Morrison-Kawamata cone conjecture for hyperkähler manifolds

Ekaterina Amerik111Partially supported by RSCF, grant number 14-21-00053.,
Misha Verbitsky222Partially supported by RSCF, grant number 14-21-00053.
Keywords: hyperkähler manifold, moduli space, period map, Torelli theorem 2010
Mathematics Subject Classification: 53C26, 32G13

Abstract

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply
connected compact holomorphically symplectic manifold of Kähler type with
$h^{2,0}=1$. Assuming $b_{2}(M)\neq 5$, we prove that the group of holomorphic
automorphisms of $M$ acts on the set of faces of its Kähler cone with finitely
many orbits. This statement is known as Morrison-Kawamata cone conjecture for
hyperkähler manifolds. As an implication, we show that any hyperkähler
manifold has only finitely many non-equivalent birational models. The proof is
based on the following observation, proven with ergodic theory. Let $M$ be a
complete Riemannian orbifold of dimension at least three, constant negative
curvature and finite volume, and $\\{S_{i}\\}$ an infinite set of complete,
locally geodesic hypersurfaces. Then the union of $S_{i}$ is dense in $M$.

###### Contents

  1. 1 Introduction
    1. 1.1 Kähler cone and MBM classes
    2. 1.2 Morrison-Kawamata cone conjecture for hyperkähler manifolds
    3. 1.3 Main results
  2. 2 Preliminaries
    1. 2.1 Hyperkähler manifolds, monodromy and MBM classes
    2. 2.2 Global Torelli theorem and deformations
  3. 3 Ergodic theory and its applications
    1. 3.1 Ergodic theory: basic definitions and facts
    2. 3.2 Lie groups generated by unipotents
  4. 4 Algebraic measures on homogeneous spaces
    1. 4.1 Limits of ergodic measures
    2. 4.2 Rational hyperplanes intersecting a compact set
    3. 4.3 Measures and rational hyperplanes in the hyperbolic space
  5. 5 The proof of Morrison-Kawamata cone conjecture
    1. 5.1 Morrison-Kawamata conjecture for the Kähler cone
    2. 5.2 Morrison-Kawamata conjecture for the ample cone

##  1 Introduction

###  1.1 Kähler cone and MBM classes

Let $M$ be a hyperkähler manifold, that is, a compact, holomorphically
symplectic Kähler manifold. We assume that $\pi_{1}(M)=0$ and
$H^{2,0}(M)={\mathbb{C}}$: the general case reduces to this by Bogomolov
decomposition (2.1). Such hyperkähler manifolds are known as simple
hyperkähler manifolds, or IHS (irreducible holomorphic symplectic) manifolds.
The known examples of such manifolds are deformations of punctual Hilbert
scheme of K3 surfaces, deformations of generalized Kummer varieties and two
sporadic ones discovered by O’Grady. In [AV] we gave a description of the
Kähler cone of $M$ in terms of a set of cohomology classes $S\subset
H^{2}(M,{\mathbb{Z}})$ called MBM classes (2.1). This set is of topological
nature, that is, it depends only on the deformation type of $M$.

Recall that on the second cohomology of a hyperkähler manifold, there is an
integral quadratic form $q$, called the Beauville-Bogomolov-Fujiki form (see
section 2 for details). This form is of signature $(+,-,\dots,-)$ on
$H^{1,1}(M)$. Let $\operatorname{Pos}\subset H^{1,1}(M)$ be the positive cone,
and $S(I)$ the set of all MBM classes which are of type (1,1) on $M$ with its
given complex structure $I$. Then the Kähler cone is a connected component of
$\operatorname{Pos}\backslash S(I)^{\bot}$, where $S(I)^{\bot}$ is the union
of the orthogonal complements to all $z\in S(I)$.

###  1.2 Morrison-Kawamata cone conjecture for hyperkähler manifolds

The Morrison-Kawamata cone conjecture for Calabi-Yau manifolds was stated in
[Mo]. For K3 surfaces it was already known since mid-eighties by the work of
Sterk [St]. Kawamata in [Ka] proved the conjecture for Calabi-Yau threefolds
admitting a holomorphic fibration over a positive-dimensional base.

In this paper, we concentrate on the following version of the cone conjecture
(see Subsection 5.2 for its relation to the classical one, formulated for the
ample cone of a projective variety).

Definition 1.1:  Let $M$ be a compact, Kähler manifold,
$\operatorname{Kah}\subset H^{1,1}(M,{\mathbb{R}})$ the Kähler cone, and
$\overline{\operatorname{Kah}}$ its closure in $H^{1,1}(M,{\mathbb{R}})$,
called the nef cone. A face of the Kähler cone is the intersection of the
boundary of $\overline{\operatorname{Kah}}$ and a hyperplane $V\subset
H^{1,1}(M,{\mathbb{R}})$ which has non-empty interior.

Conjecture 1.2: (Morrison-Kawamata cone conjecture)
Let $M$ be a Calabi-Yau manifold. Then the group $\operatorname{Aut}(M)$ of
biholomorphic automorphisms of $M$ acts on the set of faces of
$\operatorname{Kah}$ with finite number of orbits.

We shall be interested in the case when the manifold $M$ is simple hyperkähler
(that is, IHS). In [AV], we have shown that Morrison-Kawamata cone conjecture
holds whenever the Beauville-Bogomolov square of primitive MBM classes is
bounded. This is known to be the case for deformations of punctual Hilbert
schemes of K3 surfaces and for deformations of generalized Kummer varieties. A
different proof in the similar spirit for these types of hyperkähler manifolds
has been given by Markman and Yoshioka in [MY], under an extra assumption that
the manifolds in question are projective.

Let us also briefly mention that this conjecture has a birational version,
proved for projective hyperkähler manifolds by E. Markman in [M3] and
generalized in [AV] to the non-projective case. In this birational version,
the nef cone is replaced by the birational nef cone (that is, the closure of
the union of pullbacks of Kähler cones on birational models of $M$) and the
group $\operatorname{Aut}(M)$ is replaced by the group of birational
automorphisms $\operatorname{Bir}(M)$.

The key point of the proof of [AV] is the observation that the orthogonal
group $O(H^{1,1}_{{\mathbb{Z}}}(M),q)$ of the lattice
$H^{1,1}_{{\mathbb{Z}}}(M)=H^{1,1}(M)\cap H^{2}(M,{\mathbb{Z}})$, and
therefore the Hodge monodromy group $\Gamma_{\operatorname{\sf Hdg}}$ (see
2.1) which is a subgroup of finite index in $O(H^{1,1}_{{\mathbb{Z}}}(M),q)$ ,
acts with finitely many orbits on the set of classes of fixed square $r\neq
0$. When the primitive MBM classes have bounded square, we conclude that the
monodromy acts with finitely many orbits on the set of MBM classes. As those
are precisely the classes whose orthogonal hyperplanes support the faces of
the Kähler cone, it is not difficult to deduce that there are only finitely
many, up to the action of the monodromy group, faces of the Kähler cone, and
also finitely many oriented faces of the Kähler cone (an oriented face is a
face together with the choice of normal direction). An element of the
monodromy which sends a face $F$ to a face $F^{\prime}$, with both
orientations pointing towards the interior of the Kähler cone, must preserve
the Kähler cone. On the other hand, Markman proved ([M3], Theorem 1.3) that an
element of the Hodge monodromy which preserves the Kähler cone must be induced
by an automorphism, so that the cone conjecture follows.

###  1.3 Main results

The main point of the present paper is that the finiteness of the set of
primitive MBM classes of type $(1,1)$, up to the monodromy action, can be
obtained without the boundedness assumption on their Beauville-Bogomolov
square.

Our main technical result is the following

Theorem 1.3:  Let $L$ be a lattice of signature $(1,n)$ where $n\geqslant 3$,
$V=L\otimes{\mathbb{R}}$. Let $\Gamma$ be an arithmetic subgroup in $SO(1,n)$.
Let $Y:=\bigcup S_{i}$ be a $\Gamma$-invariant union of rational hyperplanes
$S_{i}$ orthogonal to negative vectors $z_{i}\in L$ in $V$. Then either
$\Gamma$ acts on $\\{S_{i}\\}$ with finitely many orbits, or $Y$ is dense in
the positive cone in $V$.

Proof: See 4.3.

Remark 1.4: The assumption $n\geqslant 3$ is important for our argument which
is based on Ratner theory. We shall see that Ratner theory applies to our
problem as soon as the connected component of the unity of $SO(1,n-1)$ is
generated by unipotents, that is, for $n\geqslant 3$.

Taking $H^{1,1}_{\mathbb{Z}}(M)=H^{1,1}(M)\cap H^{2}(M,{\mathbb{Z}})$ for $L$
and the Hodge monodromy group for $\Gamma$, we easily deduce:

Theorem 1.5:  Assume that $M$ is projective, of Picard rank at least 4. The
monodromy group acts with finitely many orbits on the set of MBM classes which
are of type $(1,1)$.

Proof: See 5.1.

Note that, by a result of Huybrechts, the projectivity assumption for $M$ is
equivalent to the signature $(1,n)$ assumption for its Picard lattice $L$.

The boundedness results as an obvious corollary.

Corollary 1.6:  On a projective $M$ with Picard number at least 4, primitive
MBM classes of type $(1,1)$ have bounded Beauville-Bogomolov square.

Proof: Indeed, the monodromy acts by isometries.

Using the deformation invariance of MBM property, we can actually drop the
assumption that $M$ is projective and has Picard rank at least four. Indeed,
if $M$ is a simple hyperkähler manifold with $b_{2}(M)\geqslant 6$, we can
always deform it to a projective manifold $M^{\prime}$ on which all classes
from $H^{1,1}_{{\mathbb{Z}}}(M)$ stay of type $(1,1)$ (see 2.2). Since the
square of a primitive MBM class is bounded on $M^{\prime}$, the same is true
for $M$.

The Morrison-Kawamata cone conjecture is then deduced as we have sketched it
above, exactly in the same way as in [AV].

Theorem 1.7:  Let $M$ be a simple hyperkähler manifold with $b_{2}(M)\geqslant
6$. The group of automorphisms $\operatorname{Aut}(M)$ acts with finitely many
orbits on the set of faces of the Kähler cone $\operatorname{Kah}(M)$.

Proof: See 5.1.

Remark 1.8: The theorem holds trivially for $M$ with $b_{2}(M)<5$, so that our
result is valid as soon as $b_{2}(M)\neq 5$. This remaining case can probably
be handled using methods of hyperbolic geometry. The general belief is,
though, that simple hyperkähler manifolds with $b_{2}=5$ do not exist.

Finally, as observed by Markman and Yoshioka, the boundedness of squares of
primitive MBM classes implies the following theorem (we thank Y. Kawamata for
indicating us the statement).

Theorem 1.9:  Let $M$ be a simple hyperkähler manifold with $b_{2}(M)\geqslant
6$. Then there are only finitely many simple hyperkähler manifolds birational
to $M$.

Proof: This is just [MY], Corollary 1.5. Indeed, the classes $e$ menitioned in
Conjecture 1.1 from [MY] (that is, the classes generating the extremal rays of
the Mori cone on the simple hyperkähler birational models of $M$) are MBM
classes in the sense of our 2.1.

The crucial tool for the proof of 1.3 is Ratner theory. We recall this and
some other relevant information from ergodic theory in section 3, after some
preliminaries on hyperkähler manifolds in section 2. In section 4 we deduce
1.3 from Mozes-Shah and Dani-Margulis theorems. Finally, in the last section
we apply this to hyperkähler manifolds and prove 1.3.

##  2 Preliminaries

###  2.1 Hyperkähler manifolds, monodromy and MBM classes

Definition 2.1: A hyperkähler manifold is a compact, Kähler, holomorphically
symplectic manifold.

Definition 2.2: A hyperkähler manifold $M$ is called simple, or IHS, if
$\pi_{1}(M)=0$, $H^{2,0}(M)={\mathbb{C}}$.

This definition is motivated by Bogomolov’s decomposition theorem:

Theorem 2.3:  ([Bo1]) Any hyperkähler manifold admits a finite covering which
is a product of a torus and several simple hyperkähler manifolds.

Remark 2.4: The Bogomolov decomposition theorem can be obtained by applying
the de Rham holonomy decomposition theorem and Berger’s classification of
manifolds with special holonomy to the Ricci-flat hyperkähler metric on a
compact holomorphically symplectic Kähler manifold. Then, a hyperkähler
manifold is simple if and only if its hyperkähler metric has maximal holonomy
group $\operatorname{Hol}(M)$ allowed by the hyperkähler structure, that is
$\operatorname{Hol}(M)=Sp(n)$, where $n=\frac{1}{2}\dim_{\mathbb{C}}M$.

Remark 2.5: Further on, we shall assume that all hyperkähler manifolds we
consider are simple.

The Bogomolov-Beauville-Fujiki form was defined in [Bo2] and [Bea], but it is
easiest to describe it using the Fujiki theorem, proved in [F1].

Theorem 2.6:  (Fujiki) Let $M$ be a simple hyperkähler manifold, $\eta\in
H^{2}(M)$, and $n=\frac{1}{2}\dim M$. Then
$\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, where $q$ is a primitive integral
quadratic form on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ a constant (depending on
$M$).

Remark 2.7: Fujiki formula (2.1) determines the form $q$ uniquely up to a
sign. For odd $n$, the sign is unambiguously determined as well. For even $n$,
one needs the following explicit formula, which is due to Bogomolov and
Beauville.

$\displaystyle\lambda q(\eta,\eta)$
$\displaystyle=\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$
(2.1)
$\displaystyle-\frac{n-1}{n}\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)$

where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$.

Definition 2.8:  A cohomology class $\eta\in H^{1,1}_{\mathbb{R}}(M)$ is
called negative if $q(\eta,\eta)<0$, and positive if $q(\eta,\eta)>0$. Since
the signature of $q$ on $H^{1,1}(M)$ is $(1,b_{2}-3)$, the set of positive
vectors is disconnected. The positive cone $\operatorname{Pos}(M)$ is the
connected component of the set $\\{\eta\in H^{1,1}_{\mathbb{R}}(M)\ \ |\ \
q(\eta,\eta)>0\\}$ which contains the classes of the Kähler forms. Using the
Cauchy-Schwarz inequality, it is easy to check that the positive cone is
convex.

Definition 2.9: Let $M$ be a hyperkähler manifold. The monodromy group of $M$
is a subgroup of $GL(H^{2}(M,{\mathbb{Z}}))$ generated by the monodromy
transforms for all Gauss-Manin local systems.

It is often enlightening to consider this group in terms of the mapping class
group action. In the following paragraphs, we recall this description.

Definition 2.10: Let $M$ be a compact complex manifold, and $\operatorname{\sf
Diff}_{0}(M)$ a connected component of its diffeomorphism group (the group of
isotopies). Denote by $\operatorname{\sf Comp}$ the space of complex
structures of Kähler type on $M$, equipped with its structure of a Fréchet
manifold (remark here that the set of complex structures of Kähler type is
open in the space of all complex structures by Kodaira-Spencer stability
theorem), and let $\operatorname{\sf Teich}:=\operatorname{\sf
Comp}/\operatorname{\sf Diff}_{0}(M)$. We call it the Teichmüller space.

For hyperkähler manifolds, this is a finite-dimensional complex non-Hausdorff
manifold ([Cat], [V2]).

Definition 2.11: The mapping class group is $\operatorname{\sf
Diff}(M)/\operatorname{\sf Diff}_{0}(M)$. It naturally acts on
$\operatorname{\sf Teich}$. The quotient of $\operatorname{\sf Teich}$ by this
action may be viewed as the “moduli space” for $M$. However, this space is too
non-Hausdorff to be useful: any two open subsets of $\operatorname{\sf
Teich}/\operatorname{\sf Diff}$ intersect ([V3], [V4]).

By a result of Huybrechts (see [H3]), in the hyperkähler case
$\operatorname{\sf Teich}$ has only finitely many connected components.
Therefore, the subgroup of the mapping class group which fixes the connected
component of our chosen complex structure is of finite index in the mapping
class group.

Definition 2.12:  The monodromy group $\Gamma$ is the image of this subgroup
in $\operatorname{Aut}H^{2}(M,{\mathbb{Z}})$. The Hodge monodromy group is the
subgroup $\Gamma_{\operatorname{\sf Hdg}}\subset\Gamma$ preserving the Hodge
decomposition.

The following theorem is crucial for the Morrison-Kawamata cone conjecture.

Theorem 2.13:  ([V2], Theorem 3.5) The monodromy group is a finite index
subgroup in $O(H^{2}(M,{\mathbb{Z}}),q)$ (and the Hodge monodromy is therefore
an arithmetic subgroup of the orthogonal group of the Picard lattice).

Next, we recall from [AV] the definition of MBM classes. Remark that any
birational map between hyperkähler manifolds $\varphi:M\dasharrow M^{\prime}$
is an isomorphism in codimension one (in general this easily follows from the
nefness of the canonical class) and therefore induces an isomorphism on the
second cohomology. We say that $M$ and $M^{\prime}$ are birational models of
each other.

Definition 2.14:  A non-zero negative rational homology class $z\in
H^{1,1}(M)$ is called monodromy birationally minimal (MBM) if for some
isometry $\gamma\in O(H^{2}(M,{\mathbb{Z}}))$ belonging to the monodromy
group, $\gamma(z)^{\bot}\subset H^{1,1}(M)$ contains a face of the pull-back
of the Kähler cone of one of birational models $M^{\prime}$ of $M$.

Remark 2.15: Here the orthogonal is taken with respect to the Beauville-
Bogomolov form. A face of $\operatorname{Kah}(M)$ is, by definition, of
maximal dimension $h^{1,1}(M)-1$. So the definition of $z$ being MBM means
that $\gamma(z)^{\bot}\cap\partial\operatorname{Kah}(M^{\prime})$ contains an
open subset of $\gamma(z)^{\bot}$. The MBM classes, or more precisely the rays
they generate, are natural analogues of “extremal rays” from projective
geometry, up to monodromy and birational equivalence; hence the name.

The following theorem has been proved in [AV].

Theorem 2.16: Let $M$ be a hyperkähler manifold, $z\in H^{1,1}(M)$ an integral
cohomology class, $q(z,z)<0$, and $M^{\prime}$ a deformation of $M$ such that
$z$ remains of type (1,1) on $M^{\prime}$. Assume that $z$ is monodromy
birationally minimal on $M$. Then $z$ is monodromy birationally minimal on
$M^{\prime}$.

The MBM classes can be used to determine the Kähler cone of $M$ explicitly.

Theorem 2.17: ([AV]) Let $M$ be a hyperkähler manifold, and $S\subset
H^{1,1}(M)$ the set of all MBM classes of type $(1,1)$. Consider the
corresponding set of hyperplanes $S^{\bot}:=\\{W=z^{\bot}\ \ |\ \ z\in S\\}$
in $H^{1,1}(M)$. Then the Kähler cone of $M$ is a connected component of
$\operatorname{Pos}(M)\backslash\cup S^{\bot}$, where $\operatorname{Pos}(M)$
is the positive cone of $M$. Moreover, for any connected component $K$ of
$\operatorname{Pos}(M)\backslash\cup S^{\bot}$, there exists $\gamma\in
O(H^{2}(M,{\mathbb{Z}}))$ in the monodromy group of $M$ and a birational model
$M^{\prime}$ such that $\gamma(K)$ is the Kähler cone of $M^{\prime}$.

Remark 2.18: The main point of this theorem is that for a negative integral
class $z\in H^{1,1}(M)$, the orthogonal hyperplane either passes through the
interiour of some Kähler-Weyl chamber and then it contains no face of a
Kähler-Weyl chamber (that is, $z$ is not MBM), or its intersection with the
positive cone is a union of faces of such chambers (when $z$ is MBM). This is
illustrated by a picture taken from [AV]:

Allowed partition Prohibited partition

###  2.2 Global Torelli theorem and deformations

In this subsection, we recall a number of results about deformations of
hyperkähler manifolds used further on in this paper. For more details and
references, see [V2].

Let $M$ be a hyperkähler manifold (as usual, we assume $M$ to be simple). Any
deformation $M^{\prime}$ of $M$ is also a simple hyperkähler manifold, because
the Hodge numbers are constant in families and thus $H^{2,0}(M^{\prime})$ is
one-dimensional. Let us view $M^{\prime}$ as a couple $(M,J)$, where $J$ is a
new complex structure on $M$, that is, a point of the Teichmüller space
$\operatorname{\sf Teich}$.

Definition 2.19: Let

$\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$

map $J$ to the line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$. The
map $\operatorname{\sf Per}$ is called the period map.

Remark 2.20: The period map $\operatorname{\sf Per}$ maps $\operatorname{\sf
Teich}$ into an open subset of a quadric, defined by

$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\
\ |\ \ q(l,l)=0,q(l,\overline{l})>0\\}.$

It is called the period domain of $M$. Indeed, any holomorphic symplectic form
$l$ satisfies the relations $q(l,l)=0,q(l,\overline{l})>0$, as follows from
(2.1).

Definition 2.21: Let $M$ be a topological space. We say that $x,y\in M$ are
non-separable (denoted by $x\sim y$) if for any open sets $V\ni x,U\ni y$,
$U\cap V\neq\emptyset$.

By a result of Huybrechts [H1], non-separable points of $\operatorname{\sf
Teich}$ correspond to birational hyperkähler manifolds.

Definition 2.22: The space $\operatorname{\sf Teich}_{b}:=\operatorname{\sf
Teich}\\!/\\!\\!\sim$ is called the birational Teichmüller space of $M$.

Remark 2.23: This terminology is slightly misleading since there are non-
separable points of the Teichmüller space which correspond to biregular, not
just birational, complex structures. Even for K3 surfaces, the Teichmüller
space is non-Hausdorff.

Theorem 2.24:  (Global Torelli theorem; [V2]) The period map
$\operatorname{\sf Teich}_{b}\stackrel{{\scriptstyle\operatorname{\sf
Per}}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf er}$ is an
isomorphism on each connected component of $\operatorname{\sf Teich}_{b}$.

By a result of Huybrechts ([H3]), $\operatorname{\sf Teich}$ has only finitely
many connected components. We shall fix the component ${\operatorname{\sf
Teich}}^{0}$ containing the parameter point for our initial complex structure,
and denote by $\widetilde{\Gamma}$ the subgroup of finite index in the mapping
class group fixing this component.

It is natural to view the quotient of $\operatorname{\sf Teich}$ by the
mapping class group as a moduli space for $M$ and the quotient of
$\operatorname{\sf Teich}_{b}$ by the mapping class group as a “birational
moduli space”: indeed its points are in bijective correspondence with the
complex structures of hyperkähler type on $M$ up to a bimeromorphic
equivalence.

Remark 2.25: The word “space” in this context is misleading. In fact, the
quotient topology on $\operatorname{\sf Teich}^{0}_{b}/\widetilde{\Gamma}$ is
extremely non-Hausdorff, e.g. every two open sets would intersect ([V3]).

The Global Torelli theorem can be stated as a result about the birational
moduli space.

Theorem 2.26:  ([V2, Theorem 7.2, Remark 7.4, Theorem 3.5]) Let $(M,I)$ be a
hyperkähler manifold, and $W$ a connected component of its birational moduli
space. Then $W$ is isomorphic to ${\operatorname{{\mathbb{P}}\sf er}}/\Gamma$,
where $\Gamma$ is an arithmetic subgroup in $O(H^{2}(M,{\mathbb{R}}),q)$,
called the monodromy group of $(M,I)$. In fact $\Gamma$ is the image of
$\widetilde{\Gamma}$ in $O(H^{2}(M,{\mathbb{R}}),q)$.

Remark 2.27:  As we have already mentioned, the monodromy group of $(M,I)$ can
be also described as a subgroup of the group $O(H^{2}(M,{\mathbb{Z}}),q)$
generated by monodromy transform maps for Gauss-Manin local systems obtained
from all deformations of $(M,I)$ over a complex base ([V2, Definition 7.1]).
This is how this group was originally defined by Markman ([M2], [M3]).

Definition 2.28: Let $z\in H^{2}(M,{\mathbb{Z}})$ be an integral cohomology
class. The space $\operatorname{\sf Teich}_{z}$ is the part of
$\operatorname{\sf Teich}$ where the class $z$ is of type $(1,1)$.

The following proposition is well-known.

Proposition 2.29:  $\operatorname{\sf Teich}_{z}$ is the inverse image under
the period map of the subset ${\operatorname{{\mathbb{P}}\sf
er}}_{z}\subset\operatorname{{\mathbb{P}}\sf er}$ which consists of $l$ with
$q(l,z)=0$.

Proof: This is clear since $H^{1,1}(M)$ is the orthogonal, under $q$, to
$H^{2,0}(M)\oplus H^{0,2}(M)$.

By a theorem of Huybrechts, a holomorphic symplectic manifold $M$ is
projective if and only if it has an integral $(1,1)$-class with strictly
positive Beauville-Bogomolov square. In this case, the Picard lattice
$H^{1,1}_{{\mathbb{Z}}}(M)=H^{2}(M,{\mathbb{Z}})\cap H^{1,1}(M)$, equipped
with the Beauville-Bogomolov form $q$, is a lattice of signature
$(+,-,-,\dots,-)$. If $M$ is not projective, the Picard lattice can be either
negative definite, or degenerate negative semidefinite with one-dimensional
kernel. In both cases, its rank cannot be maximal (i.e. equal to the dimension
of $H^{1,1}(M)$), since the signature of $q$ on $H^{1,1}(M)$ is
$(+,-,-,\dots,-)$. Together with this observation, 2.2 easily implies the
following

Proposition 2.30:  Let $M$ be an irreducible holomorphic symplectic manifold.
There exists a deformation $M^{\prime}$ of $M$ which is projective and such
that all integral $(1,1)$-classes on $M$ remain of type $(1,1)$ on
$M^{\prime}$. Moreover one can take $M^{\prime}$ of maximal Picard rank
$h^{1,1}(M)$.

Proof: By 2.2, the locus $C$ where all integral $(1,1)$-classes on $M$ remain
of type $(1,1)$ is the preimage of the intersection of $N$ complex hyperplanes
and ${\operatorname{{\mathbb{P}}\sf er}}$, where $N$ is strictly less than the
(complex) dimension of ${\operatorname{{\mathbb{P}}\sf er}}$. It is therefore
strictly positive-dimensional. For $M^{\prime}$ representing a general point
of this locus, the Picard lattice is the same as that of $M$, but at a special
point the Picard number jumps. Namely it jumps along the intersection with
each hyperplane of the form $z^{\perp}$, where $z$ is an integral
$(1,1)$-class. In particular, there are isolated points inside $C$ where the
Picard rank is maximal. By the observations above, the corresponding variety
$M^{\prime}$ must be projective.

This proposition shall be useful in reducing the cone conjecture to the
projective case with high Picard number (see 5.1).

##  3 Ergodic theory and its applications

###  3.1 Ergodic theory: basic definitions and facts

Definition 3.1: Let $(M,\mu)$ be a space with a measure, and $G$ a group
acting on $M$ preserving $\mu$. This action is ergodic if all $G$-invariant
measurable subsets $M^{\prime}\subset M$ satisfy $\mu(M^{\prime})=0$ or
$\mu(M\backslash M^{\prime})=0$.

Claim 3.2:  Let $M$ be a manifold, $\mu$ the Lebesgue measure, and $G$ a group
acting on $(M,\mu)$ ergodically. Then the set of points with non-dense orbits
has measure 0.

Proof: Consider a non-empty open subset $U\subset M$. Then $\mu(U)>0$, hence
$M^{\prime}:=G\cdot U$ satisfies $\mu(M\backslash M^{\prime})=0$. For any
orbit $G\cdot x$ not intersecting $U$, $x\in M\backslash M^{\prime}$.
Therefore the set of such points has measure 0.

Definition 3.3: Let $G$ be a Lie group, and $\Gamma\subset G$ a discrete
subgroup. Consider the pushforward of the Haar measure to $G/\Gamma$. Here, by
abuse of terminology, “taking the pushforward” of a measure means measuring
the intersection of the inverse image with a fixed fundamental domain. We say
that $\Gamma$ has finite covolume if the Haar measure of $G/\Gamma$ is finite.
In this case $\Gamma$ is called a lattice subgroup.

Remark 3.4:  Borel and Harish-Chandra proved that an arithmetic subgroup of a
reductive group $G$ is a lattice whenever $G$ has no non-trivial characters
over ${\mathbb{Q}}$ (see [BHCh], Theorem 7.8 for semisimple case and Theorem
9.4 for the general case). In particular, all arithmetic subgroups of a semi-
simple group are lattices. Therefore the monodromy and the Hodge monodromy
groups from the previous section are lattices in the corresponding orthogonal
groups, which is a very important point for us.

In this paper, we deal with the following example of an ergodic action.

Theorem 3.5:  (Calvin C. Moore, [Moo, Theorem 4]) Let $\Gamma$ be a lattice
subgroup (such as an arithmetic subgroup) in a non-compact simple Lie group
$G$ with finite center, and $H\subset G$ a Lie subgroup. Then the left action
of $H$ on $G/\Gamma$ is ergodic if and only if the closure of $H$ is non-
compact.

Let us also state the following classical result.

Theorem 3.6:  (Birkhoff ergodic theorem, see for example [W], 1.6) Let $\mu$
be a probability measure on a manifold $X$, and let $g_{t}$ be an ergodic flow
preserving $\mu$. Then for almost all $x\in X$ and any $f\in L^{1}(\mu)$, the
limit of $m_{T}(f)=\frac{1}{T}\int^{T}_{0}f(g_{t}x)dt$ as $T\to+\infty$ exists
and equals $\int_{X}fd\mu$. In particular, for any measurable subset $K$ and
almost all $x$, the part of time that the orbit of $x$ spends in $K$ is equal
to $\mu(K)$.

###  3.2 Lie groups generated by unipotents

Here we state some of the main results of Ratner theory. We follow [KSS] and
[Mor].

Definition 3.7: Let $G$ be a Lie group, and $g\in G$ any element. We say that
$g$ is unipotent if $g=e^{h}$ for a nilpotent element $h$ in its Lie algebra.
A group $G$ is generated by unipotents if $G$ is multiplicatively generated by
unipotent one-parameter subgroups.

Theorem 3.8:  (Ratner orbit closure theorem, [R1])
Let $H\subset G$ be a Lie subroup generated by unipotents, and $\Gamma\subset
G$ a lattice. Then the closure of any $H$-orbit $Hx$ in $G/\Gamma$ is an orbit
of a closed, connected subgroup $S\subset G$, such that $S\cap x\Gamma
x^{-1}\subset S$ is a lattice in $S$.

Proof: [Mor, 1.1.15 (2)].

For arithmetic groups Ratner orbit closure theorem can be stated in a more
precise way, as follows.

Theorem 3.9:  Let $G$ be a real algebraic group defined over ${\mathbb{Q}}$
and with no non-trivial characters, $W\subset G$ a subgroup generated by
unipotents, and $\Gamma\subset G$ an arithmetic lattice. For a given $g\in G$,
let $H$ be the smallest real algebraic ${\mathbb{Q}}$-subgroup of $G$
containing $g^{-1}Wg$. Then the closure of $Wg$ in $G/\Gamma$ is $Hg$.

Proof: See [KSS, Proposition 3.3.7] or [Sh1, Proposition 3.2].

Ratner orbit closure theorem is a consequence of her fundamental result on
ergodic measures [R2], known as Ratner measure classification theorem, which
we recall below.

Definition 3.10:  Let $G$ be a Lie group, $\Gamma$ a lattice, and $G/\Gamma$
the quotient space, considered as a space with Haar measure. Consider an orbit
$S\cdot x\subset G$ of a closed subgroup $S\subset G$, put the Haar measure on
$S\cdot x$, and assume that its image in $G/\Gamma$ has finite Haar measure
(this means that $S\cap x\Gamma x^{-1}$ is a lattice in $S$). A measure on
$G/\Gamma$ is called algebraic if it is proportional to the pushforward of the
Haar measure on $S\cdot x/\Gamma$ to $G/\Gamma$.

If $G$ is a non-compact simple Lie group with finite center and $H\subset G$
is a Lie subgroup with non-compact closure, as in Moore’s theorem (3.1),
consider the algebraic measure on $G/\Gamma$ which is proportional to the
pushforward of the Haar measure of $S$, where $S$ is taken from the Ratner’s
orbit closure theorem. It follows from Moore’s theorem that the action of $H$
on $G/\Gamma$ is ergodic. Ratner’s measure classification theorem states that
all invariant ergodic measures under the action of subgroups generated by
unipotents arise in this way.

Theorem 3.11:  (Ratner’s measure classification theorem, [R2])
Let $G$ be a connected Lie group, $\Gamma$ a lattice, and $G/\Gamma$ the
quotient space, considered as a space with Haar measure. Consider a finite
measure $\mu$ on $G/\Gamma$. Assume that $\mu$ is invariant and ergodic with
respect to an action of a subgroup $H\subset G$ generated by unipotents. Then
$\mu$ is algebraic.

Proof: see [Mor, 1.3.7].

Remark 3.12:  In most texts, Ratner theorems are formulated for unipotent
flows, that is, $H$ is assumed to be a one-parameter unipotent subgroup
$\\{u(t)|t\in{\mathbb{R}}\\}$. One gets rid of this assumption using the
following lemma.

Lemma 3.13:  ([MS, Lemma 2.3] or [KSS, Corollary 3.3.5]) Let $H$ be a subgroup
of $G$ generated by unipotent one-parameter subgroups. Then any finite
$H$-invariant $H$-ergodic measure is ergodic with respect to some one-
parameter unipotent subgroup of $H$.

##  4 Algebraic measures on homogeneous spaces

The main result of this section (4.3) follows from a theorem of Mozes and Shah
[MS, Theorem 1.1].

###  4.1 Limits of ergodic measures

Definition 4.1: Recall that a Polish topological space is a metrizable
topological space with countable base. Let $V$ be the set of all finite Borel
measures on a Polish topological space $M$, and $C^{0}(M)$ the space of
bounded continuous functions. Weak topology on $V$ is the weakest topology in
which for all $f\in C^{0}(M)$ the map $V{\>\longrightarrow\>}{\mathbb{R}}$
given by $\mu{\>\longrightarrow\>}\int_{M}f\mu$ is continuous. If one
identifies $V$ with a subset in $C^{0}(M)^{*}$, the weak topology is
identified with the weak-* topology on $C^{0}(M)^{*}$. This is why it is also
called the weak-* topology.

Remark 4.2: It is not hard to prove that the space of probability measures on
a compact Polish space is compact in weak topology. This explains the
usefulness of this notion.

Theorem 4.3:  (Mozes-Shah theorem)
Let $G$ be a connected Lie group, $\Gamma$ a lattice, $\\{u_{i}(t)\\}\subset
G$ a sequence of unipotent one-parameter subgroups in $G$, and $\mu_{i}$ a
sequence of $u_{i}$-invariant, $u_{i}$-ergodic probability measures on
$G/\Gamma$, associated with orbits $S_{i}\cdot x_{i}\subset G/\Gamma$ as in
3.2. Assume that $\lim\mu_{i}=\mu$ with respect to weak topology, with $\mu$ a
probability measure on $X$, and let $x\in\operatorname{Supp}(\mu)$. Then

(i)

$\mu$ is an algebraic measure, associated with an orbit $S\cdot x$ as in 3.2.

(ii)

Let $g_{i}\in G$ be elements which satisfy $g_{i}x_{i}=x$, and assume that
$g_{i}\to e$ in $G$ (so that $x_{i}$ converge to $x$). Then there exists
$i_{0}\in{\mathbb{N}}$ such that for all $i>i_{0}$, $S\cdot x\supset
g_{i}S_{i}\cdot x_{i}$.

Proof: The statement (i) follows from [MS, Theorem 1.1 (3)] and Ratner measure
classification theorem, and (ii) is [MS, Theorem 1.1 (2)].

Remark 4.4: More precisely, in [MS, Theorem 1.1] there is an additional
condition that the trajectories $\\{u_{i}(t)\\}x_{i},\ t>0$ should be
uniformly distributed with respect to $\mu_{i}$. But this is automatic by
another theorem of Ratner (Ratner equidistribution theorem, see e.g. [Mor],
Theorem 1.3.4), and in fact already by Birkhoff ergodic theorem (3.1), which
states the uniform distribution of orbits of one-parameter subgroups for
almost all starting points.

The following theorem is an interpretation of Dani-Margulis theorem as stated
in [DM, Theorem 6.1] obtained by applying Birkhoff ergodic therorem.

Theorem 4.5:  (Dani-Margulis theorem).
Let $G$ be a connected Lie group, $\Gamma$ a lattice, $X:=G/\Gamma$, $C\subset
X$ a compact subset, and $\varepsilon>0$. Then there exists a compact subset
$K\subset X$ such that for any algebraic probability measure $\mu$ on $X$,
satisfying $\mu(C)\neq 0$ and associated with a group generated by unipotents,
one has $\mu(K)\geqslant 1-\varepsilon$.

Proof: By 3.2, $\mu$ is invariant and ergodic with respect to a one-parameter
unipotent subgroup $u(t)$. Now apply [DM, Theorem 6.1] to a starting point $x$
which is one of “almost all points” of $\operatorname{Supp}(\mu)\cap C$ in the
sense of Birkhoff theorem.

Combining Dani-Margulis theorem and Mozes-Shah theorem, one gets the following
useful corollary ([MS, Corollary 1.1, Corollary 1.3, Corollary 1.4]).

Corollary 4.6:  Let $G$ be a connected Lie group, $\Gamma$ a lattice, ${\cal
P}(X)$ be the space of all probability measures on $X=G/\Gamma$, and ${\cal
Q}(X)\subset{\cal P}(X)$ the space of all algebraic probability measures
associated with subgroups $H\subset G$ generated by unipotents (as in Ratner
theorems). Then ${\cal Q}(X)$ is closed in ${\cal P}$ with respect to weak-
star topology. Moreover, let $X\cap\\{\infty\\}$ denote the one-point
compactification of $X$, so that ${\cal P}(X\cap\\{\infty\\})$ is compact. If
for a sequence $\mu_{i}\in{\cal Q}(X)$, $\mu_{i}\to\mu\in{\cal
P}(X\cap\\{\infty\\})$, then either $\mu\in{\cal Q}(X)$, or $\mu$ is supported
at infinity.

###  4.2 Rational hyperplanes intersecting a compact set

Definition 4.7: Let $V_{\mathbb{Q}}$ be an $n+1$-dimensional rational vector
space with a scalar product of signature $(+,-,-,...,-)$, and
$V:=V_{\mathbb{Q}}\otimes_{\mathbb{Q}}{\mathbb{R}}$. We consider the
projectivization of the positive cone ${\mathbb{P}}^{+}V$ as the hyperbolic
space of dimension $n$. Given a $k+1$-dimensional subspace
$W_{\mathbb{Q}}\subset V_{\mathbb{Q}}$ such that the restriction of the scalar
product to $W_{\mathbb{Q}}$ still has signature $(1,k)$, we may associate the
projectivized positive cone ${\mathbb{P}}^{+}W\subset{\mathbb{P}}^{+}V$ with
$W=W_{\mathbb{Q}}\otimes_{\mathbb{Q}}{\mathbb{R}}$. When $k=n-1$, we shall
call ${\mathbb{P}}^{+}W\subset{\mathbb{P}}^{+}V$ a rational hyperplane in
${\mathbb{P}}^{+}V$.

Let $\Gamma$ be a rational lattice in the group of isometries of
${\mathbb{P}}^{+}V$, and $\\{S_{i}\\}$ a set of rational hyperplanes. We are
interested in the images of $S_{i}$ in ${\mathbb{P}}^{+}V/\Gamma$. The
following theorem can be used to show that these images all intersect a
compact subset of ${\mathbb{P}}^{+}V/\Gamma$.

Theorem 4.8:  Let $\\{S_{i}\\}$ be a set of rational hyperplanes in
${\mathbb{P}}^{+}V$, $P_{\mathbb{Q}}\subset V_{\mathbb{Q}}$ a rational
subspace of signature $(1,2)$, and ${\mathbb{P}}^{+}P\subset{\mathbb{P}}^{+}V$
the corresponding 2-dimensional hyperbolic subspace. Consider an arithmetic
lattice $\Gamma\subset SO(V,{\mathbb{Z}})$, and let $\Gamma_{P}$ be the
stabilizer of $P_{\mathbb{Q}}$ in $\Gamma$. Then there exists a compact subset
$K\subset{\mathbb{P}}^{+}P$ such that $\Gamma_{P}\cdot K$ intersects all the
hyperplanes $S_{i}$.

Proof: Since $\Gamma$ has finite index in a lattice $O(V,{\mathbb{Z}})$,
$\Gamma_{P}$ has finite index in the lattice $O(P,{\mathbb{Z}})$. One may view
$\Gamma_{P}$ as a multi-dimensional analogue of Fuchsian or Kleinian group,
acting properly discontinuously on the hyperbolic plane. Then $\Gamma_{P}$
acts with finite stabilizers, and the quotient of the hyperbolic plane by
$\Gamma_{P}$ is a hyperbolic orbifold $X$. We must prove that there is a
compact subset of $X$ such that its intersection with the image of any line
$L_{i}=S_{i}\cap{\mathbb{P}}^{+}P$ is non-empty. But any arithmetic lattice
has a finite index subgroup which is torsion-free (for instance, the
congruence subgroup formed by integer matrices which are identity modulo $N$
for $N$ big enough). Therefore, our orbifold $X$ has a finite covering
$\widetilde{X}$ which is a hyperbolic Riemann surface, and it suffices to
prove that there is a compact $\widetilde{K}\subset\widetilde{X}$ such that
$\pi(L_{i})$ intersects $\widetilde{K}$ for any $i$, where
$\pi:{\mathbb{P}}^{+}P{\>\longrightarrow\>}\widetilde{X}$ denotes the
projection (quotient by a finite index subgroup
$\widetilde{\Gamma_{P}}\subset\Gamma_{P}$).

Let $\Gamma_{L_{i}}$ be the stabilizer of $L_{i}$ in $\Gamma_{P}$. Since
$\Gamma_{L_{i}}$ has finite index in $SO(L_{i},{\mathbb{Z}})$, the images
$\pi(L_{i})$ have finite length in $\widetilde{X}$. On the other hand,
$\pi(L_{i})$ are isometric images of $L_{i}$. Therefore, $\pi(L_{i})$ are
compact; in other words, these are closed geodesics on $\widetilde{X}$. We
have reduced 4.2 to the following well-known lemma.

Lemma 4.9:  Let $S$ be a complete hyperbolic Riemann surface (of constant
negative curvature and finite volume). Then there exists a compact subset
$K\subset X$ intersecting each closed geodesic $l\subset S$.

Proof: To obtain $K$, it suffices to remove from $S$ a neighbourhood of each
cusp: indeed, there are no closed geodesics around cusps. This is an
elementary exercise, apparently well known; see e.g. [MR, Theorem 1.2] which
is in the same spirit. For the convenience of the reader, we sketch an
argument here.

Let ${\mathbb{H}}=\\{x\in{\mathbb{C}}\ \ |\ \ \operatorname{Im}(x)>0\\}$ be a
hyperbolic half-plane, equipped with a Poincare metric, $t>0$ a real number,
and ${\mathbb{H}}_{t}=\\{x\in{\mathbb{C}}\ \ |\ \ t>\operatorname{Im}(x)>0\\}$
a strip consisting of all $x\in{\mathbb{C}}$ with $0<\operatorname{Im}(x)<t$.
In a neighbourhood of a cusp point, $S$ is isometric to a quotient
${\mathbb{H}}_{t}/{\mathbb{Z}}$, where the action of ${\mathbb{Z}}$ is
generated by the parallel transport $\gamma_{r}(x)=x+r$, where
$r\in{\mathbb{R}}$ is a fixed number. A geodesic is a half-circle
perpendicular to the line $\operatorname{Im}x=0$; closed geodesic in
${\mathbb{H}}_{t}/{\mathbb{Z}}$ is a half-circle which is mapped to itself by
a power of $\gamma_{r}$. Such half-circles clearly do not exist (see the
picture).

A neighbourhood of a cusp point in dimension 2

Remark 4.10: The result of this subsection shall be used in the next one to
justify that a certain sequence of ergodic measures does not have a
subsequence going to infinity. Since all the measures in question come from
orbits of the same subgroup, this is also a consequence of [EMS], Corollary
1.10. We prefer nevertheless to keep our simple observations on hyperbolic
geometry which might have some independent interest.

###  4.3 Measures and rational hyperplanes in the hyperbolic space

The hyperbolic space, that is, the projectivization of the positive cone in a
real vector space with a quadratic form of signature $(1,n)$, is a homogeneous
space in an obvious way. Indeed it is an orbit of any positive line by the
connected component of the unity of $SO(1,n)$, and the stabilizer is
isomorphic to $SO(n)$. If $z$ is a negative vector, then $z^{\perp}$ is a
hyperplane which intersects the positive cone; as in the previous paragraph,
by a hyperplane in the hyperbolic space we shall mean the projectivization of
this intersection.

Theorem 4.11:  Let $G$ be the connected component of the unity $SO^{+}(1,n)$
of $SO(1,n)$, where $n\geqslant 3$, $H:=SO(n)$, and $\Gamma\subset
G_{\mathbb{Z}}$ a discrete subgroup of finite index (and therefore of finite
covolume, 3.1). Consider the hyperbolic space ${\mathbb{H}}=H\backslash
G=SO(n)\backslash SO^{+}(1,n)$. Let $Y:=\bigcup S_{i}$ be a $\Gamma$-invariant
union of rational hyperplanes. Then either $\Gamma$ acts on $\\{S_{i}\\}$ with
finitely many orbits, or $Y$ is dense in ${\mathbb{H}}$.

Proof: Let $V={\mathbb{R}}^{1,n}$ be a real vector space of signature $(1,n)$,
$G=SO^{+}(V)$, and ${\mathbb{H}}=H\backslash G$, where $H\subset G$ is the
stabilizer of an oriented positive hyperplane. We may identify ${\mathbb{H}}$
with the space of positive vectors $x\in V$, $(x,x)=1$. In order to apply
ergodic theory, we replace ${\mathbb{H}}$ by the incidence variety $X$ of
pairs $({\mathbb{H}}_{W}\subset{\mathbb{H}},x\in{\mathbb{H}}_{W})$, where
${\mathbb{H}}_{W}$ is an oriented hyperplane in the hyperbolic space and
$x\in{\mathbb{H}}_{W}$. Clearly, a point of $X$ is uniquely determined by a
pair of orthogonal vectors $x,y\in V$, where $x$ is positive, $(x,x)=1$, and
$(y,y)=-1$. Therefore, $X=H_{0}\backslash G$, where $H_{0}=SO(n-1)$. The
important point is that $X$ is a quotient of $G$ by a compact group (and so is
${\mathbb{H}}$). Moreover $X$ is fibered over ${\mathbb{H}}$ in spheres of
dimension $n-1$.

We can lift our hyperplanes $S_{i}$ to $X$ in the tautological way. To make
the picture transparent, we first treat the case $n=2$, where ${\mathbb{H}}$
is a hyperplane (since in the theorem we have $n\geqslant 3$, this is just to
describe the lifting and see a certain well-known analogy). Here
$X=SO^{+}(1,2)$ is the unit tangent bundle over ${\mathbb{H}}$, and a point
$x\in S_{i}$ lifts as $(x,z)$ where $z$ is the unit tangent vector to $S_{i}$
in the direction given by the orientation. If we lift all possible hyperplanes
to $X$ in this way, we obtain a foliation known as the geodesic flow: our
liftings never intersect and are tangent to an invariant vector field on
$X=SO^{+}(1,2)$. Therefore all the lifting are orbits of a Lie subgroup
$H_{1}\subset G$ (this one-parameter subgroup, isomorphic to $SO^{+}(1,1)$,
can be identified with the group of diagonal two-by-two matrices with $e^{t}$
and $e^{-t}$ on the diagonal under an isomorphism between $SO^{+}(1,2)$ and
$PSL(2,{\mathbb{R}})$, see the first chapter of Morris’ book [Mor]).

For $n\geqslant 3$, we first tautologically lift the hyperplanes to $X$ and
then take preimages under the projection from $G$ to $X$. Again, we obtain a
translation-invariant foliation on $G$, which means that the liftings and
their preimages are orbits of a subgroup $H_{1}\subset G$ (containing
$H_{0}$). This subgroup is isomorphic to $SO^{+}(1,n-1)$, that is, generated
by unipotents, so that ergodic theory applies.

Let us denote by $R_{i}$ the preimage in $G$ of the lifting of $S_{i}$ to $X$.
Each $R_{i}$ is an orbit of $H_{1}$. By 4.2, there is a compact set $C$ such
that the $\Gamma$-orbit of any $S_{i}$ intersects $C$. Since the projection
from $G$ to ${\mathbb{H}}$ is proper, the same is true for the set of $R_{i}$.
Suppose that $\Gamma$ acts on the set of $S_{i}$ (and thus $R_{i}$) with
infinitely many orbits. Consider the homogeneous space $G/\Gamma$. Each
$\Gamma$-orbit on the set of $R_{i}$ corresponds to an algebraic probability
measure $\mu_{i}$ on $G/\Gamma$ (note that since the hyperplanes $S_{i}$ are
rational, the quotient of each $H_{1}$-orbit $R_{i}$ over its stabilizer in
$\Gamma$ has finite Haar volume by Borel and Harish-Chandra theorem). The
support of $\mu_{i}$ is the image of $R_{i}$ in $G/\Gamma$. Since the union of
$R_{i}$ is $\Gamma$-invariant, to prove 4.3, it suffices to show that the
union of $\operatorname{Supp}(\mu_{i})$ is dense in $G/\Gamma$: this will
imply the density of $R_{i}$ in $G$ and therefore the density of $S_{i}$ in
${\mathbb{H}}$.

By 4.1, the sequence $\mu_{i}$ has a limit point which is either a probability
measure, or is supported at infinity. But the latter option is impossible.
Indeed, by 4.2 all $\operatorname{Supp}(\mu_{i})$ intersect the same compact
on $G/\Gamma$. Thus there is a (slightly larger) compact $C$ such that
$\mu_{i}(C)>0$ for all $i$, and by Dani-Margulis theorem, for another compact
$K_{\varepsilon}$ and all $i$, $\mu_{i}(K)>1-\varepsilon$.

Taking a suitable subsequence, we may therefore suppose that $\lim\mu_{i}=\mu$
where $\mu$ is an algebraic probability measure.

We have reduced 4.3 to the following lemma.

Lemma 4.12:  Let $G$ be the connected component $SO^{+}(1,n)$ of $SO(1,n)$,
where $n\geqslant 3$, and $\Gamma\subset G_{\mathbb{Z}}$ a discrete subgroup
of finite index (and therefore of finite covolume). Let $H_{1}\subset G$ be
$SO^{+}(1,n-1)$. Let $\mu_{i}$ be a sequence of algebraic probability measures
on $G/\Gamma$ associated with the orbits of $H_{1}$. Suppose $\mu_{i}$
converges to an algebraic probability measure $\mu$. Then either $\mu_{i}$ are
finitely many, or $\operatorname{Supp}(\mu)$ is $G/\Gamma$, so that
$\operatorname{Supp}(\mu_{i})$ are dense in $G/\Gamma$.

Proof: By 4.1 (ii), the support of $\mu$ contains a right translate by
$g_{i}\to e$ of the support of infinitely many of $\mu_{i}$. Moreover, $\mu$
is an algebraic measure associated with an orbit of a closed subgroup
$F\subset G$. But there are no closed intermediate connected subgroups between
$G=SO^{+}(1,n)$ and $H_{1}$, which stabilizes a hyperplane. Therefore, $F$ is
either equal to $G$, or is the stabilizer $H$ of a hyperplane
${\mathbb{H}}_{W}$.

In the first case, the support of $\mu=\lim\mu_{i}$ is $G/\Gamma$ and thus
$\operatorname{Supp}(\mu_{i})$ are dense in $G/\Gamma$.

In the second case, for $i\gg 0$,
$\operatorname{Supp}(\mu)=g_{i}\operatorname{Supp}(\mu_{i})$, that is,
$Hx=g_{i}H_{1}x_{i}$ where $g_{i}x_{i}=x$ (where $x$, $x_{i}$, $g_{i}$ are as
in 4.1). That is, $Hx=H_{1}^{g_{i}}x$ and therefore $H_{1}^{g_{i}}=H=H_{1}$.
Since $H_{1}$ has finite index in its normalizer, this means that there are
only finitely many $\mu_{i}$.

##  5 The proof of Morrison-Kawamata cone conjecture

###  5.1 Morrison-Kawamata conjecture for the Kähler cone

The following theorem is an immediate consequence of 4.3.

Theorem 5.1:  Let $M$ be a projective simple hyperkähler manifold which has
Picard number at least 4. Then the Hodge monodromy group acts with finitely
many orbits on the set of MBM classes of type $(1,1)$.

Proof: This is the same as to say that the Hodge monodromy group acts with
finitely many orbits on the set of their orthogonal hyperplanes, which by [AV]
are exactly the hyperplanes supporting the faces of the Kähler chambers.

Since the Hodge monodromy group is of finite index in the orthogonal group of
the Picard lattice, which is of signature $(+,-,\dots,-)$, one can apply 4.3
to the Picard lattice, with $\Gamma$ the Hodge monodromy group. One concludes
that if the number of $\Gamma$-orbits is infinite, then the hyperplanes
orthogonal to MBM classes should be dense in the positive cone. This is
clearly absurd, as they should bound the Kähler cone (Subsection 1.1), so the
number of $\Gamma$-orbits is finite.

Corollary 5.2: On an $M$ as above, the primitive MBM classes of type $(1,1)$
have bounded Beauville-Bogomolov square.

Proof: Indeed, the monodromy acts by isometries.

Theorem 5.3:  Let $M$ be a simple hyperkähler manifold such that
$b_{2}(M)\geqslant 6$. Then the primitive MBM classes of type $(1,1)$ have
bounded Beauville-Bogomolov square.

Proof: If $M$ is not projective or the Picard number of $M$ is less than four,
apply 2.2 to get a projective deformation $M^{\prime}$ with Picard number at
least four such that all MBM classes of type $(1,1)$ on $M$ remain of type
$(1,1)$ on $M^{\prime}$. Then use the deformation invariance of MBM property
proved in [AV] to conclude that these MBM classes remain MBM on $M^{\prime}$
and therefore the primitive ones must have bounded square by the preceding
theorem.

The Morrison-Kawamata conjecture for the Kähler cone now follows in the same
way as in [AV].

Theorem 5.4:  Let $M$ be a simple hyperkähler manifold with $b_{2}(M)\geqslant
6$. Then the automorphism group of $M$ acts with finitely many orbits on the
set of faces of its Kähler cone.

Proof: The argument is the same as in [AV] where the theorem has been obtained
under the boundedness assumption on squares of primitive MBM classes, which we
have just proved: see [AV, Theorem 6.6] there for an outline of the argument
and [AV, Theorem 3.14, 3.29] for technicalities.

###  5.2 Morrison-Kawamata conjecture for the ample cone

Recall from e.g. [MY] that the classical Morrison-Kawamata cone conjecture is
formulated in the projective case and treats the ample cone rather than the
Kähler cone. It also states something a priori stronger than the finiteness of
the number of orbits of the action of automorphism group on the set of faces
of the cone, namely the existence of a finite polyhedral fundamental domain.

Conjecture 5.5: (Morrison-Kawamata cone conjecture for the ample cone)
The automorphism group $\operatorname{Aut}(M)$ has a finite polyhedral
fundamental domain on the ample cone.

We shall see in this subsection that this in fact follows from our version of
the cone conjecture, and therefore is true for all simple hyperkähler
manifolds with $b_{2}\neq 5$.

The ample cone $\operatorname{Amp}(M)$ is the convex hull of
$\operatorname{Kah}(M)\cap H^{1,1}_{{\mathbb{Q}}}(X)$ in the space
$H^{1,1}_{{\mathbb{Q}}}(X)\otimes{\mathbb{R}}=\operatorname{\sf
NS}(X)\otimes{\mathbb{R}}$, so that a priori new faces could arise from the
“circular part” of the boundary of the Kähler cone. In our case, this is not a
problem, since this part is a piece of the quadric over the rationals defining
$\operatorname{Pos}(M)$, and when it has a single rational point, it has a
dense set of them. Thus the Hodge monodromy group acts with finitely many
orbits on the set of faces of the ample cone.

Denote by ${\cal C}(M)$ the intersection of $\operatorname{Pos}(M)$ with
$\operatorname{\sf NS}(X)\otimes{\mathbb{R}}$. The Hodge monodromy group
$\Gamma$ acts on ${\mathbb{P}}{\cal C}(M)$ with finite stabilizers (since the
stabilizer of a point $x$ in ${\mathbb{P}}{\cal C}(M)$ must also stabilize the
orthogonal hyperplane to the line corresponding to $x$, and our form is
negative definite on such a hyperplane). By its arithmeticity, replacing if
necessary the group $\Gamma$ by a finite index subgroup, we may assume there
are no stabilizers at all. Indeed, an arithmetic lattice has a finite index
torsion-free subgroup, which can be obtained by taking a congruence subgroup
formed by integer matrices which are identity modulo $N$ for $N$ big enough.
Consider the quotient $S:={\cal C}(M)/\Gamma$. Since $\Gamma$ is arithmetic,
Borel and Harish-Chandra theorem implies that $S$ is a complete hyperbolic
manifold of finite volume. The image of $\operatorname{Amp}(M)$ in $S$ is a
hyperbolic manifold $T$ with finite (that is, consisting of finitely many
geodesic pieces) boundary, by 5.1. It is known (see [Bow, Proposition 4.7 and
5.6] or [K, Theorem 2.6]) that such manifolds are geometrically finite, that
is, they admit a finite cell decomposition with finite piecewise geodesic
boundary (in fact one even has a decomposition with a single cell of maximal
dimension, the Dirichlet-Voronoi decomposition). Thus $T$ becomes a union of
finitely many cells with finite piecewise geodesic boundary. Taking the union
of suitable liftings of these to $\operatorname{Amp}(M)$, we obtain a finite
polyhedron within the closure of $\operatorname{Amp}(M)$ which is a
fundamental domain for the subgroup of $\Gamma$ preserving
$\operatorname{Amp}(M)$, that is, of the automorphism group of $M$. We thus
have proved the following

Theorem 5.6: Let $M$ be a projective simple hyperkähler manifold with
$b_{2}\neq 5$. The automorphism group has a finite polyhedral fundamental
domain on the ample cone of $M$.

Acknowledgements: We are grateful to Eyal Markman for many interesting
discussions and an inspiration. Many thanks to Alex Eskin, Maxim Kontsevich,
Anton Zorich, Vladimir Fock and Sébastien Gouezel for elucidating the various
points of measure theory and hyperbolic geometry. The scheme of the proof of
4.3 is due to Alex Eskin; we are much indebted to Alex for his kindness and
patience in explaining it. Much gratitude to Mihai Paun and Sasha Anan’in for
inspiring discussions. Thanks to Martin Moeller for indicating the reference
[EMS], and to Andrey Konovalov for solving a tricky problem of hyperbolic
plane geometry which disproved one of our conjectures.

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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
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Kavli IPMU (WPI), the University of Tokyo