Parabolic automorphisms
of hyperkähler manifolds

Ekaterina Amerik111Both authors acknowledge support of HSE University basic
research program; also partially supported by ANR (France) project FANOHK
Misha Verbitsky222Partially supported by FAPERJ E-26/202.912/2018 and CNPq -
Process 313608/2017-2. Keywords: hyperkähler manifold, ergodic theory, 2010
Mathematics Subject Classification:

Abstract

A parabolic automorphism of a hyperkähler manifold is a holomorphic
automorphism acting on $H^{2}(M)$ by a non-semisimple quasi-unipotent linear
map. We prove that a parabolic automorphism which preserves a Lagrangian
fibration acts on its fibers ergodically. The invariance of a Lagrangian
fibration is automatic for manifolds satisfying the hyperkähler SYZ
conjecture; this includes all known examples of hyperkähler manifolds. When
there are two parabolic automorphisms preserving two distinct Lagrangian
fibration, it follows that the group they generate acts on $M$ ergodically.
Our results generalize those obtained by S. Cantat for K3 surfaces.

###### Contents

  1. 1 Introduction
  2. 2 Basic notions
    1. 2.1 Ergodic group action on K3 surfaces
    2. 2.2 Hyperkähler manifolds
    3. 2.3 MBM classes and existence of automorphisms
    4. 2.4 Holomorphic Lagrangian fibrations on hyperkähler manifolds
    5. 2.5 The hyperbolic space and its isometries
  3. 3 Parabolic automorphisms of hyperkahler manifolds
    1. 3.1 Hyperkähler manifolds with ergodic automorphism groups
    2. 3.2 Parabolic automorphisms and Lagrangian fibrations on hyperkähler manifolds
    3. 3.3 Parabolic automorphisms are fiberwise ergodic on Lagrangian fibrations
  4. 4 Variations of Hodge structures and Deligne’s semisimplicity theorem
    1. 4.1 Hodge structures
    2. 4.2 Variations of Hodge structures
    3. 4.3 Deligne’s semisimplicity theorem
    4. 4.4 Deligne’s global invariant cycle theorem
  5. 5 Variations of Hodge structures and Lagrangian fibrations
    1. 5.1 AM-GM inequality, products of Hermitian forms and Lagrangian fibrations
    2. 5.2 Irreducibility of the VHS associated with a Lagrangian fibration
  6. 6 Parabolic automorphisms and their orbits
    1. 6.1 Parabolic automorphisms and monodromy-invariant subtori
    2. 6.2 Ergodicity of parabolic automorphisms
  7. 7 Isotrivial Lagrangian fibrations

##  1 Introduction

Let $S$ be a projective K3 surface and let $f:S\to S$ be an automorphism. The
Neron-Severi lattice $\operatorname{\sf NS}(S)$ is of signature
$(1,\rho_{S}-1)$ where $\rho_{S}$ is the Picard number of $S$, and $f$ induces
an automorphism $f^{*}$ of $NS(S)$. When $\rho_{S}\geqslant 3$, the
projectivization of the positive cone in $\operatorname{\sf
NS}(S)\otimes{\mathbb{R}}$ can be viewed as a model of the hyperbolic space
which we denote by ${\mathbb{H}}_{S}$. According to the classification of the
isometries of the hyperbolic space, $f^{*}$ can be either elliptic, or
loxodromic, or parabolic. By definition, an elliptic isometry has a fixed
point inside ${\mathbb{H}}_{S}$. It has to be of finite order in presence of
an underlying integral structure (the Neron-Severi lattice here). A loxodromic
isometry has two fixed points on the boundary of ${\mathbb{H}}_{S}$, and a
parabolic one has a single one. In this last case, $f^{*}$ preserves a class
of self-intersection zero, which one may clearly assume nef (by convexity of
the nef cone) and integral (as $f^{*}$ is integral).

By Gizatullin’s theorem ([G, Grv]), any parabolic automorphism $p$ of a
projective surface admits an invariant elliptic fibration, or “elliptic
pencil”, that is, a fibration $\pi:S\to{\mathbb{P}}^{1}$ with general fiber an
elliptic curve and such that $p$ sends fibers to fibers. For a K3 surface this
is easy to see directly, indeed one deduces from the Riemann-Roch theorem that
the linear system of the sections of a nef line bundle of square zero is an
elliptic pencil. Moreover, the action of $p$ on the base of this fibration has
finite order, hence a power of $p$ preserves the fibers.

Serge Cantat in [C1] has shown that a parabolic automorphism of a K3 surface
has dense orbits on almost all elliptic fibers. Moreover if two automorphisms
$f_{1},f_{2}$ preserve distinct elliptic fibrations $\pi_{1},\pi_{2}$, then
the group they generate acts on $S$ ergodically, that is, every measurable
invariant subset has zero or full measure. The ergodicity statement follows
from the density of orbits by general theory, so only the density statement
requires some work to be done.

Our aim is to generalize this result to hyperkähler manifolds of arbitrary
dimension.

Let us recall some more remarks about elliptic pencils on K3 surfaces and
parabolic automorphisms from [C1], some of them well-known.

Let $S$ be a K3 surface. If the intersection form on $\operatorname{\sf
NS}(S)$ represents zero, then $S$ carries an elliptic pencil $\pi$. Moreover,
if the rank of $\operatorname{\sf NS}(S)$ is greater than two and there are no
$(-2)$-curves on $S$, then there is always a parabolic automorphism preserving
$\pi$. Indeed,111For any K3 surface $S$, the group generated by the reflexions
in the $(-2)$-curves and $\operatorname{Aut}(S)$ is a lattice in
$O(\operatorname{\sf NS}\otimes_{\mathbb{Z}}{\mathbb{R}})$ ([PSh]).
$\operatorname{Aut}(S)$ is a lattice, hence the stabilizer of an isotropic
class has infinite order: otherwise the quotient of ${\mathbb{H}}_{S}$ by
$\operatorname{Aut}(S)$ has an end of infinite volume.

As an example of a surface admitting two parabolic automorphisms with distinct
nef eigenvectors we may take a complete intersection of type $(2,2,2)$ in
$({\mathbb{P}}^{1})^{3}$, indeed each of the three projections to
${\mathbb{P}}^{1}$ is an elliptic pencil and each of the three projections to
$({\mathbb{P}}^{1})^{2}$ is a double covering. The product of any two of the
three covering involutions is parabolic (see e.g. [C1], Exemple 5, and [W] for
details).

The purpose of the present article is to generalize this picture to higher
dimension, that is, for arbitrary projective irreducible holomorphic
symplectic manifolds (IHSM), or maximal holonomy hyperkähler manifolds. As
usual, the role of the intersection form is played by the Beauville-Bogomolov-
Fujiki (BBF) form, which we denote by $q$ (see Subsection 2.2). With this
form, the projectivization of the positive cone in the real Neron-Severi group
is again viewed as a model of the hyperbolic space.

The classification of automorphisms of a hyperbolic space plays here the same
role as for complex surfaces: elliptic automorphisms have finite order,
loxodromic automorphisms have positive entropy ([O2]), and parabolic
automorphisms (conditional on SYZ conjecture, which is true in all known
examples) preserve an Abelian fibration and act ergodically in its fibers
(3.2).

The following statement is widely known as the hyperkähler SYZ conjecture
([V1]) or Lagrangian conjecture (LC):

Conjecture 1.1: Every nef Beauville-Bogomolov isotropic line bundle on an
irreducible holomorphic symplectic manifold $M$ is semiample, that is, if $L$
is such a line bundle, then $L^{\otimes N}$ is base-point-free for $N>>0$.

Conditional on SYZ conjecture (which is true for all known examples of maximal
holonomy hyperkähler manifolds), the sections of $L^{\otimes N}$ define a
fibration $\pi:M\to X$. By a famous result of Matsushita, $\pi$ is a
Lagrangian fibration, all smooth fibers of $\pi$ are tori and the base is
“very similar” to the projective space, in particular it is a Fano manifold of
Picard number 1. A priori the base might be singular; if it is smooth, it is
biholomorphic to ${\mathbb{P}}^{n}$ ([Hw]).

Assume there is an automorphism $f$ preserving the class of $L$ ($f$ is then
automatically parabolic, as follows from the classification of isometries of
the hyperbolic space). By a theorem of F. Lo Bianco (3.2), $f$ induces a
finite order automorphism of the base $X$, so that replacing $f$ by a power we
may assume that $f$ commutes with $\pi$.

Our main result shows that the orbits of $f$ on a general fiber of $\pi$ are
dense in the analytic topology. Following Cantat, we derive that if there are
two parabolic automorphisms $p_{1},p_{2}$ preserving distinct Lagrangian
fibrations $\pi_{1},\pi_{2}$, then the group generated by $p_{1}$ and $p_{2}$
acts ergodically. Preserving distinct Lagrangian fibrations is equivalent to
having distinct axes (i.e. invariant lines in $\operatorname{\sf
NS}(M)\otimes{\mathbb{R}}$ generated by nef classes).

Concerning the existence of parabolic automorphisms in the presence of
Beauville-Bogomolov isotropic nef line bundles, the role of $(-2)$-curves on a
K3 surface is played in higher dimension by MBM classes (see [AV1] for their
definition and properties). If the Neron-Severi lattice of $M$ is of rank at
least three, represents zero and contains no MBM classes, then the
automorphism group of $M$ is a lattice, as follows from the global Torelli
theorem. Since any lattice in $SO(\operatorname{\sf NS}(M))$ contains a
parabolic element, this implies the existence of a parabolic automorphism.

To construct the maximal holonomy hyperkähler (that is, ISH) manifolds with no
MBM classes, we recall from [AV4] that the MBM classes have bounded square:
$0>q(z)>-N$. Therefore to exhibit a lattice with a parabolic automorphism it
is sufficient to find one representing zero and not representing numbers
between zero and $-N$. A primitive embedding of such a lattice into
$H^{2}(M,{\mathbb{Z}})$ gives a deformation of $M$ with exactly this Neron-
Severi lattice.

Such a construction has been first carried out in [AV2], for loxodromic as
well as parabolic automorphisms. We give a particularly simple parabolic
version of these results in section 2.3: we observe that the absence of MBM
classes orthogonal to $L$, rather than the total absence, suffices to produce
parabolic automorphisms. This substantially simplifies the construction (see
also [Mats3]).

##  2 Basic notions

###  2.1 Ergodic group action on K3 surfaces

For an introduction to ergodic theory and dynamics, see, for example, [VO].

Definition 2.1: Let $\Gamma$ be a group acting on a manifold $M$ and
preserving a measure $\mu$. We say that the action of $\Gamma$ is ergodic if
any $\Gamma$-invariant measurable subset of $M$ is full measure or measure 0,
with respect to $\mu$.

Remark 2.2: Equivalently, $(M,\Gamma)$ is ergodic iff any $\Gamma$-invariant
integrable function is constant almost everywhere.

Remark 2.3: From ergodicity it follows that almost all orbits of $\Gamma$ are
dense, but the converse is not true ([Fur]). However if $g:K\to K$ is a
translation of a compact group $K$, the density of orbits implies the
ergodicity (and in fact even a stronger property of unique ergodicity, [VO],
section 6.3.3). Indeed, an action of a group on a Riemannian manifold by
isometries, with all orbits dense, is always uniquely ergodic.

A K3 surface has a natural measure coming from the volume form
$\Omega\wedge\overline{\Omega}$, where $\Omega$ generates $H^{2,0}(M)$. The
ergodicity of group actions on a K3 surface has been studied by S. Cantat in
[C1], [C2]. The article [C1] is especially concerned with parabolic
automorphisms (also called “Dehn twists along an elliptic fibration” in [C1]).

In particular, Cantat proves the following

Proposition 2.4: ([C1], Proposition 2.2, Proposition 2.1) Let
$\pi:S\to{\mathbb{P}}^{1}$ be an elliptic K3 surface and $f$ be an infinite
order automorphism of $S$ preserving $\pi$. Then for almost all
$x\in{\mathbb{P}}^{1}$ (with respect to the Lebesgue measure), $f$ acts on
$\pi^{-1}(x)$ with dense orbits.

Corollary 2.5: ([C1], Corollaire 3.2) On a K3 surface $S$ with two infinite
order automorphisms preserving two distinct elliptic fibrations,
$\operatorname{Aut}(S)$ acts ergodically.

Sketch of proof: Each pencil $\pi_{1}$, $\pi_{2}$ is preserved by a parabolic
automorphism $p_{1}$ resp. $p_{2}$, which one may assume a fiberwise
translation (after taking a power). Let $F$ be an invariant measurable
function. Then $F$ must be constant on a full measure subset along almost all
fibers of both fibrations, because $p_{1}$ resp. $p_{2}$ act ergodically on
fibers. Since a general fiber of $\pi_{1}$ intersects a general fiber of
$\pi_{2}$ (in other words, it is mapped onto ${\mathbb{P}}^{1}$ by $\pi_{2}$),
$F$ is constant almost everywhere.

Corollary 2.6: (cf. [C2], section 1.5) Let $M$ be a K3-surface which is
obtained as a degree (2,2,2)-hypersurface in
${\mathbb{C}}P^{1}\times{\mathbb{C}}P^{1}\times{\mathbb{C}}P^{1}$, and
$\Gamma$ its automorphism group. Then $\Gamma$ acts on $M$ ergodically.

Proof: Indeed the projection to each ${\mathbb{C}}P^{1}$ is an elliptic
fibration, and the product of every two of three involutions from the remark
below is parabolic, preserving one of these three fibrations (see [W]).

Remark 2.7: Since $M$ has degree 2 in each variable, it has 2:1 projection to
${\mathbb{C}}P^{1}\times{\mathbb{C}}P^{1}$. This gives an order 2 automorphism
exchanging these two preimages. We obtain 3 order 2 automorphisms acting on
$M$. It is then easy to see, according to Cantat, that they generate the free
product
$({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})$.

Remark 2.8: The group
$({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})$
is an index 6 subgroup in
$PGL(2,{\mathbb{Z}})=({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})\rtimes\Sigma_{3}$,
where the projection to the symmetric group is given by
$PGL(2,{\mathbb{Z}}){\>\longrightarrow\>}PGL(2,{\mathbb{Z}}/2)=\Sigma_{3}$
([GMST, §2.2]).

The aim of this paper is to construct, in all dimensions, hyperkähler
manifolds with ergodic automorphism groups.

###  2.2 Hyperkähler manifolds

We introduce the hyperkähler manifolds, following [Bes] and [Bea].

Definition 2.9: A hyperkähler structure on a manifold $M$ is a Riemannian
structure $g$ and a triple of complex structures $I,J,K$, satisfying
quaternionic relations $I\circ J=-J\circ I=K$, such that $g$ is Kähler for
$I,J,K$.

Remark 2.10: A hyperkähler manifold has three symplectic forms:
$\omega_{I}:=g(I\cdot,\cdot)$, $\omega_{J}:=g(J\cdot,\cdot)$,
$\omega_{K}:=g(K\cdot,\cdot)$.

Remark 2.11: This is equivalent to $\nabla I=\nabla J=\nabla K=0$: the
parallel translation along the connection preserves $I,J,K$.

Definition 2.12: Let $M$ be a Riemannian manifold, $x\in M$ a point. The
subgroup of $GL(T_{x}M)$ generated by parallel translations (along all paths)
is called the holonomy group of $M$.

Remark 2.13: A hyperkähler manifold can be defined as a manifold which has
holonomy in $Sp(n)$ (the group of all endomorphisms preserving $I,J,K$).

Definition 2.14: A holomorphically symplectic manifold is a complex manifold
equipped with non-degenerate, holomorphic $(2,0)$-form.

Remark 2.15: Hyperkähler manifolds are holomorphically symplectic. Indeed,
$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ is a holomorphic symplectic form on
$(M,I)$.

Theorem 2.16: (Calabi-Yau; [Y]) A compact, Kähler, holomorphically symplectic
manifold admits a unique hyperkähler metric in any Kähler class.

Definition 2.17: A hyperkähler manifold $M$ is called maximal holonomy, or
IHS, if $\pi_{1}(M)=0$, $H^{2,0}(M)={\mathbb{C}}$.

Bogomolov’s decomposition: ([Bo1]) Any hyperkähler manifold admits a finite
covering which is a product of a torus and several hyperkähler manifolds of
maximal holonomy.

Remark 2.18: For the rest of this paper a hyperkähler manifold means “a
compact, Kähler, holomorphically symplectic manifold of maximal holonomy”; it
is also called “an IHS manifold”, from “irreducible holomorphically
symplectic”.

Theorem 2.19: (“Bochner’s vanishing”, [Bes])
Let $M$ be a maximal holonomy hyperkähler manifold. Then $H^{p,0}=0$ for $p$
odd, and $H^{p,0}={\mathbb{C}}$ for $p$ even.

Theorem 2.20: (Fujiki, [Fuj]). Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where
$M$ is hyperkähler. Then $\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, for some
primitive integer quadratic form $q$ on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ a
rational number.

Definition 2.21: This form is called Bogomolov-Beauville-Fujiki form. It is
defined by the Fujiki’s relation uniquely if we also ask that the square of a
Kahler class should be positive.

Remark 2.22: The form $q$ has signature $(3,b_{2}-3)$, and signature
$(1,b_{2}-3)$ on the space of $(1,1)$-classes (see [Bea]) . It is negative
definite on primitive forms, and positive definite on
$\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega,\omega\rangle$, where
$\omega$ is a Kähler form. If $M$ is projective, it has signature $(1,\rho-1)$
on the space $\operatorname{\sf NS}(M)\otimes{\mathbb{R}}$, where
$\operatorname{\sf NS}(M)$ is the Neron-Severi group and $\rho=\rho(M)$ the
Picard number.

Remark 2.23: The polarized form of Fujiki’s relation which we shall use in
section 5.1 reads as follows:

$\int_{M}\eta_{1}\wedge\dots\wedge\eta_{2n}=K\sum_{\sigma\in
S_{2n}}q(\eta_{\sigma(1)},\eta_{\sigma(2)})\dots
q(\eta_{\sigma(2n-1)},\eta_{\sigma(2n)})$

for a positive rational constant $K$ (one can establish a relation between $K$
and $c$, but we shall not need it here and leave it as an exercize).

###  2.3 MBM classes and existence of automorphisms

Let $M$ be projective. We denote by
$\operatorname{Pos}_{{\mathbb{Q}}}(M)$111The notation $\operatorname{Pos}(M)$
usually refers to the positive cone in $H^{1,1}(M)$, so that
$\operatorname{Pos}_{{\mathbb{Q}}}(M)$ is its “algebraic version”. the
connected component of the cone of classes of positive square in
$\operatorname{\sf NS}(M)\otimes{\mathbb{R}}$ which contains the ample
classes. The projectivization
${\mathbb{P}}(\operatorname{Pos}_{{\mathbb{Q}}}(M))$ shall be viewed as a
model for the hyperbolic space ${\mathbb{H}}^{k}$, where $k=\rho-1$.

Definition 2.24: ([Mar]) Let the monodromy group $\operatorname{\sf Mon}$ be
the subgroup of $\operatorname{Aut}(H^{2}(M,{\mathbb{Z}}))$ generated by the
parallel transports along the Gauss-Manin connection in families. The Hodge
monodromy group, which we denote by $\Gamma$ here, is the image in
$\operatorname{Aut}(\operatorname{\sf NS}(M))$ of the subgroup of
$\operatorname{\sf Mon}$ preserving the Hodge decomposition.

Theorem 2.25: ([V2]) The group $\operatorname{\sf Mon}$ is a finite index
subgroup of the lattice $O(H^{2}(M,{\mathbb{Z}}),q)$, and $\Gamma$ is a finite
index subgroup of $O(\operatorname{\sf NS}(M),q)$. In particular, $\Gamma$
acts on ${\mathbb{H}}^{k}={\mathbb{P}}(\operatorname{Pos}_{{\mathbb{Q}}}(M))$
with finite covolume.

Recall that for a K3 surface, the ample cone is cut out inside the positive
cone by the hyperplane orthogonal to the classes of $-2$-curves. In the same
way, for an arbitrary $M$ the ample cone is bounded by the orthogonals to the
MBM classes.

Definition 2.26: A class $z\in\operatorname{\sf NS}(M)$ is MBM if for some
$\gamma\in\Gamma$, the orthogonal to $\gamma z$ contains a face of the ample
cone of some IHS birational model of $M$. A class $z\in H^{2}(M,{\mathbb{Z}})$
is MBM if it is MBM in some complex structure where it is of type $(1,1)$.

An MBM class $z$ always satisfies $q(z)<0$, and the dual homology class to
$z$, up to a scalar, is represented by a rational curve on a deformation of
$M$ ([AV1]).

MBM classes have been studied in [AV1], [AV3], [AV4]. The following theorem
summarizes the results that we use in this paper.

Theorem 2.27: 1) The ample cone is a connected component of the complement in
$\operatorname{Pos}(M)$ to the union of $z^{\bot}$ for all MBM $(1,1)$-classes
$z$.

2) Suppose $b_{2}(M)\geqslant 5$, then $\Gamma$ acts on the set of primitive
MBM classes of type (1,1) with finitely many orbits.

3) For a primitive MBM class $z$, the number $q(z)$ is bounded from below.
Moreover, one can choose a bound which depends only on the deformation type of
$M$.

One can produce examples of hyperkähler manifolds with many automorphisms
using the following theorem by Markman (“the Hodge-theoretic Torelli
theorem”):

Theorem 2.28: ([Mar]) Let $M$ be a hyperkähler manifold and $\gamma\in\Gamma$
an element of the Hodge monodromy such that $\gamma(\alpha)$ is Kähler for
some Kähler class $\alpha$. Then there exists an automorphism $f:M\to M$,
inducing $\gamma$ on $H^{2}(M)$.

The following theorem is a restatement of results proven in [Mar], [AV5] and
[AV3].

Theorem 2.29:  Let $M$ be a projective hyperkähler manifold, and
${\mathbb{H}}_{\mathbb{Q}}:={\mathbb{P}}(\operatorname{Pos}_{{\mathbb{Q}}}(M))$
the hyperbolic space associated with its Picard lattice. Denote by
$S_{\alpha}\subset{\mathbb{H}}_{\mathbb{Q}}$ the hyperplane obtained as
orthogonal complement to an MBM class $\alpha$ of type (1,1), and consider the
set $\\{S_{\alpha}\\}$ of all such hyperplanes. Let $\Gamma\subset
O(\operatorname{\sf NS}(M))$ be the group of Hodge monodromy of $M$. Then
$\Gamma$ acts on ${\mathbb{H}}_{\mathbb{Q}}$ properly discontinuously, with
finite covolume. Moreover, $\Gamma$ acts on the set $\\{S_{\alpha}\\}$ with
finitely many orbits. Consider the quotient
${\mathfrak{K}}:=\frac{H_{\mathbb{Q}}\backslash\bigcup_{\alpha}S_{\alpha}}{\Gamma}$.
Since the set $\frac{\\{S_{\alpha}\\}}{\Gamma}$ is finite, the complement
${\mathfrak{K}}$ has finitely many connected components, denoted
$K_{1},...,K_{n}$. Then

(i)

For each $K_{i}$, the universal orbifold covering of $K_{i}$ is a locally
finite convex hyperbolic polyhedron in
${\mathbb{P}}\operatorname{Pos}_{{\mathbb{Q}}}(M)$, obtained by
projectivization of the ample cone of a hyperkähler birational model of $M$.

(ii)

The set of isomorphism classes of birational models of $M$ is in bijective
correspondence with the set $\\{K_{i}\\}$.

(iii)

Let $M_{i}$ be a birational model of $M$ such that $K_{i}$ is the image of the
projectivization of its ample cone. Denote by $\pi_{1}(K_{i})$ the orbifold
fundamental group, that is, the deck group of the universal cover of $K_{i}$.
Then $\pi_{1}(K_{i})=\operatorname{Aut}(M_{i})/G$, where $G$ is the group of
automorphisms of $M_{i}$ acting trivially on the Neron-Severi group. Moreover,
the group $G$ is finite.

Proof of 2.3 (i): The set of orthogonal complements to MBM classes is locally
finite, and its $\Gamma$-quotient is finite, as shown in [AV3]. In [AV1] it
was proven that the Kähler cones are of birational models of $M$ are locally
polyhedral, and a their codimension 1 faces are precisely orthogonal
complements to the MBM classes.

Let $\operatorname{Pos}(M)$ be the positive cone (the set of positive vectors
in $H^{1,1}(M,{\mathbb{R}})$) and $K\subset\operatorname{Pos}(M)$ the Kähler
cone of a birational model of $M$. A Kähler chamber is a $\Gamma$-image of
$K$. In [Mar], E. Markman has shown that $\operatorname{Pos}(M)$ is a closure
of the union of all Kähler chambers, and if two Kähler chambers intersect,
they coincide.222For a K3 surface, Kähler chambers are sometimes called “Weyl
chambers” ([BF]); indeed, they are in many ways similar to Weyl chambers
associated with the root decomposition of a simple Lie group, with the
(-2)-curves playing the role of root vectors. This takes care of 2.3 (i).

Proof of 2.3 (ii): In [Mar] it was shown that the set $U:=\frac{\text{The set
of K\"{a}hler chambers in $\operatorname{Pos}(M)$}}{\Gamma}$ is in bijective
correspondence with the set of equivalence classes of birational models of
$M$. However, the set $U$ by construction coincides with the set of connected
components of ${\mathfrak{K}}$.

Proof of 2.3 (iii): To see that $\pi_{1}(K_{i})=\operatorname{Aut}(M_{i})/G$,
let $\widetilde{K}_{i}\subset H^{1,1}(M,{\mathbb{Q}})$ be the ample cone of
$M_{i}$. Then $\widetilde{K}_{i}{\>\longrightarrow\>}K_{i}$ is the orbifold
universal covering. As shown in [Mar], an element $u\in\Gamma$ corresponds to
an automorphism of $M_{i}$ if and only if $u$ maps its Kähler chamber to
itself. This gives $\pi_{1}(K_{i})=\operatorname{Aut}(M_{i})/G$. To see that
$G$ is finite, we use the Calabi-Yau theorem, which gives a bijection between
the set of hyperkähler metrics on $M_{i}$ and the set of Kähler classes in
$H^{1,1}(M)$. If $u\in\operatorname{Aut}(M)$ fixes a Kähler class $\omega\in
H^{1,1}(M,{\mathbb{Q}})$, it preserves the hyperkähler metric. Since the group
of isometries of a compact manifold is compact, this implies that $G$ is
finite.

Corollary 2.30:  (see also [Mats3])
Let $M$ be a projective hyperkähler manifold with the Picard rank
$\rho\geqslant 3$, carrying a nef isotropic line bundle $L$. Assume that no
MBM class is Beauville-Bogomolov orthogonal to $L$. Then there exists an
automorphism of $M$ preserving $L$, acting on the corresponding hyperbolic
space ${\mathbb{H}}={\mathbb{P}}\operatorname{Pos}(M)$ parabolically.

Proof: Let $H:={\mathbb{P}}\operatorname{Pos}_{{\mathbb{Q}}}(M)/\Gamma$ be the
hyperbolic manifold associated with $M$ as in 2.3, and $K\subset H$ a
hyperbolic polyhedron obtained as the image of the Kähler cone. As shown e. g.
in [Ka2], the cusps of $H$ are in bijective correspondence with the set
$R/\Gamma$ of orbits of $\Gamma$ on $R$, where $R$ is the set of rational
poins on the absolute. 333Absolute is the boundary of
${\mathbb{P}}\operatorname{Pos}_{{\mathbb{Q}}}(M)$ in
${\mathbb{P}}(H^{1,1}(M))$, which we identify with the projectivization of the
set of all isotropic vectors in $H^{1,1}(M,{\mathbb{Q}})$. The statement “no
MBM class is orthogonal to $L$,” is translated, in this language, as “the
hyperplanes which cut $H$ into $K_{1},...,K_{n}$ don’t pass through the cusp
associated with the isotropic vector $c_{1}(L)\in H^{1,1}(M,{\mathbb{Z}})$”.
This means that the hyperbolic polyhedron $K$ contains an open neighbourhood
of a cusp.

Recall now the “thick-thin decomposition theorem”, well-known in hyperbolic
geometry. It allows one to describe the cusps geometrically.

Theorem 2.31: (Thick-thin decomposition)
Any $n$-dimensional complete hyperbolic manifold ${\mathbb{H}}/\Gamma$ of
finite volume can be represented as a union of a “thick part”, which is a
compact manifold with a boundary, and a “thin part”, which is a finite union
of quotients of form $B/{\mathbb{Z}}^{n-1}$, where $B$ is a horoball tangent
to the boundary at a cusp point, and
${\mathbb{Z}}^{n-1}=\operatorname{St}_{\Gamma}(B)$.

Proof: See [Th, Section 5.10] or [Ka1, page 491].

Now, consider the thick-thin decomposition of $K$ near the cusp. Then the map
$\pi_{1}(B/{\mathbb{Z}}^{n-1}){\>\longrightarrow\>}\pi_{1}(K)$ is injective,
because $\pi_{1}(K)$ acts on $\widetilde{K}\subset{\mathbb{H}}$ in a
neighbourhood of a cusp, and the quotient is isometric to
$B/{\mathbb{Z}}^{n-1}$. The corresponding elements in $O(NS(M))\supset\Gamma$
are parabolic by construction. Since $\operatorname{Aut}(M)/G=\pi_{1}(K)$,
this implies that $\operatorname{Aut}(M)$ contains parabolic automorphisms.

To finish this section, we give a simple method to construct such hyperkähler
manifolds, inspired by [AV2] but substantially simpler.

Theorem 2.32: Let $M$ be hyperkähler manifold with $b_{2}\geqslant 5$. Then
$M$ admits a deformation which has a parabolic automorphism.

Proof: When $b_{2}\geqslant 5$, the lattice $H^{2}(M,{\mathbb{Z}})$ contains
an isotropic vector by Meyer’s theorem [Me]. Let $y$ be a primitive element of
$H^{2}(M,{\mathbb{Z}})$ of square zero. Let $N$ be any number which exceeds
the MBM bound for $M$ (that is, $q(z)>-N$ for any MBM class $z$ on a
deformation of $M$). There exists a primitive element $x\in
H^{2}(M,{\mathbb{Z}})$ such that $q(x)\leqslant-N$ and $x$ is orthogonal to
$y$. Indeed one can find a rank-two primitive sublattice of signature $(1,1)$
of $H^{2}(M,{\mathbb{Z}})$ orthogonal to $y$, and any lattice of signature
$(1,1)$ has primitive vectors of arbitrarily negative length444This elementary
lemma can be checked as follows: if true for a lattice $\Lambda$, it is also
true for $\Lambda^{\prime}\subset\Lambda\otimes{\mathbb{Q}}$ commensurable to
$\Lambda$; any $\Lambda$ is commensurable to a diagonal lattice, for which the
statement is obvious.. Take any $a\in H^{2}(M,{\mathbb{Z}})$ with positive
square. By the surjectivity of the period map, there exists a deformation
$M^{\prime}$ of $M$ which has Neron-Severi lattice generated by $a,x,y$.
Moreover by [AV1], the positive cone of $M^{\prime}$ is decomposed by the
orthogonal hyperplanes to the MBM classes into the union of Kähler chambers.
The Kähler chambers are, by definition, the Kähler cones of the hyperkähler
birational models of $M^{\prime}$ and their monodromy images. The hyperkähler
birational models $M^{\prime}$ are still deformations of $M$ by [H]; in fact
such a birational model is unseparable from $M^{\prime}$ in the parameter
space for hyperkähler manifolds, which is not Hausdorff. So replacing
$M^{\prime}$ by an appropriate birational model, we may assume that $y$ is the
class of a nef line bundle $L$ on $M^{\prime}$.

Now the statement follows from 2.3. Indeed the Neron-Severi lattice generated
by $a,x,y$ does not contain MBM classes orthogonal to $y$, since all integer
negative classes $\eta\in\langle a,x,y\rangle$ orthogonal to $x$ are of the
form $\eta=ny+mx$, and satisfy $q(\eta,\eta)=n^{2}q(y,y)$. This implies that
$q(\eta,\eta)\leqslant-N$.

###  2.4 Holomorphic Lagrangian fibrations on hyperkähler manifolds

Theorem 2.33: (Matsushita, [Mats1])
Let $\pi:\;M{\>\longrightarrow\>}X$ be a proper surjective holomorphic map
with connected fibers from a hyperkähler manifold $M$ to a projective variety
$X$, with $0<\dim X<\dim M$. Then $\dim X=1/2\dim M$, and the fibers of $\pi$
are Lagrangian (this means that the holomorphic symplectic form vanishes on
the fibers $\pi^{-1}(x)$). Moreover, $X$ is a Fano variety with
${\mathbb{Q}}$-factorial singularities and Picard number 1.

Definition 2.34: A proper surjective holomorphic map with connected fibers is
called a fibration; if its fibers are Lagrangian, it is a holomorphic
Lagrangian fibration.

Remark 2.35:  It is well-known that a smooth fiber $F$ of a Lagrangian
fibration is a torus, indeed it has trivial tangent bundle (the symplectic
form gives an isomorphism between the tangent and the conormal bundle). When
$M$ is hyperkähler, $F$ is projective even if $M$ itself is not (see
Proposition 2.1 of [Cam], attributed to Voisin by the author; see also [SV,
Corollary 3.4]).

Remark 2.36: The base $X$ of a Lagrangian fibration on a hyperkähler manifold
is conjectured to be rational, and smooth when it is normal. Matsushita
([Mats2]) proved that $X$ has the same Dolbeault cohomology as
${\mathbb{P}}^{n}$ if it is smooth. Hwang ([Hw]) proved
$X\cong{\mathbb{P}}^{n}$ whenever $X$ is smooth.

Definition 2.37: A cohomology class $\eta\in H^{1,1}(M)$ is nef if it lies in
the closure of the Kähler cone.

A trivial observation: Let $\pi:\;M{\>\longrightarrow\>}X$ be a holomorphic
Lagrangian fibration, and $\omega_{X}$ a Kähler class on $X$. Then
$\eta:=\pi^{*}\omega_{X}$ is nef, and satisfies $q(\eta,\eta)=0$.

Definition 2.38: A line bundle is called semiample if for a sufficiently big
$N$, the bundle $L^{\otimes N}$ is generated by its holomorphic sections (i.e.
they have no common zeros).

The hyperkähler SYZ conjecture: Let $L$ be a line bundle on a hyperkähler
manifold, with $q(c_{1}(L),c_{1}(L))=0$, and $c_{1}(L)$ nef. Then $L$ is
semiample.

Remark 2.39: For $N$ sufficiently big, the corresponding projective morphism
$M{\>\longrightarrow\>}{\mathbb{P}}(H^{0}(M,L^{\otimes N})^{*})$ is a
Lagrangian fibration onto its image, as follows from Matsushita’s theorem.

Remark 2.40: Hyperkähler SYZ conjecture has been proved for all known examples
of hyperkähler manifolds. See [BM] for Hilbert schemes of K3 surfaces, and
implicitly [Yo] for generalized Kummer varieties, as well as [MR], [MO] for
the two sporadic O’Grady’s examples.

The following proposition shall be used to show that, in the same way as for
K3 surfaces, the fiberwise density of orbits for two parabolic automorphisms
implies the ergodicity for the group they generate.

Proposition 2.41:  Let $\pi_{1}:\;M{\>\longrightarrow\>}X_{1}$,
$\pi_{2}:\;M{\>\longrightarrow\>}X_{2}$ be two distinct Lagrangian fibrations.
Then the intersection of general fibers of $\pi_{1},\pi_{2}$ is 0-dimensional.
Moreover, the restriction of $\pi_{2}$ on any fiber of $\pi_{1}$ is
surjective.

Proof: Let $\pi_{1}$, $\pi_{2}$ be two distinct Lagrangian fibrations given by
the sections of nef line bundles $L_{1}$ resp. $L_{2}$. The Hodge index
theorem implies $q(L_{1},L_{2})>0$. By Fujiki’s formula this means
$L_{1}^{n}L_{2}^{n}>0$ and the latter implies that the restriction of $L_{1}$
to a smooth fiber of $\pi_{2}$ is ample, because a nef and big line bundle on
an abelian variety is ample. Hence the restriction of $\pi_{2}$ to a smooth
fiber of $\pi_{1}$ is a finite map.

###  2.5 The hyperbolic space and its isometries

For an introduction to geometry of the hyperbolic space, see [Ka2, Chapter 3].

Remark 2.42: The group $O(m,n),m,n>0$ has 4 connected components. We denote
the connected component of 1 by $SO^{+}(m,n)$. We call a vector $v$ positive
if its square is positive.

Definition 2.43: Let $V$ be a vector space with a quadratic form $q$ of
signature $(1,n)$, $n\geqslant 2$, $\operatorname{Pos}(V)=\\{x\in V\ \ |\ \
q(x,x)>0\\}$ its positive cone, and ${\mathbb{P}}^{+}V$ projectivization of
$\operatorname{Pos}(V)$. Denote by $g$ any $SO(V)$-invariant Riemannian
structure on ${\mathbb{P}}^{+}V$. Then $({\mathbb{P}}^{+}V,g)$ is called the
hyperbolic space, and the group $SO^{+}(V)$ the group of oriented hyperbolic
isometries.

Theorem-definition: Let $n>0$, and $\alpha\in SO^{+}(1,n)$ a non-trivial
oriented isometry acting on $V={\mathbb{R}}^{1,n}$. Then one and only one of
these three cases occurs

(i)

$\alpha$ has an eigenvector $x$ with $q(x,x)>0$ ($\alpha$ is an elliptic
isometry)

(ii)

$\alpha$ has two eigenvectors $x,y$ with $q(x,x)=0$ and real eigenvalues
$\lambda_{x}$, $\lambda_{y}=\lambda_{x}^{-1}$ satisfying $|\lambda_{x}|>1$
($\alpha$ is a loxodromic, or hyperbolic, isometry)

(iii)

$\alpha$ has a unique eigenvector $x$ with $q(x,x)=0$. In this case the
eigenvalue $\lambda_{x}=1$, and $\alpha$ is called a parabolic isometry).

Remark 2.44: All eigenvalues of elliptic and parabolic isometries have
absolute value 1. Hyperbolic and elliptic isometries are semisimple (that is,
the corresponding linear operators are diagonalizable over ${\mathbb{C}}$),
parabolic are not.

Remark 2.45: The quadric $\\{l\in{\mathbb{P}}V\ \ |\ \ q(l,l)=0\\}$ is called
the absolute. It is realized as the boundary of the hyperbolic space
${\mathbb{P}}^{+}V$. When $\dim V=3$, the classification of hyperbolic
isometries can be reformulated in terms of the fixed point on the absolute:
elliptic isometries have no fixed points on the absolute, parabolic isometries
have exactly 1 fixed point on the absolute, and hyperbolic isometries have 2.

Remark 2.46: When $(V,q)$ has some underlying integral structure, for example
if $V=H^{2}(M,{\mathbb{R}})$ and $q$ is the BBF form, the elliptic isometries
preserving the integral structure are of finite order. Parabolic ones are,
clearly, of infinite order; indeed, any linear homomorphism of finite order is
semisimple.

##  3 Parabolic automorphisms of hyperkahler manifolds

###  3.1 Hyperkähler manifolds with ergodic automorphism groups

Remark 3.1: Let $M$ be a hyperkähler manifold. Then the BBF form has signature
$(1,b_{2}-3)$ on $H^{1,1}(M)$.

Definition 3.2: An automorphism of a hyperkähler manifold $(M,I)$ is called
elliptic (parabolic, hyperbolic) if it is elliptic (parabolic, hyperbolic) as
an element of $SO(H^{1,1}_{I}(M,{\mathbb{R}}))$.

Remark 3.3: Let $p$ be a parabolic automorphism of a hyperkähler manifold, and
$\eta$ its fixed point in $H^{1,1}(M)$ associated with the fixed point in the
absolute. Then $\eta$ is proportional to an integer cohomology class which
lies on the boundary of the Kähler cone, that is, the class of a nef line
bundle $L$. Indeed the nef cone is a convex cone invariant under the action of
$p$. One may also remark that $\eta$ can be obtained as a limit $p^{i}(w)$ for
any Kähler class $w$ on $M$.

One of the main results of this paper can be stated as follows

Theorem 3.4:  Let $M$ be a hyperkähler manifold admitting two parabolic
automorphisms $p_{1},p_{2}$ which preserve nef line bundles $L_{1}$, $L_{2}$,
with $c_{1}(L_{1})$ not proportional to $c_{1}(L_{2})$.111Equivalently, the
fixed points of $p_{1}$, $p_{2}$ on the absolute of the hyperbolic space
${\mathbb{P}}\operatorname{Pos}(M)$ are not equal. Assume the SYZ conjecture
holds for $L_{1}$, $L_{2}$. Then $p_{1},p_{2}$ generate a group acting on $M$
ergodically.

Remark 3.5: If $M$ admits a parabolic automorphism $p$ and
$\operatorname{Aut}(M)$ is not virtually abelian,222A group is called
virtually abelian if it contains an abelian subgroup of finite index. $M$
admits parabolic automorphisms which have different fixed points on the
absolute. Indeed it is well-known that that the stabilizer of a rational point
on the absolute in the group of isometries of a hyperbolic lattice is
virtually abelian. So if $\operatorname{Aut}(M)$ is not virtually abelian,
then not all automorphisms of $M$ fix the same point $x$ of the absolute as
$p$ does. Conjugating $p$ by an automorphism $g$ which does not fix $x$ we
obtain a parabolic automorphism with a different fixed point $x^{\prime}$.
Note that $x^{\prime}$ is also on the boundary of the Kähler cone, since the
latter is preserved by $g$. In particular, $x$ and $x^{\prime}$ are classes of
two different nef line bundles $L$, $L^{\prime}$.

Remark 3.6: In [AV2] it was shown that that any hyperkähler manifold with
$b_{2}\geqslant 14$ admits a projective deformation $M_{p}$ with a parabolic
automorphism; moreover, the automorphism group of $M_{p}$ is arithmetic, hence
$M_{p}$ admits parabolic automorphisms with different fixed points in the
absolute. If we are looking for a single automorphism, rather than for two
automorphisms with different fixed points on the absolute, we now have a
simpler construction in Subsection 2.3.

###  3.2 Parabolic automorphisms and Lagrangian fibrations on hyperkähler
manifolds

When a parabolic endomorphism $p$ preserves a nef line bundle $L$ and the
hyperkähler SYZ conjecture holds, i.e. the sections of a power of $L$ define a
Lagrangian fibration $\pi:M\to X$, then obviously $p$ takes fibers to fibers,
inducing an automorphism of the base $B$. It turns out that this automorphism
is of finite order, that is, a certain power of $p$ commutes with $\pi$.

This fact is due to Federico Lo Bianco, see [LoB1, Theorem B] for the case
when $X={\mathbb{P}}^{n}$, [LoB2] for the general case.

Theorem 3.7:  (Lo Bianco) Let $p$ be a parabolic automorphism of a projective
hyperkähler manifold $M$, and $\pi:\;M{\>\longrightarrow\>}X$ a Lagrangian
fibration such that for a Kähler class $\omega$ on $X$, its pulback
$\pi^{*}\omega$ is the class on the boundary of the Kähler cone fixed by $p$.
Then a certain power of $p$ preserves the fibers of $\pi$.

Remark 3.8: If $p$ commutes with $\pi$ (which is always the case after
replacing $p$ by a power), we say that $p$ preserves the Lagrangian fibration
$\pi$. Such a fibration is unique, because $\pi$ is uniquely determined by the
cohomology class $\pi^{*}\omega$, and $p$ fixes one and only one point in the
absolute.

Proposition 3.9: Let $p$ be a parabolic automorphism preserving a Lagrangian
fibration $\pi:M\to X$. Then some power of $p$ acts as a translation on a
general fiber.

Proof: Let $T$ be a smooth fiber of $\pi$, a complex algebraic torus. Its
group of automorphisms is the extension of the translation group by the group
automorphisms, infinite in general. We remark nevertheless that the
restriction of $p$ to $T$ must preserve a polarization. Indeed it is well-
known ([O1], see also section 5.1 here) that the rank of the restriction map
$H^{2}(M,{\mathbb{Z}})\stackrel{{\scriptstyle
r}}{{{\>\longrightarrow\>}}}H^{2}(T,{\mathbb{Z}})$ is one. The map induced by
$p$ must preserve the image of $r$. This implies that the restriction of a
Kähler class from $M$ to $T$ is also preserved by $p$. Hence
$p{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$ is an isometry with respect to
the Calabi-Yau metric (which is actually flat, because $T$ is a torus). The
connected component of the unity of the group of holomorphic isometries of a
flat torus coincides with the translation group, and the set of connected
components is finite, because this group is compact.

Remark 3.10: A complex torus $T$ is not a group, unless you fix the origin.
However, its translation group, denoted by $\operatorname{Tr}(T)$, is a
complex, commutative Lie group, and it is isomorphic to $T$ as a manifold.

Claim 3.11: A translation $\tau_{x}$ of a torus $T$ by an element
$x\in\operatorname{Tr}(T)$ has all its orbits dense if and only if $x$ is not
contained in a smaller real torus $T^{\prime}\subset\operatorname{Tr}(T)$. In
this case, $\tau_{x}$ is ergodic ([Wil]).

###  3.3 Parabolic automorphisms are fiberwise ergodic on Lagrangian
fibrations

The following theorem is the main result of this paper, used to obtain
ergodicity of the parabolic actions.

Theorem 3.12:  Let $p$ be a parabolic automorphism of a hyperkähler manifold
preserving a Lagrangian fibration $\pi:\;M{\>\longrightarrow\>}X$. Then there
exists a full measure, Baire second category subset $R\subset X$, such that
for all $r\in R$ the fibers $\pi^{-1}(r)$ are tori, and the automorphism $p$
acts on $\pi^{-1}(r)$ with dense orbits.

Remark 3.13: 3.3 immediately implies 3.1, in the same way as for K3 surfaces.
Indeed, let $\Gamma$ be the group generated by two parabolic automorphisms
$p_{1},p_{2}$, $\varphi$ a $\Gamma$-invariant measurable function, and
$\pi_{1},\pi_{2}$ the Lagrangian fibrations associated with $p_{1},p_{2}$.
Ergodicity of the action of $\Gamma$ means that any $\Gamma$-invariant
measurable function is constant almost everywhere. Since $\varphi$ is
$p_{i}$-invariant, it is constant almost everywhere on almost all fibers of
$\pi_{i}$. However, a function which is constant on fibers of $\pi_{i}$ is
constant on $M$, because these fibers are transversal and complementary (2.4).

The proof of 3.3 is based on Hodge theory, and we give it in Section 6 after
the relevant results about variations of Hodge structures are introduced.

##  4 Variations of Hodge structures and Deligne’s semisimplicity theorem

We give a brief introduction to the variations of Hodge structures. For more
detail, see [Voi1, Grf, PS].

###  4.1 Hodge structures

Definition 4.1: Let $V_{\mathbb{R}}$ be a real vector space. A (real) Hodge
structure of weight $w$ on a vector space
$V_{\mathbb{C}}=V_{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}$ is a
decomposition $V_{\mathbb{C}}=\bigoplus_{p+q=w}V^{p,q}$, satisfying
$\overline{V^{p,q}}=V^{q,p}$. It is called integer or rational Hodge structure
if one fixes an integer or rational lattice $V_{\mathbb{Z}}$ or
$V_{\mathbb{Q}}$ in $V_{\mathbb{R}}$. A Hodge structure is equipped with
$U(1)$-action, with $u\in U(1)$ acting as $u^{p-q}$ on $V^{p,q}$. A morphism
of integer/rational Hodge structures is an integer/rational map which is
$U(1)$-invariant.

Definition 4.2: A polarization on a rational Hodge structrure of weight $w$ is
a $U(1)$-invariant non-degenerate 2-form $h\in V_{\mathbb{Q}}^{*}\otimes
V^{*}_{\mathbb{Q}}$ (symmetric or antisymmetric depending on parity of $w$)
which satisfies

$-\sqrt{-1}\>^{p-q}h(x,\overline{x})>0$ (4.1)

(“Riemann-Hodge relations”) for each non-zero $x\in V^{p,q}$.

Definition 4.3: Two complex tori $T_{1},T_{2}$ are called isogeneous if there
exists a surjective finite holomorphic map $T_{1}{\>\longrightarrow\>}T_{2}$.

Remark 4.4: The category of complex tori with a group structure (that is, the
zero point fixed) is equivalent to the category of integer Hodge structures of
weight 1. The category of complex tori with a group structure up to isogeny is
equivalent to the category of rational Hodge structures of weight 1. Under
this correspondence, abelian varieties correspond to Hodge structures
admitting a polarization.

Claim 4.5: The category ${\cal C}$ of rational Hodge structures admitting a
polarization is semisimple, that is, any object of ${\cal C}$ is a direct sum
of irreducible ones.

Remark 4.6: Equivalently, the category of abelian varieties up to isogeny is
semisimple.

###  4.2 Variations of Hodge structures

Definition 4.7: Let $M$ be a complex manifold. A variation of Hodge structures
(VHS) on $M$ is a complex vector bundle $(B,\nabla)$ with a flat connection
equipped with a parallel anti-complex involution and Hodge structures at each
point, $B=\bigoplus_{p+q=w}B^{p,q}$ which satisfy the following conditions:

(a)

the polarization, the rational lattice and the real structure are preserved by
$\nabla$.

(b)

(“Griffiths transversality condition”) $\nabla^{1,0}(B^{p,q})\subset
B^{p,q}\oplus B^{p+1,q-1}$.

Example 4.8: Let $\pi:\;M{\>\longrightarrow\>}X$ be a proper holomorphic
submersion. Consider the bundle $V:=R^{k}\pi_{*}({\mathbb{C}}_{M})$ with the
fiber in $x$ the $k$-th cohomology of $\pi^{-1}(x)$, the Hodge decomposition
coming from the complex structure on $\pi^{-1}(x)$, and the Gauss-Manin
connection. This defines a variation of Hodge structures. However, to produce
a polarization, we need $M$ to be projective, indeed, in the Kähler case the
polarization does not always exist.

Claim 4.9: Polarized integer variations of Hodge structures of weight 1 are in
a functorial bijection with holomorphic Abelian fibrations. More precisely,
polarized variations of Hodge structures of weight 1 correspond to fibrations
in complex tori $X{\>\longrightarrow\>}Y$ with a holomorphic section, giving
the fiberwise choice of zero, and a line bundle $L$ on $X$ which is ample on
all fibers, inducing a polarization. Given a fibration in abelian varieties
without a section, one can associate to it the “relative Albanese”, or
“translation fibration”, taking $\operatorname{Tr}(T)$ instead of a fiber $T$.
The variation of Hodge structures defined as in the example above with $k=1$
is the same for a fibration and its relative Albanese.

###  4.3 Deligne’s semisimplicity theorem

The following important theorem is one of the main tools of our argument.

Theorem 4.10:  (Deligne’s semisimplicity theorem, [D, Wr])
Let $V$ be a polarized, rational variation of Hodge structures over a
quasiprojective base $M$. Then the underlying flat bundle can be decomposed as
$V=\bigoplus_{i}W_{i}\otimes L_{i}$, where $L_{i}$ correspond to pairwise non-
isomorphic irreducible representations of $\pi_{1}(M)$, and $W_{i}$ are
trivial representations. Moreover, each $W_{i}$ is equipped with a Hodge
structure, each $L_{i}$ is equipped with the structure of a VHS, and the
decomposition $V=\bigoplus_{i}W_{i}\otimes L_{i}$ is compatible with the Hodge
structures on $W_{i},L_{i}$.

Applying this result to a VHS of weight 1, we notice that one of two summands
in $W_{i}\otimes L_{i}$ has weight 0, and another has weight 1. Separating the
sum in two parts according to whether $W_{i}$ or $L_{i}$ is of weight 0, we
obtain the following corollary. Recall that a fibration in abelian varieties
is called isotrivial if all its fibers are isomorphic.

Corollary 4.11: Let $V$ be a quasiprojective manifold fibered in abelian
varieties. Then, after passing to an isogeneous fibration $V_{1}$, we can
decompose $V_{1}$ into a product of abelian fibrations with irreducible
monodromy of the Gauss-Manin connection and an isotrivial abelian fibration.

###  4.4 Deligne’s global invariant cycle theorem

Further on, we shall use the following important theorem, also due to Deligne.

Theorem 4.12:  (see for example [dCM, Theorem 1.2.4]).
Let $M{\>\longrightarrow\>}X$ be an algebraic family of projective manifolds,
$X_{0}$ its smooth locus, $M_{0}:=\pi^{-1}(X_{0})$, and
$V=R^{i}\pi_{*}({\mathbb{C}}_{M_{0}})$ the corresponding Gauss-Manin local
system. Consider a monodromy invariant vector $\alpha\in
R^{i}\pi_{*}({\mathbb{C}}_{M_{0}}){\left|{}_{{\phantom{|}\\!\\!}_{x}}\right.}$,
and let $[\alpha]\in\pi^{-1}(x)$ be the corresponding cohomology class in the
fiber. Then $[\alpha]$ belongs to the image of the restriction map
$H^{i}(M){\>\longrightarrow\>}H^{i}(\pi^{-1}(x))$.

##  5 Variations of Hodge structures and Lagrangian fibrations

In this section, we prove some preliminary results about the Hodge structures
associated with a Lagrangian fibration. We start with the following basic
observation.

###  5.1 AM-GM inequality, products of Hermitian forms and Lagrangian
fibrations

In this section we give an elementary proof of the following useful
observation by Oguiso ([O1], p.3) who combined results by Voisin [Voi2] and
Matsushita [Mats2].

Proposition 5.1:  Let $\pi:\;M{\>\longrightarrow\>}X$ be a Lagrangian
fibration on a hyperkähler manifold and $T$ a smooth fiber111A projective
torus, see 2.4 of $\pi$. Then the natural restriction map
$H^{2}(M){\>\longrightarrow\>}H^{2}(T)$ has rank 1.

We use the following basic lemma, found in Cauchy’s “Cours d’analyse de
l’École Royale Polytechnique, premiére partie, Analyse algébrique” (1821). In
high school, it is usually called the AM-GM222Short for “arithmetic mean -
geometric mean” inequality.

Lemma 5.2: Let $\alpha_{1},...,\alpha_{n}$ be positive real numbers. Then

$\frac{\sum\alpha_{i}}{n}\geqslant\sqrt[n]{\prod_{i}\alpha_{i}},$

and the equality holds if and only if all $\alpha_{i}$ are equal.

Corollary 5.3: Let $\alpha_{1},...,\alpha_{n}$ be positive numbers such that
$\sum\alpha_{i}=n$ and $\prod\alpha_{i}=1$. Then all $\alpha_{i}=1$.

Proposition 5.4:  Let $\omega_{1},\omega_{2}$ be Hermitian forms on a vector
space $V={\mathbb{C}}^{n}$, and $h_{1},h_{2}$ the corresponding Hermitian
metrics. Suppose that
$\omega_{1}\wedge\omega_{2}^{n-1}=\omega_{1}^{n}=\omega_{2}^{n}$. Then
$\omega_{1}=\omega_{2}$.

Proof: Simultaneous diagonalization theorem implies that $h_{1}$ can be
diagonalized in an orthonormal basis for $h_{2}$. Let
$A=\omega_{1}\omega_{2}^{-1}$ be the corresponding diagonal matrix. Clearly,
$\frac{\omega_{1}\wedge\omega_{2}^{n-1}}{\omega_{2}^{n}}=\frac{1}{n}\operatorname{Tr}(A)$
and $\frac{\omega_{1}^{n}}{\omega_{2}^{n}}=\det A$. In other words,
$\frac{\omega_{1}^{n}}{\omega_{2}^{n}}=\det A$ is the product
$\prod_{i=1}^{n}\alpha_{i}$ of all eigenvalues of $A$, and
$\frac{\omega_{1}\wedge\omega_{2}^{n-1}}{\omega_{2}^{n}}$ the arithmetic mean
$\frac{1}{n}\sum_{i=1}^{n}\alpha_{i}$ of all these eigenvalues. On the other
hand, $\omega_{1}^{n}=\omega_{2}^{n}$ implies that the product of all
eigenvalues of $A$ is 1, hence the geometric mean of these eigenvalues is also
1. Then $\omega_{1}\wedge\omega_{2}^{n-1}=\omega_{1}^{n}=\omega_{2}^{n}$
implies that

$\frac{1}{n}\sum_{i=1}^{n}\alpha_{i}=\sqrt[n]{\prod_{i=1}^{n}\alpha_{i}}=1.$

The AM-GM inequality implies that all $\alpha_{i}$ are equal to 1, which gives
$\omega_{1}=\omega_{2}$.

Proof of 5.1: Let $\eta_{0}\in H^{2}(X)$ be a class on $X$ satisfying
$\int_{X}\eta_{0}^{n}=1$, where $n=\dim_{\mathbb{C}}X$,
$\omega_{1},\omega_{2}\in H^{2}(M)$ any cohomology classes, and
$\eta:=\pi^{*}\eta_{0}$. We need to show that
$\omega_{1}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$ is proportional to
$\omega_{2}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$. Since $T$ is
Lagrangian, the (2,0)-forms are restricted to zero, and we may assume that
$\omega_{1},\omega_{2}\in H^{1,1}(M)$. Picking $\omega_{1}$ Kähler and
replacing $\omega_{2}$ by an appropriate linear combination of $\omega_{1}$
and $\omega_{2}$, we may assume that $\omega_{2}$ is also Kähler. Clearly,
$q(\eta,\eta)=0$. On the other hand, $\eta^{n}$ is cohomologous to the
fundamental class $[T]$ of the fiber of $\pi$, because $\int_{X}\eta^{n}=1$.
By Fujiki formula (polarized form),
$\int_{T}\omega_{i}^{n}=\int_{M}\omega_{i}^{n}\wedge\eta^{n}=Cq(\omega_{i},\eta)^{n}$
and

$\int_{T}\omega_{2}\wedge\omega_{1}^{n-1}=\int_{M}\omega_{2}\wedge\omega_{1}^{n-1}\wedge\eta^{n}=Cq(\omega_{2},\eta)q(\omega_{1},\eta)^{n-1}.$

Rescaling $\omega_{i}$ in such a way that
$q(\omega_{1},\eta)=q(\omega_{2},\eta)=C^{-1/n}$, we obtain that

$\omega_{1}\wedge\omega_{2}^{n-1}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}=\omega_{1}^{n}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}=\omega_{2}^{n}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$

for two Kähler classes restricted to the fiber. Representing these Kähler
classes by translation-invariant Hermitian forms on $T$, we obtain two
Hermitian forms on ${\mathbb{C}}^{n}$ satisfying
$\omega_{1}\wedge\omega_{2}^{n-1}=\omega_{1}^{n}=\omega_{2}^{n}$. By 5.1,
these Hermitian forms are equal, hence
$\omega_{1}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$ is cohomologous to
$\omega_{2}{\left|{}_{{\phantom{|}\\!\\!}_{T}}\right.}$.

###  5.2 Irreducibility of the VHS associated with a Lagrangian fibration

We start with the following lemma.

Lemma 5.5:  Let $\pi:\;M{\>\longrightarrow\>}X$ be a Lagrangian fibration on a
hyperkähler manifold of maximal holonomy. Let $X_{0}\subset X$ be the smooth
locus of $\pi$ and $V$ the variation of Hodge structures over $X_{0}$
associated with $R^{1}\pi_{*}({\mathbb{C}}_{M})$. Then the space of parallel
sections of $\Lambda^{2}(V)$ is 1-dimensional.

Proof: Let $\omega$ be the polarization on $V$. It is a parallel section of
$\Lambda^{2}V^{*}$, corresponding to the restriction of the Riemann-Hodge form
form to the fibers of $\pi$. Since the smooth fibers of $\pi$ are tori, the
second cohomology VHS of $\pi$ is $\Lambda^{2}V$. By Deligne’s global
invariant cycle theorem (4.4), any monodromy invariant element in a fiber of
$R^{2}\pi_{*}({\mathbb{C}}_{M})$ is obtained as a restriction of a globally
defined cohomology class in $H^{2}(M)$. Applying this to
$\Lambda^{2}V=R^{2}\pi_{*}({\mathbb{C}}_{M})$, we obtain that any
antisymmetric, monodromy-invariant 2-tensor on $V$ is obtained as a
restriction of a globally defined cohomology class in $H^{2}(M)$.

By Voisin’s theorem (5.1), the restriction map from $H^{2}(M)$ to a smooth
fiber of $\pi$ has rank 1, hence any parallel 2-form on $V$ is equal to the
polarization. Dualizing this statement, we obtain that the space of parallel
sections of $\Lambda^{2}V^{*}$ is 1-dimensional and generated by $\omega$.

Corollary 5.6:  Let $\pi:\;M{\>\longrightarrow\>}X$ be a Lagrangian fibration
on a hyperkähler manifold, $X_{0}\subset X$ a smooth locus of $\pi$, and
$V:=R^{1}\pi_{*}({\mathbb{C}}_{M})$ the corresponding Gauss-Manin local
system. Then $V$ is irreducible as a variation of Hodge structures. Moreover,
the monodromy representation is irreducible, unless $\pi$ is isotrivial.

Proof: Suppose that $V$ is not irreducible as a rational variation of Hodge
structures: $V=V_{1}\oplus V_{2}$. Restricting the polarization $\omega$ to
$V_{1}$ and $V_{2}$, we obtain that the space of parallel sections of
$\Lambda^{2}(V^{*})=\Lambda^{2}(V)$ is at least 2-dimensional, which is
impossible by 5.2.

To prove the second part, apply Deligne’s semisiplicity 4.3 to obtain the
decomposition $V=\bigoplus_{i}W_{i}\otimes L_{i}$. Since $V$ is irreducible as
a VHS, only one of the summands is non-trivial: $V=W\otimes L$. Then either
$W$ has weight 0, that is, $W={\mathbb{Q}}$, and $L$ is irreducible of weight
1, or $W$ has weight 1 and $L$ has weight 0. In the second case $\pi$ is
isotrivial.

Remark 5.7: More geometrically, one can argue that a Hodge substructure of $V$
yields a fibration in subtori on each smooth fiber of $\pi$. This in turn
gives a rational map $\varphi$ from $M$ to the component of the Chow variety
of $M$ which parameterizes such subtori. Considering the inverse image by
$\varphi$ of an ample line bundle (or a Kähler class) on this component, one
again sees that the rank of the restriction of $H^{2}(M)$ to a smooth fiber of
$\pi$ must be greater than one: indeed this inverse image is semiample but not
ample on a smooth fiber of $\pi$.

##  6 Parabolic automorphisms and their orbits

In this section and the next one, we prove 3.3 which claims that a generic
orbit of a parabolic automorphism of a hyperkähler manifold is dense in the
corresponding fiber of a Lagrangian fibration.

###  6.1 Parabolic automorphisms and monodromy-invariant subtori

Let $p$ be a parabolic automorphism of a hyperkähler manifold, preserving a
Lagrangian fibration $\pi:\;M{\>\longrightarrow\>}X$. Recall that for a
regular value $x\in X$, the fiber $\pi^{-1}(x)$ is a compact complex (in fact
algebraic) torus. Consider the group of its translations
$\operatorname{Aut}_{0}(\pi^{-1}(x))=\frac{H^{1}(\pi^{-1}(x),{\mathbb{R}})}{H^{1}(\pi^{-1}(x),{\mathbb{Z}})}$
understood as the connected subgroup of its group of automorphisms. Consider
the element $p_{x}\in\operatorname{Aut}_{0}(\pi^{-1}(x))$ induced by $p$, and
let $P_{x}$ be the closure of the group it generates in in
$\operatorname{Aut}_{0}(\pi^{-1}(x))$. This is a subtorus, not necessarily
compex. Clearly the orbits of $p$ are dense in the fibers of $\pi$ if and only
if $P_{x}$ is equal to $\operatorname{Aut}_{0}(\pi^{-1}(x))$, and otherwise
their closures are sutori of the same dimension as that of $P_{x}$.

Since $\pi$ is differentiably locally trivial outside of its singularities,
the toric bundle of all translations of the fibers of $\pi$ (also known as the
relative Albanese variety $\operatorname{Alb}_{\pi}$, with the projection
$\widetilde{\pi}:\operatorname{Alb}_{\pi}\to X_{0}$ where $X_{0}$ is the part
of $X$ over which $\pi$ is smooth) is equipped with a natural connection,
induced by the Gauss-Manin connection on $R^{1}\pi_{*}({\mathbb{C}}_{M})$.
Denote by $\Gamma$ its monodromy group.

Proposition 6.1:  In the setting described above, let

$r:={\rm max}_{z\in X_{0}}\dim P_{z}$

and assume that the dimension of $\dim P_{x}$ is equal to $r$. Then $P_{x}$ is
a $\Gamma$-invariant subtorus of $\operatorname{Aut}_{0}(\pi^{-1}(x))$.

Proof. Step 1: Let $X_{0}\subset X$ be the smooth locus of $\pi$. For each
$x\in X_{0}$, the fiber $\pi^{-1}(x)$ is a torus and
$p_{x}\in\operatorname{Tr}(\pi^{-1}(x))$ its translation. Denote by $P_{x}$
the closure of the group generated by $p_{x}$. It is clearly a subtorus in
$\operatorname{Tr}(\pi^{-1}(x))$. The map $x{\>\longrightarrow\>}p_{x}$ gives
a section of the relative Albanese fibration of $\pi$. This fibration is
locally trivial in real analytic category. Then the map
$x{\>\longrightarrow\>}p_{x}$ can be interpreted as a map from a neighbourhood
of $x$ to a fixed real torus $T^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$.

Step 2: The dimension of $P_{x}$ can be expressed through the smallest number
of integer relations between the coordinate components of $p_{x}\in T_{n}$.
Since $x{\>\longrightarrow\>}p_{x}$ is real analytic, $\dim P_{x}$ is real
analytically Zariski lower semicontinuous. When the dimension is maximal, the
torus $P_{x}\subset T^{n}$ is locally constant. Let $Y\subset X_{0}$ be the
set of all $x$ where rank of $P_{x}$ is not maximal. $Y$ is a real analytic
subset, so a closed real analytic path crosses $Y$ in a finite set. Clearly,
it suffices to show that $P_{x}$ stays constant along such a real analytic
path.

We reduced 6.1 to the following:

Step 3: Statement. Let $\gamma\subset X_{0}$ be a real analytic path
intersecting $Y$ in a finite set, and $x_{1},x_{2}\in\gamma\backslash Y$.
Using the Gauss-Manin connection, we trivialize the bundle $\pi$ along
$\gamma$ real-analytically, and this gives a trivialization of the
corresponding automorphism bundle with the fiber
$\operatorname{Aut}_{0}(\pi^{-1}(x))$ at each regular value $x\in X$. Then the
tori $P_{x_{1}}$, $P_{x_{2}}$ are equal as subgroups of
$\operatorname{Aut}_{0}(\pi^{-1}(x_{1}))=\operatorname{Aut}_{0}(\pi^{-1}(x_{2}))$.

Step 3: Proof. Passing to a universal cover of the fiber, we can consider $p$
as a real analytic map from $\gamma$ to ${\mathbb{R}}^{n}$. Under this
interpretation, $P_{x}$ is given by the smallest rational subspace $V$ of
${\mathbb{R}}^{n}$ containing $p(x)$. The map
$p:\;\gamma{\>\longrightarrow\>}{\mathbb{R}}^{n}$ belongs to $V$ on an open
subset of $\gamma\backslash Y$. By analytic continuation this means that
$p(\gamma)\subset V$ (indeed if some integer relation holds on one side of a
critical point $y\in Y\cap\gamma$, it also holds on another side, because of
the real analytic nature of $p$). Since the rank of $P_{x}$ is constant
outside of $Y$, this implies that $P_{x}=V$ for all $x\in\gamma\backslash Y$.

###  6.2 Ergodicity of parabolic automorphisms

Now we can prove the following corollary, which immediately implies 3.3 when
$\pi$ is not isotrivial.

Corollary 6.2: Let $\pi:\;M{\>\longrightarrow\>}X$ be a non-isotrivial
Lagrangian fibration on a hyperkähler manifold of maximal holonomy, and $p$ a
parabolic automorphism preserving $\pi$. Then $p$ acts with dense orbits on a
general fiber of $\pi$.

Proof: The closure of an orbit of maximal dimension is given by a monodromy-
invariant subspace in $H^{1}(\pi^{-1}(x))$. However, for a non-isotrivial
Lagrangian fibration the monodromy is irreducible by 5.2.

Remark 6.3: In this proof we use the following general statement. Let
$\pi:\;M{\>\longrightarrow\>}X$ be a Lagrangian fibration, and $p$ a parabolic
automorphism preserving $\pi$. Let $X_{0}$ be the smooth locus of $\pi$, and
$M_{0}:=\pi^{*}(X_{0})$. Then the corresponding principal bundle of fiberwise
translations
$\widetilde{\pi}:\;\operatorname{Alb}_{\pi}{\>\longrightarrow\>}X_{0}$,
considered as a smooth principal fibration, contains a $p$-invariant principal
sub-bundle $F\subset M_{0}$, and the general orbit of $p$ is dense in the
fibers of the projection $\widetilde{\pi}:\;F{\>\longrightarrow\>}X_{0}$.

##  7 Isotrivial Lagrangian fibrations

Up to this point, to prove non-existence of subtori in a holomorphic
Lagrangian fibration, we have used the irreducibility of the corresponding
Gauss-Manin local system. This works when the Lagrangian fibration is non-
isotrivial; for isotrivial Lagrangian fibrations, an extra effort is needed.

Note that the monodromy of the weight one variation of Hodge structures $V$
associated with an isotrivial Lagrangian fibrations is unitary, and therefore
finite. Indeed, when the complex structure is constant in the family, the
complex structure operator $I$ acting on $V^{0,1}$ as $-\sqrt{-1}\>$ and on
$V^{1,0}$ as $\sqrt{-1}\>$ is parallel, and it defines a positive definite
Hermitian metric (4.1), which is also parallel, hence preserved by the
monodromy.

Further on, we shall need the following trivial lemma.

Lemma 7.1:  Let $V$ be a local system equipped with a lattice
$V_{\mathbb{Z}}$, and suppose that the monodromy of the corresponding torus
bundle $V/V_{\mathbb{Z}}$ is trivial. Then the monodromy of $V$ is also
trivial.

Proof: The group of automorphisms of the lattice is $GL(n,{\mathbb{Z}})$, and
this group acts faithfully on the torus, indeed already the action on its
tangent space is faithful.

Lemma 7.2:  Let $\pi:\;M{\>\longrightarrow\>}X$ be an isotrivial Lagrangian
fibration on a hyperkähler manifold of maximal holonomy, $p$ a parabolic
automorphism, $X_{0}\subset X$ the smooth locus of $\pi$,
$M_{0}:=\pi^{-1}(X_{0})$, and $P\subset M_{0}$ the sub-torus obtaines as the
closure of orbits of $p$ (6.1). Let $V:=R^{1}\pi_{*}({\mathbb{C}}_{M})$ be the
variation of Hodge structures associated with $\pi$ and ${\mathbb{P}}\subset
V$ a parallel sub-bundle associated with
$R^{1}\pi_{*}({\mathbb{C}}_{P})$.111Note that ${\mathbb{P}}\subset V$ is a
parallel sub-bundle, but not necessarily complex-linear. Then ${\mathbb{P}}$
is locally generated by the holomorphic sections of $V$ which belong to
${\mathbb{P}}$.222We mean that each fiber $P_{x}$ is generated, as a real
vector space, by the images of local holomorphic sections at $x$.

Proof: Clearly, $V/V_{\mathbb{Z}}$ is the relative Albanese space
$\operatorname{Alb}_{\pi}$ of $\pi:\;M_{0}{\>\longrightarrow\>}X_{0}$, that
is, the space of fiberwise translations of the fibers of $\pi$, which are
tori. Therefore, $p$ can be considered as a section of the relative Albanese
map mapping $X_{0}$ to the total space of
${\mathbb{P}}/{\mathbb{P}}_{\mathbb{Z}}$. It is holomorphic by definition; the
iterations of $p$ are dense in $P$, hence generate ${\mathbb{P}}$.

The following proposition takes care of isotrivial Lagrangian fibrations,
finishing the proof of 3.3. Indeed, any parabolic automorphism whose orbits
are not dense in the orbits of the Lagrangian projection defines a parallel
sub-bundle ${\mathbb{P}}=V_{1}\subset V=R^{1}\pi_{*}({\mathbb{C}}_{M})$ (6.1),
and it is generated by holomorphic sections (7).

Proposition 7.3: Let $\pi:\;M{\>\longrightarrow\>}X$ be an isotrivial
Lagrangian fibration on a hyperkähler manifold of maximal holonomy. Suppose
that the corresponding flat bundle $V:=R^{1}\pi_{*}({\mathbb{C}}_{M})$
contains a parallel sub-bundle $V_{1}$. Assume, moreover, that $V_{1}$
generated by sections which are holomorphic in $V$. Then $V_{1}=V$.

Proof. Step 1: We prove that any proper parallel sub-bundle of $V$ is totally
real (that is, contains no holomorphic subspaces) and Lagrangian with respect
to the polarization.

Let $X_{0}\subset X$ be the smooth locus of $\pi$, $M_{0}:=\pi^{-1}(X_{0})$,
and $V$ the variation of Hodge structures associated with
$R^{1}\pi_{*}({\mathbb{C}}_{M_{0}})$. Since $\pi$ is isotrivial, the complex
structure operator on $V$ is parallel with respect to the flat connection. Let
$\omega$ be the polarization on $V$. It is a parallel section of
$\Lambda^{2}V^{*}$, corresponding to the restriction of the Riemann-Hodge form
to the fibers of $\pi$. Then the bilinear symmetric form
$g(x,y)=\omega(I(x),y)$ is positive definite and parallel on $V$.

Any parallel sub-bundle $V_{1}\subset V$ is Lagrangian with respect to
$\omega$. To see this, consider the orthogonal decomposition $V=V_{1}\oplus
V_{1}^{\bot}$ with respect to the positive definite form $g$ on $V$. This
decomposition is parallel, because $g$ is parallel. Restricting $\omega$ to
$V_{1}$, we obtain a parallel 2-form on $V_{1}$; using the decomposition
$V=V_{1}\oplus V_{1}^{\bot}$, we can consider this 2-form as a parallel
section $\omega_{1}\in\Lambda^{2}(V)$. Since the rank of $\omega_{0}$ is
strictly less than $\operatorname{rk}\omega$, we have $\omega_{1}=0$ by 5.2.
Therefore, $V_{1}$ is isotropic. Similarly, the restriction of $\omega$ to
$V_{1}^{\bot}$ also vanishes. This implies that these two sub-bundles have
rank $1/2\operatorname{rk}V$, hence are Lagrangian.

The fact that $V_{1}\subset V$ is Lagrangian with respect to the polarization
implies that $V_{1}$ is totally real, indeed the polarization would be
positive on a holomorphic subspace.

Step 2: Let $\lambda$ be a local section of $V_{1}$. Since $V_{1}\subset V$ is
parallel, $\nabla_{v}\lambda\in V_{1}$ for any vector field $v\in TX_{0}$.
However, when $\lambda$ is holomorphic, the space $\bigcup_{v\in
T_{x}X_{0}}\nabla_{v}\lambda$ is complex linear, which is impossible, because
$V_{1}$ is Lagrangian, unless $\nabla\lambda=0$. Applying this to $p$ and its
powers, considered locally as sections of $V_{1}$, we obtain that they are all
parallel. Since $\\{p^{n}\\}$ are dense in general fibers of the corresponding
torus $V_{1}/(V_{1}\cap V_{\mathbb{Z}})$, its monodromy is trivial. By 7, the
monodromy of $V_{1}$ is also trivial.

This is impossible, again, by Deligne’s global invariant cycle theorem (4.4),
because $H^{1}(M)=0$ whenever $M$ is a hyperkähler manifold of maximal
holonomy.

Remark 7.4: It is easy to see that the monodromy of an isotrivial Lagrangian
fibration on the Kummer K3 surface associated to the product of elliptic
curves is $\\{\pm 1\\}$, hence the corresponding local system of real rank 2
is not irreducible.

Acknowledgements: We are grateful to Serge Cantat and Ilyas Bairamov for
interesting discussions.

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Ekaterina Amerik
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Paris-11,
Laboratoire de Mathématiques,
Campus d’Orsay, Bâtiment 425, 91405 Orsay, France

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
<EMAIL_ADDRESS>
also:
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str.
Moscow, Russia.