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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Teichm\"uller space and Ratner theory, lecture 2:\\[4mm]
Global Torelli theorem for hyperk\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf 
Hyperbolicity 2015\\[2mm]
Holomorphic dynamics school\\[2mm]
Hyperbolicity in algebraic geometry conference 
\\[2mm] 
Ilhabela, 07.01.2015
}
\end{center}

\vfil

{\bf \red The first lecture is available at:\\
\url{http://verbit.ru/MATH/TALKS/Ergo/Ilhabela/} (with corrections)}

\vfil

\newpage

{\bf \blue Complex manifolds}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}



{\bf\green DEFINITION:} Let $M$ be a smooth manifold. 
An {\bf \blue almost complex structure} is an operator
$I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$.

The eigenvalues of this operator are $\pm \1$.
The corresponding eigenvalue 
decomposition is denoted $TM\otimes \C=T^{0,1}M\oplus T^{1,0}(M)$.

{\bf\green DEFINITION:}
An almost complex structure is {\bf \blue 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 {\bf \blue a complex structure operator}.
A manifold with an integrable almost complex structure
is called {\bf \blue a complex manifold}. 

{\bf\green THEOREM:} (Newlander-Nirenberg)\\
{\bf \purple This definition is equivalent to the usual one.}

\definition
{\bf \blue The space of almost complex structures}
is an infinite-dimensional Fr\'echet manifold $X_M$  of all tensors
$I^2 = - \Id_{TM}$, equipped with the natural Fr\'echet topology.

\claim
The space $\Comp$ of integrable almost complex structures
{\bf \red is a submanifold in $X_M$} (also infinite-dimensional).


\newpage

{\bf\blue Teichm\"uller space}


{\bf \green Definition:} Let $M$ be a compact complex manifold, and 
$\Diff_0(M)$ a connected component of its diffeomorphism group
({\bf\blue the group of isotopies}). Denote by $\Comp$
the space of complex structures on $M$, and let
$\Teich:=\Comp/\Diff_0(M)$. We call 
it {\bf \blue the Teichm\"uller space.}

\remark $\Teich$ is {\bf \blue a finite-dimensional
complex space} (Kodaira-Spencer-Kuranishi-Douady),
but often {\bf \red non-Hausdorff}.

\definition Let $\Diff_+(M)$ be the group of oriented
diffeomorphisms of $M$. We call $\Gamma:=\Diff_+(M)/\Diff_0(M)$ {\bf \blue the
mapping class group}. The {\bf \blue moduli space of complex
structures on $M$} is a connected component of $\Teich/\Gamma$.

\remark This terminology is {\bf \purple standard for curves.}

\remark
The topology of the moduli space $\Teich/\Gamma$ is often
bizzarre. However, {\bf \purple its points are in bijective correspondence
with equivalence classes of complex structures.}

\remark To describe the moduli space, we shall
compute $\Teich$ and $\Gamma$. 


\newpage

{\bf \blue Hyperk\"ahler manifolds}

\definition
A {\bf \blue hyperk\"ahler 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 K\"ahler
for $I,J,K$.

\remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
$\omega_K:=  g(K\cdot, \cdot)$.}


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


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

{\bf\green REMARK:}
{\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has
holonomy in $Sp(n)$ } (the group of all endomorphisms preserving
$I,J,K$).


\newpage


{\bf \blue Holomorphically symplectic manifolds}

{\bf\green DEFINITION:} A holomorphically symplectic manifold 
is a complex manifold equipped with non-degenerate, holomorphic
$(2,0)$-form.

\remark Hyperk\"ahler manifolds are holomorphically symplectic.
Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic
form on $(M,I)$.

{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
admits a unique hyperk\"ahler metric in any K\"ahler class.

{\bf\green DEFINITION:}  For the rest of this talk, 
{\bf \red a hyperk\"ahler manifold
is a compact, K\"ahler, holomorphically symplectic manifold.}

{\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called
{\bf \blue simple} if $\pi_1(M)=0$, $H^{2,0}(M)=\C$.

{\bf \purple Bogomolov's decomposition:} Any 
hyperk\"ahler manifold admits a finite covering
which is a product of a torus and several 
simple hyperk\"ahler manifolds.

{\bf \red Further on, all hyperk\"ahler manifolds
are assumed to be simple}.


\newpage


{\bf \blue EXAMPLES.}

\example An even-dimensional complex vector space.

\example An even-dimensional complex torus.

\example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi).

\remark $T^*\C P^1$ {\bf \blue
is a resolution of a singularity $\C^2/{\pm1}$.}

\example Take a 2-dimensional complex torus $T$,
then the singular locus of $T/{\pm1}$ is of form 
$(\C^2/{\pm1}) \times T$.
Its resolution $\widetilde {T/{\pm1}}$ is called 
{\bf \green a Kummer surface}. {\bf \red
It is holomorphically symplectic}.

\remark Take a symmetric square $\Sym^2 T$, with a natural
action of $T$, and let $T^{[2]}$ be a blow-up of a singular
divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the
Kummer surface $\widetilde {T/{\pm1}}$.}

\definition A complex surface is called {\bf \blue K3 surface}
if it a deformation of the Kummer surface.

\theorem {\bf \blue (a special case of Enriques-Kodaira classification)}\\
Let $M$ be a compact complex surface which is hyperk\"ahler.
{\bf \red Then $M$ is either a torus or a K3 surface.}

\newpage 

{\bf \blue Hilbert schemes} 

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

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

\theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of
a hyperk\"ahler surface is hyperk\"ahler.}

\example
{\bf\blue A Hilbert scheme of K3} is hyperk\"ahler.

\example
Let $T$ be a torus. Then it 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
{\bf \blue a generalized Kummer variety}. 

\remark There are 2 more ``sporadic'' examples
of compact hyperk\"ahler manifolds, constructed by K. O'Grady.
{\bf \purple All known compact hyperkaehler manifolds are
these 2 and the three series:} tori, Hilbert schemes of K3, and
generalized Kummer.

\newpage

{\bf \blue The Bogomolov-Beauville-Fujiki form}

{\bf \green THEOREM:} (Fujiki). 
Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
hyperk\"ahler. Then $\int_M \eta^{2n}=c q(\eta,\eta)^n$,
for some primitive integer quadratic form $q$ on $H^2(M,\Z)$,
and $c>0$ an integer number.


{\bf \green Definition:} This form is called
{\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined
by the Fujiki's relation uniquely, up to a sign.} The sign is determined
from the following formula (Bogomolov, Beauville)
\begin{align*}  \lambda q(\eta,\eta) &=
   \int_X \eta\wedge\eta  \wedge \Omega^{n-1}
   \wedge \bar \Omega^{n-1} -\\
 &-\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)
\end{align*}
where $\Omega$ is the holomorphic symplectic form, and 
$\lambda>0$.

{\bf \green Remark:}  {\bf \red $q$ 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 K\"ahler form. 

\newpage

{\bf \blue Automorphisms of cohomology.}

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


{\bf \green Proof. Step 1:} Fujiki formula 
$v^{2n}= q(v,v)^n$ implies that
$\Gamma_0$ {\bf \purple preserves the Bogomolov-Beauville-Fujiki
up to a sign. } The sign is fixed, if $n$ is odd.


{\bf \green Step 2:} For even $n$, the sign is also fixed. 
Indeed, $G$ preserves $p_1(M)$, and (as Fujiki has shown)
$v^{2n-2}\wedge p_1(M)= q(v,v)^{n-1} c$, 
for some $c\in \R$. The constant $c$ is positive, 
{\bf \purple because the degree of $c_2(B)$ is positive}
for any Yang-Mills bundle with $c_1(B)=0$.

{\bf \green Step 3:} ${\goth o}(H^2(M, \R), q)$
acts on $H^*(M, \R)$ by derivations preserving 
Pontryagin classes (V., 1995). Therefore 
$\Lie(G)$ surjects to ${\goth o}(H^2(M, \R), q)$.


{\bf \green Step 4:} {\bf \purple The kernel $K$ of the map
$G \arrow G\restrict{H^2(M,\R)}$ is compact,}
because it com