<EMAIL_ADDRESS>Vavilova Str., Moscow, Russia, 117312

# Teichmüller spaces, ergodic theory
and global Torelli theorem

Misha Verbitsky

###### Abstract

A Teichmüller space $\operatorname{Teich}$ is a quotient of the space of all
complex structures on a given manifold $M$ by the connected components of the
group of diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group
of connected components of the diffeomorphism group. The moduli problems can
be understood as statements about the $\Gamma$-action on
$\operatorname{Teich}$. I will describe the mapping class group and the
Teichmüller space for a hyperkähler manifold. It turns out that this action is
ergodic. We use the ergodicity to show that a hyperkähler manifold is never
Kobayashi hyperbolic.

###### :

P

###### keywords:

Torelli theorem, hyperkähler manifold, moduli space, mapping class group,
Teichmüller space, ergodicity

rimary 32G13, Secondary 53C26.

###### Contents

  1. 1 Teichmüller spaces
    1. 1.1 Teichmüller spaces and period maps
    2. 1.2 Marked moduli spaces
  2. 2 Torelli theorem
    1. 2.1 Torelli theorem: an introduction
    2. 2.2 Birational Teichmüller space
  3. 3 Hyperkähler manifolds and Bogomolov-Beauville-Fujiki form
    1. 3.1 Hyperkähler manifolds: definition and examples
    2. 3.2 Bogomolov-Beauville-Fujiki form and the mapping class group
  4. 4 Global Torelli theorem
    1. 4.1 Period map
    2. 4.2 Moduli of hyperkähler structures and twistor curves
  5. 5 Teichmüller spaces and ergodic theory
    1. 5.1 Ergodic complex structures
    2. 5.2 Ergodicity of the monodromy group action
  6. 6 Applications of ergodicity
    1. 6.1 Ergodic complex structures, Gromov-Hausdorff closures, and semicontinuity
    2. 6.2 Kobayashi non-hyperbolicity of hyperkähler manifolds
    3. 6.3 Symplectic packing and ergodicity

This talk is based on two papers, [V:2013] and [V]. In these papers one can
find details, examples, and rigorous proofs omitted here.

##  1 Teichmüller spaces

###  1.1 Teichmüller spaces and period maps

The notion of Teichmüller spaces has a long history since its discovery by
Teichmüller in 1944 ([Te:1944]) and further development by Ahlfors, Bers and
others. However, it is rarely applied to complex manifolds of dimension $>1$.
It turns out that this notion is interesting and useful for many purposes of
complex geometry in any dimension.

######  Definition 1.1.

Let $M$ be a smooth manifold. An almost complex structure is an operator
$I:\;TM{\>\longrightarrow\>}TM$ which satisfies
$I^{2}=-\operatorname{Id}_{TM}$. An almost complex structure is integrable if
$\forall X,Y\in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$. In this case $I$ is
called a complex structure operator. A manifold with an integrable almost
complex structure is called a complex manifold.

######  Definition 1.2.

The space of almost complex structures is an infinite-dimensional Fréchet
manifold $X_{M}$ of all tensors $I\in\operatorname{End}(TM)$ satisfying
$I^{2}=-\operatorname{Id}_{TM}$. Similarly, one considers the group of
diffeomorphisms as a Fréchet Lie group.

######  Remark 1.3.

Definition of Fréchet manifolds and Fréchet spaces and many results on the
geometry of infinite-dimensional manifolds can be found in [Ha].

######  Definition 1.4.

Let $M$ be a compact complex manifold, and $\operatorname{Diff}_{0}(M)$ a
connected component of its diffeomorphism group (the group of isotopies).
Denote by $\operatorname{Comp}$ the space of complex structures on $M$,
considered with the topology induced from a Fréchet manifold of almost complex
structures, and let
$\operatorname{Teich}:=\operatorname{Comp}/\operatorname{Diff}_{0}(M)$ be the
quotient space with the quotient topology. We call it the Teichmüller space.

######  Remark 1.5.

When the complex manifold $M$ admits a certain geometric structure, such as
Kähler, or hyperkähler structure, it is natural to consider the Teichmüller
space of complex structure compatible with (say) Kähler structure. Consider
the open subset $\operatorname{Comp}_{K}\subset\operatorname{Comp}$ of all
complex structures $I$ such that $(M,I)$ admits a Kähler structure. The
corresponding Teichmüller space is
$\operatorname{Teich}_{K}:=\operatorname{Comp}_{K}/\operatorname{Diff}_{0}$.
When working with the Teichmüller space of hyperkähler manifolds, or a torus,
we shall always restrict ourselves to $\operatorname{Comp}_{K}$ and
$\operatorname{Teich}_{K}$.

Results of Kuranishi about local structure of deformation spaces can be
summarized as a statement about local structure of $\operatorname{Comp}$ as
follows ([Ku1, Ku2, Dou]).

###### Theorem 1.6

Let $M$ be a compact complex manifold and $I\in\operatorname{Comp}$. Then
there exists an open neighbourhood $U\ni I$ in $\operatorname{Comp}$ and a
neighbourhood $R$ of unit in $\operatorname{Diff}_{0}$ satisfying the
following. Consider the quotient $U/R$ of $U$ by an equivalence relation
generated by $x\sim wy$, for all $x,y\in U$ and $w\in R$. Then $U/R$ is a
complex variety, equipped with a natural holomorphic embedding to $H^{1}(TM)$.

######  Remark 1.7.

The quotient space $U/R$ obtained by Kuranishi is called the Kuranishi space.
Let $\operatorname{Teich}(U)$ be an image of $U$ in the Teichmüller space.
Clearly, the Kuranishi space admits a surjective, continuous map to
$\operatorname{Teich}(U)$. It is not entirely clear whether this map is always
a homeomorphism. However, if it is always a homeomorphism, for a given $M$,
the space $\operatorname{Teich}$ acquires a structure of a complex variety.

As shown by F. Catanese ([C, Proposition 15]), for Kähler manifolds with
trivial canonical bundle, e.g. for the hyperkähler manifolds, the Teichmüller
space is locally isomorphic to the Kuranishi moduli space, hence it is a
complex variety. In this case it is actually a complex manifold, by Bogomolov-
Tian-Todorov theorem ([B:1981, Ti, Tod2]).

It is not clear if this is true for a general complex manifold; in the present
work we deal with hyperkähler manifolds, which are Calabi-Yau.

######  Question 1.8.

Consider a compact complex manifold $M$, and let $\operatorname{Teich}$ be its
Teichmüller space. Can we equip $\operatorname{Teich}$ with a structure of a
complex variety (possibly non-Hausdorff), in a way which is compatible with
the local charts obtained from the Kuranishi theorem?

When $M$ is a torus, or a hyperkähler manifold, $\operatorname{Teich}$ is a
complex manifold which can be described explicitly (Theorem 2.1, Theorem 4.5).
However, even for a hyperkähler manifold, $\operatorname{Teich}$ is not
Hausdorff.

###### Claim 1.9

Assume that $M$ is Kähler, and $\operatorname{Teich}$ the Teichmüller space of
all complex structures of Kähler type on $M$. For a given
$I\in\operatorname{Teich}$, choose a representative
$\tilde{I}\in\operatorname{Comp}$. Then the Hodge decomposition
$H^{*}(M)=\bigoplus H^{p,q}(M,\tilde{I})$ is independent from the choice of
$\tilde{I}$.

Proof: The ambiguity of a choice of $\tilde{I}$ lies in
$\operatorname{Diff}_{0}$. However, $\operatorname{Diff}_{0}$ acts trivially
on $H^{*}(M)$.

This elementary claim allows one to define the period map.

######  Definition 1.10.

Lety $M$ be a Kähler manifold, $\operatorname{Teich}$ its Teichmüller space,
and $\operatorname{\sf Per}$ the map associating to $I$ the Hodge
decomposition $H^{*}(M)=\bigoplus H^{p,q}(M,I)$. Then $\operatorname{\sf Per}$
is called the period map of $M$.

######  Remark 1.11.

Consider the product $\operatorname{Comp}\times M$ trivially fibered over
$\operatorname{Comp}$. The fibers of
$\pi:\;M\times\operatorname{Comp}{\>\longrightarrow\>}\operatorname{Comp}$ can
be considered as complex manifolds, with complex structure at
$I\in\operatorname{Comp}$ given by $I$. This complex structure is clearly
$\operatorname{Diff}_{0}$-invariant, giving a complex structure on the fibers
of the quotient fibration
$(M\times\operatorname{Comp})/\operatorname{Diff}_{0}{\>\longrightarrow\>}\operatorname{Teich}$.
At each $I\in\operatorname{Teich}$, the fiber of this fibration (called the
universal fibration) is isomorphic to $(M,I)$.

###  1.2 Marked moduli spaces

A more conventional approach to the moduli problem goes as follows. Given a
complex manifold $M$, one defines the deformation functor from marked complex
spaces to sets as a functor mapping a complex space $(B,x)$ to the set of
equivalence classes of deformations $\pi:\;{\cal X}{\>\longrightarrow\>}B$ of
$M$ over $B$ with $M$ identified with the fiber of $\pi$ at $x$.

If the deformation functor is representable by a complex space, this space is
called the fine moduli space of deformations of $M$.

Usually, the fine moduli space does not exist. In this case, one considers the
category of natural transformations from the deformation functor to
representable functors. The initial object in this category is called the
coarse moduli space. The points of coarse moduli are identified with
equivalence classes of deformations of $M$.

In this setup, an analogue of Teichmüller space can be defined as follows. Fix
an abelian group which is isomorphic to $H^{*}(M,{\mathbb{Z}})$, and define a
marked manifold as a pair
$(M,\varphi:\;V\tilde{\>\longrightarrow\>}H^{*}(M,{\mathbb{Z}}))$, where $M$
is a complex manifold, and $\varphi$ a group isomorphism. In the same way as
above, one defines a coarse moduli space of deformations of marked manifolds.
To compare this space with Teichmüller space, consider a subgroup group
$\Gamma_{0}$ of mapping class group which acts trivially on cohomology.
Clearly, the points of $\operatorname{Teich}/\Gamma_{0}$ are in bijective
correspondence with the equivalence classes of marked complex structures on
$M$.

Given a coarse marked moduli space $W$, one obtains the tautological map
$W{\>\longrightarrow\>}\operatorname{Teich}/\Gamma_{0}$, by construction
continuous. For hyperkähler manifolds (or compact tori), this map is a
diffeomorphism on each connected component ([V:2013, Corollary 4.31]).

##  2 Torelli theorem

###  2.1 Torelli theorem: an introduction

Torelli theorems are a broad class of results which describe the Teichmüller
spaces in terms of the period maps (Definition 1.10).

The name originates with Ruggiero Torelli, who has shown that it is possible
to reconstruct a Riemann surface from its Jacobian ([To:1913]). The term
“Torelli theorems” is due to André Weil ([W:1957]), who gave a modern proof of
this classical result, and explained its possible generalizations.

One may distinguish between the “local Torelli theorem”, where a local
structure of deformation space is described in terms of periods, and “global
Torelli”, where the Teichmüller space is described globally.

Weil, who was the first to define and study K3 surfaces, spent much time
trying to prove the Torelli theorem for K3 surfaces, but it was notoriously
difficult. Its local version is due to Tjurina, Piatetski-Shapiro and
Shafarevich ([Tj, PS]). The local Torelli was generalized by Bogomolov to
hyperkähler manifolds ([B:1978]) and by Bogomolov-Tian-Todorov to Calabi-Yau
manifolds ([B:1981, Ti, Tod2]), building foundation for the theory of Mirror
Symmetry.

In dimension $>1$, the global Torelli theorem was known only for compact tori
(where it is essentially trivial) and the K3 surfaces, where it was proven by
Kulikov in 1977 ([Kul]), and then improved many times during the 1980-ies
([Tod1, Bea2, L, Si, Fr]).

###  2.2 Birational Teichmüller space

In what follows, a hyperkähler manifold is a compact complex manifold
admitting a Kähler structure and a holomorphic symplectic form.

In generalizing global Torelli to more general hyperkähler manifolds, two
problems were apparent. First of all, bimeromorphic hyperkähler manifolds have
the same periods, hence the period map cannot distinguish between them.
However, for $\dim_{\mathbb{C}}M>0$, birational holomorphically symplectic
manifolds can be non-isomorphic ([De]).

Another (mostly psychological) difficulty is based on attachment to moduli
spaces, as opposed to marked moduli or Teichmüller spaces. For a K3, one can
reconstruct a K3 from its Hodge structure, and this gives an identification
between the moduli and the space of Hodge structures. In bigger dimension, one
has to use the Teichmüller space. Indeed, for some classes of hyperkähler
manifolds, the group $O(H^{2}(M,{\mathbb{Z}})$ of Hodge isometries of
cohomology is strictly bigger than the image of the mapping class group. This
gives elements $\gamma\in O(H^{2}(M,{\mathbb{Z}})$ acting non-trivially on the
Teichmüller space in such a way that the complex manifolds $(M,I)$ and
$(M,\gamma(I))$ are not birationally equivalent ([Na, Ma2]). However, their
Hodge structures are equivalent, by construction. This example explains the
necessity of using the Teichmüller spaces (or marked moduli) to state the
Torelli theorem: its Hodge-theoretic version is often false.

For Teichmüller spaces, the Torelli theorem is a statement about the period
map (Definition 1.10). Ideally, we want the period map to give a
diffeomporphism between $\operatorname{Teich}$ and the corresponding space of
Hodve structures. This is what happens for a compact torus.

###### Theorem 2.1

Let $M$ be a compact torus, $\dim_{\mathbb{R}}M=2n$, and
$\operatorname{Teich}$ the Teichmüller space of all complex structures of
Kähler type on $M$. Denote by $\operatorname{{\mathbb{P}}\sf er}$ the space
$SL(2n,{\mathbb{R}})/SL(n,{\mathbb{C}})$ of all Hodge structures of weight one
on $H^{1}(M,{\mathbb{C}})$, that is, the space of all complex operators on
$H^{1}(M,{\mathbb{R}})$ compatible with the orientation. Then the period map
$\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is a diffeomorphism on each connected component of
$\operatorname{{\mathbb{P}}\sf er}$.

Unfortunately, this ideal situation is almost never realized. Even in the
simplest cases (such as for hyperkähler manifolds), the Teichmüller space is
no longer Hausdorff. However, in some situations it is still possible to deal
with non-Hausdorff points.

######  Remark 2.2.

A non-Hausdorff manifold is a topological space locally diffeomorphic to
${\mathbb{R}}^{n}$ (but not necessarily Hausdorff).

######  Definition 2.3.

Let $X$ be a topological space, and
$X\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}X_{0}$ a continuous
surjection. The space $X_{0}$ is called a Hausdorff reduction of $X$ if any
continuous map $X{\>\longrightarrow\>}X^{\prime}$ to a Hausdorff space is
factorized through $\varphi$.

######  Definition 2.4.

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 $V\ni x,U\ni y$, one has $U\cap
V\neq\emptyset$.

######  Remark 2.5.

Suppose that $\sim$ is an equivalence relation, and the quotient $M/\sim$ is
Hausdorff. Then $M/\sim$ is a Hausdorff reduction of $M$.

Unfortunately, this notion cannot be applied universally. Firstly, $\sim$ is
not always an equivalence relation; and secondly, even if $\sim$ is
equivalence, the $M/\sim$ is not always Hausdorff. Fortunately, for
Teichmüller space of a hyperkähler manifold, Hausdorff reduction can be
defined, using the following theorem due to D. Huybrechts ([Hu1]).

###### Theorem 2.6

If $I_{1}$, $I_{2}\in\operatorname{Teich}$ are non-separate points, then
$(M,I_{1})$ is birationally equivalent to $(M,I_{2})$.

Using this result and geometry of the period map (Bogomolov’s local Torelli
theorem), it is elementary to show that the quotient
$\operatorname{Teich}_{b}:=\operatorname{Teich}/\sim$ is a Hausdorff manifold.
This quotient is called the birational Teichmüller space of a hyperkähler
manifold.

Global Torelli theorem implies that for hyperkähler manifolds the period map
induces a diffeomorphism between the Hausdorff reduction of the Teichmüller
space and the appropriate period domain.

##  3 Hyperkähler manifolds and Bogomolov-Beauville-Fujiki form

###  3.1 Hyperkähler manifolds: definition and examples

The standard definition of hyperkähler manifolds is rather differential
geometric. It is, indeed, synonymous with “holomorphic symplectic”, but this
synonymity follows from Calabi-Yau theorem. For more details about hyperkähler
manifolds, please see [Bea1] or [Bes].

######  Definition 3.1.

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 3.2.

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

######  Remark 3.3.

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

######  Definition 3.4.

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 3.5.

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 3.6.

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

######  Remark 3.7.

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

######  Theorem 3.8 ((Calabi-Yau))

A compact, Kähler, holomorphically symplectic manifold admits a unique
hyperkähler metric in any Kähler class.

######  Remark 3.9.

For the rest of this talk, a hyperkähler manifold means a compact complex
manifold admitting a Kähler structure and a holomorphically symplectic
structure.

######  Definition 3.10.

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

The rationale for this terminology comes from Bogomolov’s decomposition
theorem.

######  Theorem 3.11 ((Bogomolov, [B:1974]))

Any hyperkähler manifold admits a finite covering which is a product of a
torus and several simple hyperkähler manifolds.

Further on, all hyperkähler manifolds are assumed to be simple.

######  Remark 3.12.

A hyperkähler manifold is simple if and only if its holonomy group is $Sp(n)$,
and not a proper subgroup of $Sp(n)$ ([Bes]).

######  Example 3.13.

Take a 2-dimensional complex torus $T$, then the singular locus of $T/{\pm 1}$
is 16 points locally of form ${\mathbb{C}}^{2}/{\pm 1}$. Its resolution by
blow-up is called a Kummer surface. It is not hard to see that it is
holomorphically symplectic.

######  Definition 3.14.

A K3 surface is a hyperkähler manifold which is diffeomorphic to a Kummer
surface.

In real dimension 4, the only compact hyperkähler manifolds are tori and K3
surfaces, as follows from the Kodaira-Enriques classification.

######  Definition 3.15.

A Hilbert scheme $M^{[n]}$ of a complex surface $M$ is a classifying space of
all ideal sheaves $I\subset{\cal O}_{M}$ for which the quotient ${\cal
O}_{M}/I$ has dimension $n$ over ${\mathbb{C}}$.

######  Remark 3.16.

A Hilbert scheme is obtained as a resolution of singularities of the symmetric
power $\operatorname{Sym}^{n}M$.

######  Theorem 3.17 ((Fujiki, Beauville))

A Hilbert scheme of a hyperkähler manifold of real dimension 2 is hyperkähler.

######  Example 3.18.

Let $T$ be a torus. Then $T$ acts on its Hilbert scheme freely and properly by
translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For
$n>2$, a universal covering of $T^{[n]}/T$ is called a generalized Kummer
variety.

######  Remark 3.19.

There are 2 more “sporadic” examples of compact hyperkähler manifolds,
constructed by K. O’Grady ([O]). All known simple hyperkaehler manifolds are
these 2 and the two series: Hilbert schemes of K3 and generalized Kummer.

###  3.2 Bogomolov-Beauville-Fujiki form and the mapping class group

###### Theorem 3.20

(Fujiki, [Fu])
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 positive rational number,
called Fujiki constant.

######  Definition 3.21.

This form is called Bogomolov-Beauville-Fujiki form. It is defined by the
Fujiki’s relation uniquely, up to a sign. The sign is determined from the
following formula (Bogomolov, Beauville)

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

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

######  Remark 3.22.

The BBF form $q$ has signature $(b_{2}-3,3)$. It is negative definite on
primitive forms, and positive definite on
$\langle\Omega,\bar{\Omega},\omega\rangle$, where $\omega$ is a Kähler form.

Using the BBF form, it is possible to describe the automorphism group of
cohomology in a very convenient way.

###### Theorem 3.23

Let $M$ be a simple hyperkähler manifold, and $G\subset GL(H^{*}(M))$ a group
of automorphisms of its cohomology algebra preserving the Pontryagin classes.
Then $G$ acts on $H^{2}(M)$ preserving the BBF form. Moreover, the map
$G{\>\longrightarrow\>}O(H^{2}(M,{\mathbb{R}}),q)$ is surjective on a
connected component, and has compact kernel.

Proof. Step 1: Fujiki formula $v^{2n}=q(v,v)^{n}$ implies that $\Gamma_{0}$
preserves the Bogomolov-Beauville-Fujiki up to a sign. The sign is fixed, if
$n$ is odd.

Step 2: For even $n$, the sign is also fixed. Indeed, $G$ preserves
$p_{1}(M)$, and (as Fujiki has shown in [Fu]), $v^{2n-2}\wedge
p_{1}(M)=q(v,v)^{n-1}c$, for some $c\in{\mathbb{R}}$. The constant $c$ is
positive, because the degree of $c_{2}(B)$ is positive for any non-trivial
Yang-Mills bundle with $c_{1}(B)=0$.

Step 3: ${\mathfrak{o}}(H^{2}(M,{\mathbb{R}}),q)$ acts on
$H^{*}(M,{\mathbb{R}})$ by derivations preserving Pontryagin classes
([V:1996]). Therefore $\operatorname{Lie}(G)$ surjects to
${\mathfrak{o}}(H^{2}(M,{\mathbb{R}}),q)$.

Step 4: The kernel $K$ of the map
$G{\>\longrightarrow\>}G{\left|{}_{{H^{2}(M,{\mathbb{R}})}}\right.}$ is
compact, because it commutes with the Hodge decomposition and Lefschetz
${\mathfrak{s}l}(2)$-action, hence preserves the Riemann-Hodge form, which is
positive definite.

Using this result, the mapping class group can also be computed. We use a
theorem of D. Sullivan, who expressed the mapping group in terms of the
rational homotopy theory, and expressed the rational homotopy in terms of the
algebraic structure of the de Rham algebra.

###### Theorem 3.24

(Sullivan, [Su, Theorem 10.3, Theorem 12.1, Theorem 13.3])
Let $M$ be a compact, simply connected Kähler manifold,
$\dim_{\mathbb{C}}M\geqslant 3$. Denote by $\Gamma_{0}$ the group of
automorphisms of an algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin
classes $p_{i}(M)$. Then the natural map
$\operatorname{Diff}(M)/\operatorname{Diff}_{0}{\>\longrightarrow\>}\Gamma_{0}$
has finite kernel, and its image has finite index in $\Gamma_{0}$.

As a corollary of this theorem, we obtain a similar result about hyperkähler
manifolds.

###### Theorem 3.25

Let $M$ be a simple hyperkähler manifold, and $\Gamma_{0}$ the group of
automorphisms of an algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin
classes $p_{i}(M)$. Then

(i)

$\Gamma_{0}{\left|{}_{{H^{2}(M,{\mathbb{Z}})}}\right.}$ is a finite index
subgroup of $O(H^{2}(M,{\mathbb{Z}}),q)$.

(ii)

The map $\Gamma_{0}{\>\longrightarrow\>}O(H^{2}(M,{\mathbb{Z}}),q)$ has finite
kernel.

We obtained that the mapping group is arithmetic (commensurable to a subgroup
of integer points in a rational Lie group).

As follows from [Hu2, Theorem 2.1], there are only finitely many connected
components of $\operatorname{Teich}$. Let $\Gamma^{I}$ be the group of
elements of mapping class group preserving a connected component of
Teichmüller space containing $I\in\operatorname{Teich}$. Then $\Gamma^{I}$ is
also arithmetic. Indeed, it has finite index in $\Gamma$.

######  Definition 3.26.

The image of $\Gamma^{I}$ in $GL(H^{2}(M,{\mathbb{Z}}))$ is called monodromy
group of a manifold.

######  Remark 3.27.

The monodromy group can also be obtained as a group generated by monodromy of
all Gauss-Manin local system for all deformations of $M$ ([V:2013, Theorem
7.2]). This explains the term. This notion was defined and computed in many
special cases by E. Markman ([Ma1], [Ma2]).

##  4 Global Torelli theorem

###  4.1 Period map

To study the moduli problem, one should understand the mapping class group
(described above) and the Teichmüller space. It turns out that the birational
Teichmüller space has a very simple description in terms of the period map,
inducing a diffeomorphism
$\operatorname{Teich}_{b}{\>\longrightarrow\>}\frac{SO(b_{2}-3,3)}{SO(b_{2}-3,1)\times
SO(2)}$ on each connected component of $\operatorname{Teich}_{b}$.

######  Definition 4.1.

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

######  Remark 4.2.

$\operatorname{\sf Per}$ maps $\operatorname{Teich}$ into an open subset of a
quadric, defined by

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

The manifold $\operatorname{{\mathbb{P}}\sf er}$ is called the period space of
$M$.

As follows from Proposition 4.8 below, $\operatorname{{\mathbb{P}}\sf
er}=\frac{SO(b_{2}-3,3)}{SO(b_{2}-3,1)\times SO(2)}$.

###### Theorem 4.3

(Bogomolov, [B:1978])
Let $M$ be a simple hyperkähler manifold, and $\operatorname{Teich}$ its
Teichmüller space. Then the period map $\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{\sf Per}$ is
etale (has invertible differential everywhere).

######  Remark 4.4.

Bogomolov’s theorem implies that $\operatorname{Teich}$ is smooth. It is non-
Hausdorff even in the simplest examples.

Now the global Torelli theorem can be stated as follows. Recall that the
birational Teichmüller space $\operatorname{Teich}_{b}$ is a Hausdorff
reduction of the Teichmüller space of the holomorphic symplectic manifolds of
Kähler type.

###### Theorem 4.5

Let $M$ be a simple hyperkähler manifold, and $\operatorname{\sf
Per}:\;\operatorname{Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ the period map. Then $\operatorname{\sf Per}$ is a diffeomorphism on each
connected component.

The following proposition is proven in a straghtforward manner using 1950-ies
style arguments of geometric topology.

######  Proposition 4.6 ((The Covering Criterion))

Let $X\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}Y$ be an etale
map of smooth manifolds. Suppose that each $y\in Y$ has a neighbourhood $B\ni
y$ diffeomorphic to a closed ball, such that for each connected component
$B^{\prime}\subset\varphi^{-1}(B)$, $B^{\prime}$ projects to $B$ surjectively.
Then $\varphi$ is a covering.

Now, the Global Torelli implied by the following result, which is proven in
Subsection 4.2 using hyperkähler structures.

###### Proposition 4.7

In assumptions of Theorem 4.5, the period map satisfies the conditions of the
Covering Criterion.

###  4.2 Moduli of hyperkähler structures and twistor curves

###### Proposition 4.8

The period space

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

is identified with $\frac{SO(b_{2}-3,3)}{SO(2)\times SO(b_{2}-3,1)}$, which is
a Grassmannian of positive oriented 2-planes in $H^{2}(M,{\mathbb{R}})$.

Proof. Step 1: Given $l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$, the space
generated by $\operatorname{Im}l,\operatorname{Re}l$ is 2-dimensional, because
$q(l,l)=0,q(l,\bar{l})$ implies that $l\cap H^{2}(M,{\mathbb{R}})=0$.

Step 2: This 2-dimensional plane is positive, because
$q(\operatorname{Re}l,\operatorname{Re}l)=q(l+\bar{l},l+\bar{l})=2q(l,\bar{l})>0$.

Step 3: Conversely, for any 2-dimensional positive plane $V\in
H^{2}(M,{\mathbb{R}})$, the quadric $\\{l\in
V\otimes_{\mathbb{R}}{\mathbb{C}}\ \ |\ \ q(l,l)=0\\}$ consists of two lines;
a choice of a line is determined by orientation.

######  Remark 4.9.

Two hyperkähler structures $(M,I,J,K,g)$ and
$(M,I^{\prime},J^{\prime},K^{\prime},g)$ are called equivalent if there exists
a unitary quaternion $h$ such that $I^{\prime}=hIh^{-}1$,
$J^{\prime}=hJh^{-}1$, $K^{\prime}=hKh^{-}1$. From the holonomy
characterization of simple hyperkähler manifolds (Remark 3.12) it follows that
two hyperkähler structures are isometric if and only if they are equivalent.

######  Definition 4.10.

Let $(M,I,J,K,g)$ be a hyperkähler manifold. A hyperkähler 3-plane in
$H^{2}(M,{\mathbb{R}})$ is a positive oriented 3-dimensional subspace $W$,
generated by three Kähler forms $\omega_{I},\omega_{J},\omega_{K}$.

######  Definition 4.11.

Similarly to the Teichmüller space and period map of complex structures, one
can define the period space of hyperkähler metrics. Denote it by
$\operatorname{Teich}_{H}$. The corresponding period map is

$\operatorname{\sf
Per}:\;\operatorname{Teich}_{H}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}_{H},$

where $\operatorname{{\mathbb{P}}\sf er}_{H}=\frac{SO(b_{2}-3,3)}{SO(3)\times
SO(b_{2}-3)}$ is the space of positive, oriented 3-planes, and
$\operatorname{\sf Per}$ maps a hyperkähler structure to the corresponding
hyperkähler 3-plane.

######  Remark 4.12.

There is one significant difference between $\operatorname{Teich}$ and the
hyperkähler Teichmüller space $\operatorname{Teich}_{H}$: the latter is
Hausdorff, and, in fact, metrizable. Indeed, we could equip the space
$\operatorname{Teich}_{H}$ of hyperkähler metrics with the Gromov-Hausdorff
metric.

Let $I\in\operatorname{Teich}$ be a complex structure, and ${\cal K}(I)$ its
Kähler cone. The set of hyperkähler metrics compatible with $I$ is
parametrized by ${\cal K}(I)$, by Calabi-Yau theorem. The corresponding
3-dimensional subspaces are generated by $\operatorname{\sf Per}(I)+\omega$,
where $\omega\in{\cal K}(I)$. The local Torelli theorem implies that locally
$I\in\operatorname{Teich}$ is uniquely determined by the 2-plane generated by
$\omega_{J}$ and $\omega_{K}$; Calabi-Yau theorem implies that the hyperkähler
metric is uniquely determined by the complex structure and the Kähler
structure. This gives the following hyperkähler version of the local Torelli
theorem.

###### Theorem 4.13

Let $M$ be a simple hyperkähler manifold, and $\operatorname{Teich}_{H}$ its
hyperkähler Teichmüller space. Then the period map $\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{\sf Per}_{H}$
mapping an equivalence class of hyperkähler structures to is its 3-plane is
etale (has invertible differential everywhere).

######  Remark 4.14.

Let $W\subset H^{2}(M,{\mathbb{R}})$ be a positive 3-dimensional plane. The
set $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ of oriented 2-dimensional
planes in $W$ is identified with $S^{2}={\mathbb{C}}P^{1}$. When $W$ is a
hyperkähler 3-plane, $S_{W}$ is called the twistor family of a hyperkähler
structure. A point in the twistor family corresponds to a complex structure
$aI+bJ+cK\in{\mathbb{H}}$, with $a^{2}+b^{2}+c^{2}=1$. We call the
corresponding rational curves ${\mathbb{C}}P^{1}\subset\operatorname{Teich}$
the twistor lines. It is not hard to see that the twistor lines are
holomorphic.

######  Definition 4.15.

Let $W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ be a positive 3-plane,
$S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ the corresponding rational
curve, and $x\in S_{W}$ be a point. It is called liftable if for any point
$y\in\operatorname{\sf Per}^{-1}(x)\subset\operatorname{Teich}$ there exists
${\cal H}\in\operatorname{Teich}_{H}$ such that the corresponding twistor line
contains $y$.

When $W$ is generic, the corresponding line $S_{W}$ is liftable, as indicated
below.

######  Definition 4.16.

The Neron-Severi lattice $NS(I)$ of a hyperkähler manifold $(M,I)$ is
$H^{1,1}(M,I)\cap H^{2}(M,{\mathbb{Z}})$.

The following theorem, based on results of [DP], was proven by D. Huybrechts.

######  Theorem 4.17 (([Hu1]))

Let $M$ be a hyperkaehler manifold with $\operatorname{\sf NS}(M)=0$. Then its
Kaehler cone is one of two components of the set

$\\{\nu\in H^{1,1}(M,{\mathbb{R}})\ |\ q(\nu,\nu)\geqslant 0\\}.$

######  Definition 4.18.

Let $S\subset\operatorname{Teich}$ be a ${\mathbb{C}}P^{1}$ associated with a
twistor family. It is called generic if it passes through a point
$I\in\operatorname{Teich}$ with $\operatorname{\sf NS}(M,I)=0$. Clearly, a
hyperkähler 3-plane $W\subset H^{2}(M,{\mathbb{R}})$ corresponds to a generic
twistor family if and only if its orthogonal complement $W^{\bot}\subset
H^{2}(M,{\mathbb{R}})$ does not contain rational vectors. A 3-plane
$W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ is called generic if
$W^{\bot}\subset H^{2}(M,{\mathbb{R}})$ does not contain rational vectors. The
corresponding rational curve $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$
is called a GHK line. GHK lines are liftable, which is very useful for many
purposes, including the proof of Torelli theorem (see also [AV], where GHK
lines were used to study Kähler cones of hyperkähler manifolds).

The following theorem immediately follows from the Calabi-Yau theorem and the
description of the Kähler cone given in Theorem 4.17.

###### Theorem 4.19

Let $W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ be a generic plane,
$S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ the corresponding rational
curve, and $x\in S_{W}$ a generic point. Then $(S_{W},x)$ is liftable.

Assumptions of the covering criterion (Proposition 4.7) immediately follow
from Theorem 4.19. Indeed, it is not hard to see that any two points on a
closed ball $B\subset\operatorname{{\mathbb{P}}\sf er}$ can be connected
inside $B$ by a sequence of GHK curves intersecting in generic points of $B$.
Since these curves are liftable, any connected component of $\operatorname{\sf
Per}^{-1}(B)$ is mapped to $B$ surjectively.

##  5 Teichmüller spaces and ergodic theory

###  5.1 Ergodic complex structures

After the Teichmüller space and the mapping class group are understood, it is
natural to consider the quotient space
$\operatorname{Comp}/\operatorname{Diff}=\operatorname{Teich}/\Gamma$ of the
Teichmüller space by the mapping class group
$\Gamma:=\operatorname{Diff}/\operatorname{Diff}_{0}$.

###### Claim 5.1

Let $M$ be a simple hyperkähler manifold, $\Gamma$ its mapping class group,
and $\operatorname{Teich}_{b}$ the birational Teichmüller space. Then the
quotient $\operatorname{Teich}_{b}/\Gamma$ parametrizes the birational classes
of deformations of $M$.

One could call the quotient $\operatorname{Teich}_{b}/\Gamma$ “the moduli
space”, but, unfortunately, this is not a space in any reasonable sense.
Indeed, as we shall see, non-trivial closed subsets of
$\operatorname{Teich}_{b}/\Gamma$ are at most countable, making
$\operatorname{Teich}_{b}/\Gamma$ terribly non-Hausdorff. This means that the
concept of “moduli space” has no meaning, and all interesting information
about moduli problems is hidden in dynamics of $\Gamma$-action on
$\operatorname{Teich}$.

Let $I\in\operatorname{Teich}$ be a point, and
$\operatorname{Teich}^{I}\subset\operatorname{Teich}$ its connected component.
Since $\operatorname{Teich}$ has finitely many components, a subgroup mapping
class group fixing $\operatorname{Teich}$ has finite index. Its image in
$\operatorname{Aut}(\operatorname{Teich}^{I})$ is called monodromy group and
denoted $\Gamma^{I}$ (Definition 3.26). It is a finite index subgroup in
$SO(H^{2}(M,{\mathbb{Z}}))$.

All that said, we find that the moduli problem for hyperkähler manifold is
essentially reduced to the dynamics of the $\Gamma^{I}$-action on the space
$\operatorname{{\mathbb{P}}\sf er}$, which is understood as a Grassmannian of
positive, oriented 2-planes in $H^{2}(M,{\mathbb{R}})$.

It is natural to study the dynamics of a group action from the point of view
of ergodic theory, ignoring measure zero subsets. However, the quotient map
$\operatorname{Teich}{\>\longrightarrow\>}\operatorname{Teich}_{b}$ is
bijective outside of a union of countably many divisors, corresponding to
complex structures $I$ with $NS(M,I)$ non-zero. This set has measure 0.
Therefore, the quotient map
$\operatorname{Teich}{\>\longrightarrow\>}\operatorname{Teich}_{b}$ induces an
equivalence of measured spaces. For the purposes of ergodic theory, we shall
identify $\operatorname{Teich}^{I}$ with the corresponding homogeneous space
$\operatorname{{\mathbb{P}}\sf er}$.

Ths first observation, based on a theorem of C. Moore, implies that the
monodromy action on $\operatorname{{\mathbb{P}}\sf er}$ is ergodic.

######  Definition 5.2.

Let $(M,\mu)$ be a space with measure, and $G$ a group acting on $M$. 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 5.3

Let $M$ be a manifold, $\mu$ a Lebesgue measure, and $G$ a group acting on
$(M,\mu)$ ergodically. Then the set of 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 $U$, one has $x\in M\backslash M^{\prime}$.
Therefore, the set of such orbits has measure 0.

######  Definition 5.4.

Let $I\in\operatorname{Comp}$ be a complex structure on a manifold. It is
called ergodic if its $\operatorname{Diff}$-orbit is dense in its connected
component of $\operatorname{Comp}$.

######  Remark 5.5.

This is equivalent to density of $\Gamma$-orbit of $I$ in its Teichmüller
component.

###  5.2 Ergodicity of the monodromy group action

######  Definition 5.6.

Let $G$ be a Lie group, and $\Gamma\subset G$ a discrete subgroup. Consider
the pushforward of the Haar measure to $G/\Gamma$. 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 5.7.

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 e.g. [VGS]). In particular, all arithmetic subgroups of a
semi-simple group are lattices.

###### Theorem 5.8

(Calvin C. Moore, [Mo:1966, Theorem 7])
Let $\Gamma$ be an arithmetic subgroup in a non-compact simple Lie group $G$
with finite center, and $H\subset G$ a non-compact subgroup. Then the left
action of $\Gamma$ on $G/H$ is ergodic.

###### Theorem 5.9

Let ${\operatorname{Teich}}$ be a connected component of a Teichmüller space,
and $\Gamma^{I}$ its monodromy group. Then the set of all non-ergodic points
of $\operatorname{Teich}$ has measure 0.

Proof: Global Torelli theorem identifies $\operatorname{Teich}$ (as a measured
space) and $G/H$, where $G=SO(b_{2}-3,3)$, $H=SO(2)\times SO(b_{2}-3,1)$.
Since $\Gamma^{I}$ is an arithmetic lattice, $\Gamma^{I}$-action on $G/H$ is
ergodic, by Moore’s theorem.

Moore’s theorem implies that outside of a measure zero set, all complex
structures on $\operatorname{Teich}$ are ergodic. If we want to determine
which exactly complex structures are ergodic, we have to use Ratner’s theorem,
giving precise description of a closure of a $\Gamma^{I}$-orbit in a
homogeneous space.

Now I will state some basic results of Ratner theory. For more details, please
see [KSS] and [Mor].

######  Definition 5.10.

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
elements.

###### Theorem 5.11

([Mor, 1.1.15 (2)])
Let $H\subset G$ be a Lie subroup generated by unipotents, and $\Gamma\subset
G$ a lattice. Then a closure of any $H$-orbit in $G/\Gamma$ is an orbit of a
closed, connected subgroup $S\subset G$, such that $S\cap\Gamma\subset S$ is a
lattice.

When this lattice is arithmetic, one could describe the group $S$ very
explicitly.

###### Claim 5.12

( [KSS, Proposition 3.3.7] or [Sh, Proposition 3.2])
Let $x\in G/H$ be a point in a homogeneous space, and $\Gamma\cdot x$ its
$\Gamma$-orbit, where $\Gamma$ is an arithmetic lattice. Then its closure is
an orbit of a group $S$ containing stabilizer of $x$. Moreover, $S$ is a
smallest group defined over rationals and stabilizing $x$.

For the present purposes, we are interested in a pair $SO(3,k)\supset
SO(1,k)\times SO(2)\subset G$ (or, rather, their connected components
$G=SO^{+}(3,k)$ and $H=SO(1,k)\times SO(2)\subset G$). In this case, there are
no intermediate subgroups.

###### Claim 5.13

Let $G=SO^{+}(3,k)$, and $H\cong SO^{+}(1,k)\times SO(2)\subset G$. Then any
closed connected Lie subgroup $S\subset G$ containing $H$ coincides with $G$
or with $H$.

###### Corollary 5.14

Let $J\in\operatorname{{\mathbb{P}}\sf er}=G/H$. Then either $J$ is ergodic,
or its $\Gamma$-orbit is closed in $\operatorname{{\mathbb{P}}\sf er}$.

By Ratner’s theorem, in the latter case the $H$-orbit of $J$ has finite volume
in $G/\Gamma$. Therefore, its intersection with $\Gamma$ is a lattice in $H$.
This brings

###### Corollary 5.15

Let $J\in\operatorname{{\mathbb{P}}\sf er}$ be a point such that its
$\Gamma$-orbit is closed in $\operatorname{{\mathbb{P}}\sf er}$. Consider its
stabilizer $\operatorname{St}(J)\cong H\subset G$. Then
$\operatorname{St}(J)\cap\Gamma$ is a lattice in $\operatorname{St}(J)$.

###### Corollary 5.16

Let $J$ be a non-ergodic complex structure on a hyperkähler manifold, and
$W\subset H^{2}(M,{\mathbb{R}})$ be a plane generated by
$\operatorname{Re}\Omega,\operatorname{Im}\Omega$. Then $W$ is rational.
Equivalently, this means that $Pic(M)$ has maximal possible dimension.

Similar results are true for a torus of dimension $>1$; we refer the reader to
[V] for precise statements and details of the proof.

##  6 Applications of ergodicity

###  6.1 Ergodic complex structures, Gromov-Hausdorff closures, and
semicontinuity

The ergodicity theorem (Subsection 5.2) has some striking and even paradoxical
implications. For instance, consider a Kähler cone $\operatorname{Kah}$ of a
hyperkähler manifold (or a torus of dimension $>1$) equipped with an ergodic
complex structure. By Calabi-Yau theorem, each point of $\operatorname{Kah}$
corresponds to a Ricci-flat metric on $M$. If we restrict ourselves to those
metrics which satisfy $\operatorname{\sf diam}(M,g)\leqslant d$ (with bounded
diameter), then, by Gromov’s compactness theorem ([Gr]), the set $X_{d}$ of
such metrics is precompact in the Gromov’s space of all metric spaces,
equipped with the Gromov-Hausdorff metric. It is instructive to see what kind
of metric spaces occur on its boundary (that is, on $\bar{X}_{d}\backslash
X_{d}$). To see this, let $\nu_{i}$ be a sequence of diffeomorphisms
satisfying $\lim_{i}\nu_{i}(I)=I^{\prime}$. By Kodaira stability theorem, the
Kähler cone of $(M,I)$ is lower continuous on $I$. Therefore, there exists a
family of Kähler classes $\omega_{i}$ on $(M,\nu_{i}(I))$ which converge to a
given Kähler class $\omega^{\prime}$ on $(M,I^{\prime})$. This implies
convergence of the corresponding Ricci-flat metrics. We obtain that any Ricci-
flat metric on $(M,I^{\prime})$ (for any $I^{\prime}$ in the same deformation
class as $I$) is obtained as a limit of Ricci-flat metrics on $(M,I)$.

This gives the following truly bizzarre theorem.

###### Theorem 6.1

Let $(M,I)$ be an ergodic complex structure on a hyperkähler manifold,
$X\cong\operatorname{Kah}$ the set of all Ricci-flat Kähler metrics on
$(M,I)$, and $g^{\prime}$ another Ricci-flat metric on $M$ in the same
deformation class. Then $g^{\prime}$ lies in the closure of $X$ with respect
to the Gromov topology on the space of all metrics.

This result is very strange, because $\operatorname{Kah}$ is a smooth manifold
of dimension $b_{2}(M)-2$. By Theorem 4.13, the space
$\operatorname{Teich}_{H}$ of all hyperkähler metrics is a smooth manifold of
dimension $\frac{b_{2}(b_{2}-1)(b_{2}-2)}{6}$, clearly much bigger than
$\dim\operatorname{Kah}$. Obviously, the boundary of $X$ is highly irregular
and chaotic.

For another application, consider some numerical quantity $\mu(I)$ associated
with an equivalence class of complex structures. Suppose that $\mu$ is
continuous or semi-continuous on $\operatorname{Teich}$. Then $\mu$ is
constant on ergodic complex structures. To see this, suppose that $\mu$ is
upper semicontinuous, giving

$\mu(\lim_{k}I_{k})\geqslant\lim_{k}(\mu(I_{k})).$ (6.1)

Given an ergodic complex structire $I$, find a sequence $I_{k}=\nu_{k}(I)$
converging to a complex structure $I^{\prime}$. Then (6.1) gives
$\mu(I)\leqslant\mu(I^{\prime})$. This implies that any ergodic complex
structure satisfies
$\mu(I)=\inf_{I^{\prime}\in\operatorname{Teich}}\mu(I^{\prime})$.

This observation can be applied to Kobayashi pseudometric and Kobayashi
hyperbolicity.

###  6.2 Kobayashi non-hyperbolicity of hyperkähler manifolds

######  Definition 6.2.

Pseudometric on $M$ is a function $d:\;M\times
M{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$ which is symmetric:
$d(x,y)=d(y,x)$ and satisfies the triangle inequality $d(x,y)+d(y,z)\geqslant
d(x,z)$.

######  Remark 6.3.

Let ${\mathfrak{D}}$ be a set of pseudometrics. Then $d_{{\operatorname{\sf
max}}}(x,y):=\sup_{d\in{\mathfrak{D}}}d(x,y)$ is also a pseudometric.

######  Definition 6.4.

The Kobayashi pseudometric on a complex manifold $M$ is $d_{\operatorname{\sf
max}}$ for the set ${\mathfrak{D}}$ of all pseudometrics such that any
holomorphic map from the Poincaré disk to $M$ is distance-non-increasing.

In other words, a Kobayashi pseudo-distance between two points $x,y$ is an
infimum of distance from $x$ to $y$ in Poincare metric for any sequence of
holomorphic disks connecting $x$ to $y$.

The following observation is not difficult to see.

###### Claim 6.5

Let $\pi:\;{\cal M}{\>\longrightarrow\>}X$ be a smooth holomorphic family,
which is trivialized as a smooth manifold: ${\cal M}=M\times X$, and $d_{x}$
the Kobayashi metric on $\pi^{-1}(x)$. Then $d_{x}(m,m^{\prime})$ is upper
continuous on $x$.

###### Corollary 6.6

Denote the diameter of the Kobayashi pseudometric by $\operatorname{\sf
diam}(d_{x}):=\sup_{m,m^{\prime}}d_{x}(m,m^{\prime})$. Then the Kobayashi
diameter of a fiber of $\pi$ is an upper continuous function:
$\operatorname{\sf diam}:\;X{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$.

For a projective K3 surface, the Kobayashi pseudometric vanishes ([Vo, Lemma
1.51]). However, all non-projective K3 surfaces are ergodic (Corollary 5.16).
This proves the vanishing of Kobayashi pseudodistance for all K3 surfaces. A
more general version of this result is due to due to Kamenova-Lu-Verbitsky.

######  Theorem 6.7 (([KLV]))

Let $M$ be a Hilbert scheme of K3. Then the Kobayashi pseudometric on $M$
vanishes

######  Definition 6.8.

A complex manifold is called Kobayashi hyperbolic if the Kobayashi
pseudometric is a metric.

######  Definition 6.9.

An entire curve is a non-constant map ${\mathbb{C}}{\>\longrightarrow\>}M$.

Brody has shown that a compact manifold is Kobayashi hyperbolic if and only if
it admits no entire curves. The same argument also proves semicontinuity.

###### Theorem 6.10

([Br:1978]) Let $I_{i}$ be a sequence of complex structures on $M$ which are
not hyperbolic, and $I$ its limit. Then $(M,I)$ is also not hyperbolic.

With ergodicity, this can be used to prove that all hyperkähler manifolds are
non-hyperbolic.

Recall that a twistor family of complex structures on a hyperkähler manifold
$(M,I,J,K)$ is a family of complex structures of form
$S^{2}\cong\\{L:=aI+bJ+cK,\ \ \ a^{2}+b^{2}+c^{2}=1\\}$. F. Campana has
obtained a remarkable partial result towards non-hyperbolicity.

###### Theorem 6.11

([Cam]) Let $M$ be a hyperkähler manifold, and $S\subset\operatorname{Teich}$
a twistor family. Then there exists an entire curve in some $I\in S$.

###### Claim 6.12

There exists a twistor family which has only ergodic fibers.

Proof: There are only countably many complex structures which are not ergodic;
however, twistor curves move freely through the Teichmüller space of a
hyperkähler manifold, as seen from Theorem 4.19.

Applying Campana’s theorem to the family constructed in Claim 6.12, we obtain
an ergodic complex structure which is non-hyperbolic. Then the Brody’s theorem
implies that all complex structures in the same deformation class are non-
hyperbolic.

###### Theorem 6.13

All hyperkähler manifolds are non-hyperbolic.

###  6.3 Symplectic packing and ergodicity

I will finish this talk with a list of open problems of hyperkähler and
holomorphically symplectic geometry which might be solvable with ergodic
methods.

######  Question 6.14.

Let $M$ be a hyperkähler manifold, and $\operatorname{Teich}$ its Teichmüller
space. Consider the universal fibration ${\cal
X}{\>\longrightarrow\>}\operatorname{Teich}$ (Remark 1.11). The mapping class
group $\Gamma$ acts on ${\cal X}$ in a natural way. Is this action ergodic?

This question (suggested by Claire Voisin) seems to be related to the
following conjecture.

###### Conjecture 6.15

Let $M$ be a K3 surface. Then for each $x\in M$ and $v\in T_{x}M$ there exists
an entire curve $C\ni x$ with $T_{x}C\ni v$.

The symplectic packing problem is a classical subject of symplectic geometry
([MP]). However, its holomorphically symplectic version seems to be completely
unexplored.

######  Definition 6.16.

A holomorphic symplectic ball $B_{r}$ of radius $r$ is a complex
holomorphically symplectic manifold admitting a holomorphic symplectomorphism
to an open ball in ${\mathbb{C}}^{2n}$ of radius $r$ with the standard
holomorphic symplectic form $\sum_{i=1}^{n}dz_{2i-1}\wedge dz_{2i}$.

Notice that by a holomorphic symplectic version of Darboux theorem, any
holomorphically symplectic manifold is locally symplectomorphic to a
holomorphic symplectic ball.

######  Definition 6.17.

Let $M$ be a holomorphically symplectic manifold. Symplectic packing of radii
$r_{1},...,r_{k}$ of $M$ is a set of holomorphic symplectomorphisms
$\varphi_{i}:\;B_{r_{i}}\hookrightarrow M$ with images of $\varphi_{i}$ not
intersecting.

Obviously, in these assumptions,
$\sum\operatorname{Vol}(B_{r_{i}})\leqslant\operatorname{Vol}_{M}$, where
$\operatorname{Vol}$ denotes the symplectic volume of a holomorphic symplectic
manifold $(M,\Omega_{M})$:

$\operatorname{Vol}(M)=\int_{M}(\Omega_{M}\wedge\bar{\Omega}_{M}),\ \
2n=\dim_{\mathbb{C}}M.$

The volume inequality puts certain restrictions on the possible symplectic
packing. Are there any other restrictions?

For the usual (smooth) symplectic packing, some additional restrictions are
obtained from the Gromov’s symplectic capacity theorem and from the study of
pseudoholomorphic curves. However, it seems that in holomorphic symplectic
situation these restrictions are also trivial. For a general compact torus of
real dimension 4, volume is known to be the only restriction to existence of
symplectic packing ([LMS]). It seems that a similar result about the smooth
symplectic packings is true for K3 surfaces as well, and, possibly, for any
hyperkähler manifold.

The arguments used to treat the usual (smooth) symplectic packings don’t work
for the holomorphic symplectic case. However, the set of possible radii for
symplectic packing is obviously semicontinous, hence it can be studied by
ergodic methods, in the same way as one studies the Kobayashi pseudometric.

The following classical question was treated Buzzard and Lu in [BL].

######  Definition 6.18.

A complex manifold $M$ of dimension $n$ is called dominated by
${\mathbb{C}}^{n}$ if there exists a holomorphic map
$\varphi:\;{\mathbb{C}}^{n}{\>\longrightarrow\>}M$ which has non-degenerate
differential in generic point.

Buzzard and Lu proved that Kummer K3 surfaces are dominated by
${\mathbb{C}}^{2}$. So far, there is not a single example of a hyperkähler
manifold $M$ for which it is proven that $M$ is not dominated. This leads to
the following conjecture

###### Conjecture 6.19

Any compact hyperkähler manifold is dominated by ${\mathbb{C}}^{n}$.

There is no semicontinuity in dominance, because Brody lemma fails to produce
dominating maps ${\mathbb{C}}^{n}{\>\longrightarrow\>}M$ for $n>1$ as limits
of sequences of dominating maps. In the proof of Brody’s lemma (showing that a
limit of a sequence of entire curves contains an entire curve) one takes a
reparametrizations of each of the curves in the sequence. Starting from a
sequence of dominating maps, one could apply the same argument, but each
subsequent reparametrization can lead to smaller Jacobian of the differential,
and the differential of the limit could be zero.

It seems that more of the Brody’s argument can be retained if we restrict
ourselves to symplectomorphisms.

######  Question 6.20.

Consider a flat holomorphically symplectic structure on ${\mathbb{C}}^{2}$. Is
there a holomorphic map ${\mathbb{C}}^{2}{\>\longrightarrow\>}M$ to a K3
surface which is compatible with the holomorhic symplectic form?

Probably not. However, a quantitative version of this question makes sense.
Let $M$ be a hyperkähler manifold, and $K(M)$ the supremum of all $r$ such
that there exists a symplectic immersion from a symplectic ball of radius $r$
to $M$. It is not hard to see that $K(M)$ is semicontinuous in families, hence
constant on ergodic complex structures.

######  Question 6.21.

For a given hyperkähler manifold, find $K(M)$.

It is not clear if $K(M)$ is finite or infinite, even for a K3 surface (it is
clearly infinite for a torus).

## References

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