenous. Hence, there exists a
third $\tilde{G}$-abelian variety $F_{3}^{\flat}$ with $\tilde{G}$-equivariant
isogenies dominated by both $F_{1}^{\flat}$ and $F_{2}^{\flat}$, giving a
commutative diagram

${F\times
Y}$${F_{1}^{\flat}}$${F_{2}^{\flat}}$${F_{3}^{\flat}.}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

The (equivariant) uniqueness of the Stein factorization then implies that
$p_{1}=p_{2}$. ∎

###  3.2. Kodaira dimension of the intermediate trivialization

We now return to the study of the birational type of the base $U$ of the
intermediate trivialization. Namely, we establish that the Kodaira dimension
$\kappa(U)$ is equal to $0$ or $n$, as stated in Theorem 3.2 of the
introduction. The proof of the fact that if $\kappa(U)=0$ then $U$
compactifies to an abelian torsor will appear in the next section (Proposition
4.4).

###### Proof of Theorem 3.2.

We first prove that $U$ is either of general type or birational to a
$G$-abelian torsor. Consider the equivariant morphism $U\longrightarrow A$
from Proposition 2.19 and fix an equivariant smooth projective
compactification $U\subset\tilde{Y}$. Then the morphism $U\longrightarrow A$
extends to a morphism $\tilde{Y}\longrightarrow A$ because $\tilde{Y}$ is
smooth. Its Stein factorization gives an equivariant fibration
$\tilde{Y}\longrightarrow Y$ together with an equivariant finite morphism
$f:Y\longrightarrow A$. The fibration $\tilde{Y}\longrightarrow Y$ is a
birational morphism since $\tilde{Y}\longrightarrow A$ is generically finite
on its image by Proposition 2.19. It thus suffices to prove that the normal
projective $G$-variety $Y$ is either of general type or an abelian torsor.

By an equivariant version of a theorem of Ueno and Kawamata [Kaw81, Thm 13],
which we include as Theorem C.2 for completeness, there exists a $G$-abelian
subvariety $A_{1}\subset A$ such that

$\kappa(Y)=n-\dim A_{1}.$

Since $A$ is a simple $G$-abelian variety, $A_{1}\subset A$ is either a point
or $A$ itself. If $A_{1}$ is a point then we obtain $\kappa(Y)=n$. If
$A_{1}=A$, since $\dim A\geq n$ we have $\kappa(Y)=n-\dim A\leq 0$. This
forces $\kappa(Y)=0$ and $\dim A=n$. A theorem of Kawamata–Viehweg [KV80]
[Kaw81, Thm 4] states that if a morphism $f:Y\longrightarrow A$ to an abelian
torsor is finite surjective and $\kappa(Y)=0$, then $f$ is étale and $Y$ is an
abelian torsor. One shows that it is isogenous to the $n$-th power of an
elliptic curve by the same reasoning as in Theorem 2.14: use Corollary 2.20 to
deduce that $Y$ is isogenous to a power of a simple abelian torsor, and the
fact that $Y/G$ is $\mathbb{Q}$-Fano to conclude that it is a power of an
elliptic curve.

Next, we apply to same methods to study the Kodaira dimension of $Z$. In
Theorem 3.1, we constructed a finite morphism $Z\longrightarrow\bar{F}\times
U$ and in Proposition 2.19 a morphism $U\longrightarrow A$, which is
generically finite on its image. We thus obtain a morphism
$Z\longrightarrow\bar{F}\times A$ that is generically finite on its image. As
above, considering a smooth equivariant compactification of $Z$ and a Stein
factorization gives an equivariant finite morphism
$f:Y\longrightarrow\bar{F}\times A$ from a normal projective $G$-variety $Y$
that is birational to $Z$. Applying Theorem C.2 to
$f:Y\longrightarrow\bar{F}\times A$, we find three possibilities:
$\kappa(Z)=2n$, $n$ or $0$ (because $\dim A$ is a multiple of $n$ by
Proposition 2.19). However $\kappa(Z)\leq n$, because the morphism
$p:Z\longrightarrow U$ with general fiber $F$ gives an upper bound on the
Kodaira dimension (e.g., [Mor87, Cor 2.3])

$\kappa(Z)\leq\kappa(F)+\dim U=n.\vskip 3.0pt plus 1.0pt minus 1.0pt$

If $\kappa(Z)=0$ then Kawamata’s characterization of abelian torsors [Kaw81,
Cor 2] shows that $Z$ is birational to an abelian torsor. If $\kappa(Z)=n$,
notice that the dimension of $U$ coincides with the Kodaira dimension of $Z$,
and that $Z\longrightarrow U$ is a fibration whose very general fiber has
Kodaira dimension $0$. These two properties characterize the Iitaka fibration
of $Z$ (see [Mor87, Def-Thm 1.11]). In fact, the diagram

${Z}$${\bar{F}\times
A}$${U}$${A}$$\scriptstyle{p}$$\scriptstyle{\operatorname{pr}_{2}}$

is precisely the Ueno–Kawamata diagram from Theorem C.2. ∎

###  3.3. The second isotrivial fibration

Another interesting byproduct of Theorem 3.1 is the existence of a second
fibration of the hyper-Kähler manifold $X$. Since
$p^{\prime}:Z\longrightarrow\bar{F}$ is $G$-equivariant, taking its
$G$-quotient yields a new fibration $X_{0}\longrightarrow
B^{\prime}:=\bar{F}/G$, or a rational fibration $\pi^{\prime}:X\dashrightarrow
B^{\prime}$.

######  Proposition 3.12.

There exists a preferred choice of a second rational isotrivial Lagrangian
fibration $\pi^{\prime}:X\dashrightarrow B^{\prime}$ to a projective
$\mathbb{Q}$-Fano variety $B^{\prime}$.

###### Proof.

As mentioned, $\pi^{\prime}$ is constructed by taking the quotient of
$p^{\prime}:Z\longrightarrow\bar{F}$ in Theorem 3.1. Let us first show its
general fiber is Lagrangian. Let $\sigma$ be a symplectic form on $Z$ pulled
back from that on $X$. A general fiber of $\pi^{\prime}$ is a general fiber of
$p^{\prime}:Z\longrightarrow\bar{F}$, so it suffices to prove the latter is
Lagrangian with respect to $\sigma$. Decompose $p^{\prime}$ into
$Z\longrightarrow\bar{F}\times U\longrightarrow\bar{F}$, where the first map
$(p^{\prime},p)$ is finite étale by Theorem 3.1 and the second map is the
projection onto the first factor. The pullback homomorphism
$H^{*}(\bar{F}\times U,\mathbb{Q})\longrightarrow H^{*}(Z,\mathbb{Q})$ is an
isomorphism of $G$-equivariant Hodge structures by Proposition 2.18. By Lemma
2.22, the general fiber of $\operatorname{pr}_{1}:\bar{F}\times
U\longrightarrow\bar{F}$ is Lagrangian with respect to $\text{Tr}(\sigma)\in
H^{2,0}(\bar{F}\times U)$. Hence the general fiber of $p^{\prime}$ is
Lagrangian with respect to $\sigma$. Finally, the base $B^{\prime}$ is
$\mathbb{Q}$-Fano by Lemma 2.15. ∎

###### Proof of Theorem 1.3.

Corollary 2.12 shows that $F$ is isogenous to a power of a simple abelian
variety. It therefore suffices to show that $F$ has an isogeny factor of
dimension $1$. Let $\pi^{\prime}:X\dashrightarrow B^{\prime}$ be the second
fibration from Proposition 3.12. We proved that the variety
$B^{\prime}=\bar{F}/G$ is $\mathbb{Q}$-Fano. If a finite quotient $B^{\prime}$
of an abelian torsor $\bar{F}$ is $\mathbb{Q}$-Fano, then $\bar{F}$ is
isogenous to a product of elliptic curves by [Shi21, Lem 4.2]. This concludes
the proof of the theorem.

For the sake of completeness, let us also present a simple argument which does
not rely on [Shi21, Lem 4.2]. Let
$q:\bar{F}\longrightarrow\bar{F}/G=B^{\prime}$ be the quotient map. Since
$K_{\bar{F}}$ is trivial and $K_{B^{\prime}}$ is anti-ample, $q$ is
necessarily ramified at some codimension $1$ points in $\bar{F}$.
Equivalently, there exists a prime divisor $D\subset\bar{F}$ whose generic
point has a non-trivial stabilizer $H\subset G$. The restriction map
$H^{1}(\bar{F},\mathbb{Q})\longrightarrow H^{1}(D,\mathbb{Q})$ cannot be
injective since $G$ acts faithfully on $H^{1}(\bar{F},\mathbb{Q})$ and $H$
acts trivially on $H^{1}(D,\mathbb{Q})$. The following lemma ensures that
$\bar{F}$ has an isogeny factor of dimension one. ∎

######  Lemma 3.13.

Let $\bar{F}$ be an abelian torsor and $D\subset\bar{F}$ a prime divisor such
that $H^{1}(\bar{F},\mathbb{Q})\longrightarrow H^{1}(D,\mathbb{Q})$ is not
injective. Then $D$ is an abelian subtorsor of $\bar{F}$. In particular,
$\bar{F}$ has a 1-dimensional isogeny factor.

###### Proof.

Factorize the inclusion $D\subset\bar{F}$ through the Albanese morphism
$D\longrightarrow\operatorname{Alb}^{\operatorname{tor}}_{D}\longrightarrow\bar{F}$.
Since $H^{1}(\bar{F},\mathbb{Q})\longrightarrow H^{1}(D,\mathbb{Q})$ is not
injective, the morphism
$\operatorname{Alb}^{\operatorname{tor}}_{D}\longrightarrow\bar{F}$ is not
surjective. Its image $D$ is therefore an abelian subtorsor containing $D$.
This implies that $D$ is an abelian subtorsor of codimension $1$.∎

######  Remark 3.14.

Let $M$ be a general fiber of the rational (isotrivial) Lagrangian fibration
$\pi^{\prime}:X\dashrightarrow B^{\prime}$ above. Then one can prove that $M$
is either birational to an abelian variety or compactifies to a variety of
general type. Moreover, the following are equivalent:

  * •

$\pi$ is of type A.1.

  * •

$M$ is birational to an abelian variety.

  * •

There exists a birational hyper-Kähler model $X^{\prime}$ of $X$ such that the
rational map $\pi^{\prime}:X^{\prime}\dashrightarrow B^{\prime}$ is a
morphism.

This can be deduced, for example, from Proposition 5.1 below. We omit the
proof as this will not be needed in the sequel. We will later discuss yet
another characterization of type A fibrations in Proposition 4.24.

##  4\. General singular fibers and the canonical bundle formula

So far, our analysis has been focused on the smooth fibration
$\pi_{0}:X_{0}\longrightarrow B_{0}$ obtained by restricting $\pi$ to the
complement of the singular fibers, and in particular on the cohomological
properties of the general fiber $F$ and of the base $U$ of the intermediate
trivialization. We now extend our study to a fibration
$\pi_{1}:X_{1}\longrightarrow B_{1}$ obtained by restricting $\pi$ to the
complement of a codimension two locus pulled back from the base $B$. There are
two general tools that apply to this situation. First, Hwang–Oguiso [HO09,
HO11] have analyzed the behavior of general singular fibers using the notion
of characteristic cycle. The second tool is the canonical bundle formula of
Kawamata [Kaw98], Ambro [Amb04, Amb05], and Fujino [Fuj03], which is a higher
dimensional generalization of Kodaira’s canonical bundle formula for elliptic
fibrations. It takes the form

$K_{X}\sim_{\mathbb{Q}}\pi^{*}(K_{B}+M_{B}+D_{\pi}),$ (4.1)