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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Teichm\"uller spaces and moduli of geometric structures (2)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf TWENTY-THIRD G\"OKOVA
GEOMETRY / TOPOLOGY CONFERENCE\\
May 30 - June 4 (2016)\\ G\"okova, Turkey}
\end{center}


\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Gr}{\operatorname{\sf Gr}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Symp}{\operatorname{Symp}}
\newcommand{\Hyp}{\operatorname{Hyp}}



\newpage

{\bf \blue Plan:}

1. Hyperk\"ahler manifolds: basic facts

2. Teichm\"uller space of hyperk\"ahler structures

3. Teichm\"uller space of symplectic structures
of a hyperk\"ahler manifold.

4. Ergodic action of the mapping class group 
on the space of symplectic structures.

\newpage



{\bf \blue Teichm\"uller space for symplectic structures (reminder)}


\definition
Let $\Gamma(\Lambda^2 M)$ be the space of all
2-forms on a manifold $M$,
and $\Symp\subset \Gamma(\Lambda^2 M)$ the space of 
all symplectic 2-forms. We equip $\Gamma(\Lambda^2 M)$
with $C^\infty$-topology of uniform convergence 
on compacts with all derivatives. 
Then $\Gamma(\Lambda^2
M)$ is a Frechet vector space, and $\Symp$ a Frechet
manifold. 


\definition
Consider the group of diffeomorphisms, denoted
$\Diff$ or $\Diff (M)$ as a Frechet Lie group, and denote
its connected component (``group of isotopies'') by $\Diff_0$. The quotient group
$\Gamma:=\Diff/\Diff_0$ is called {\bf\blue the mapping class group}
of $M$.

\definition
{\bf \blue Teichm\"uller space of symplectic structures on 
$M$} is defined as a quotient $\Teich_s:= \Symp/\Diff_0$.
The quotient $\Teich_s/\Gamma=\Symp/\Diff$,
is called {\bf\blue the moduli space of symplectic structures.}


\newpage

{\bf \blue Moser's theorem (reminder)}

\definition
Define {\bf \blue the period map} 
$\Per:\; \Teich_s \arrow H^2(M,\R)$ mapping 
a symplectic structure to its cohomology class.

\theorem {\bf \blue (Moser, 1965)}\\
The {\bf \red Teichm\"uler space $\Teich_s$ is a
manifold} (possibly, non-Hausdorff), and 
the {\bf \red period map $\Per:\; \Teich_s \arrow H^2(M,\R)$
is locally a diffeomorphism.}

The proof is based on another theorem of Moser.

{\bf \green Theorem 1:} {\bf \blue (Moser)}\\
Let $\omega_t$, $t\in S$ be a smooth family of symplectic
structures, parametrized by a connected manifold $S$.
Assume that the cohomology class $[\omega_t]\in H^2(M)$ is constant in $t$.
{\bf \purple Then all $\omega_t$ are diffeomorphic.}

\newpage

{\bf \blue Non-Hausdorff points on symplectic Teichm\"uller space}

Example of D. McDuff found in 
Salamon, Dietmar, {\em\green Uniqueness of symplectic structures}, 
Acta Math. Vietnam. 38 (2013), no. 1, 123-144.


Let $M= S^1\times Z^1 \times S^2 \times S^2$ with coordinates
$\theta_1, \theta_2 \in S^1\subset \C^*$ and $z_1, z_2\in S^2$.
Let $\phi_{\theta, z}\; \C P^1 \arrow \C P^1$ be a
rotation around the axis $z\in \C P^1$ by the angle $\theta$.
{\bf \blue Consider the diffeomorphism $\Psi:\; M \arrow M$
mapping $(\theta_1, \theta_2, z_1, z_2)$ to
$(\theta_1, \theta_2, z_1, \phi_{\theta_1, z_1}(z_2))$.}

\theorem 
Let $\omega_\lambda$ be the product symplectic form on $M = T^2 \times \C
P^1 \times \C P^1$ obtained as a  product of symplectic forms of
volume 1, 1, $\lambda$ on $T^2$, $\C P^1$, $\C P^1$.
{\bf \red The form $\Psi^*(\omega_1)$ is homologous, but not diffeomorphic to 
$\omega_1$.} However, {\bf \red the form $\Psi^*(\omega_\lambda)$
is diffeomorphic to $\omega_\lambda$ for any $\lambda\neq
1$.}

{\small (D. McDuff, {\em\green Examples of symplectic structures},
 Invent. Math. 89 (1987), 13-36.)}

\newpage


{\bf \blue Complex manifolds}


{\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.}

\newpage

{\bf \blue The Hodge decomposition in linear algebra}

\definition
{\bf \blue The Hodge decomposition} 
$V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that
$V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace.

\remark 
Let $V_\C:= V \otimes_\R \C$.
The Grassmann algebra of skew-symmetric forms
$\Lambda^n V_\C:=\Lambda^n_\R V \otimes _\R C$
admits a decomposition
\[ 
  \Lambda^n V_\C= \bigoplus_{p+q=n} \Lambda^p V^{1,0} 
  \otimes \Lambda^q V^{0,1}
\]
We denote $\Lambda^p V^{1,0} \otimes \Lambda^q V^{0,1}$ by $\Lambda^{p,q}V$.
The resulting decomposition $\Lambda^n V_\C= \bigoplus_{p+q=n}\Lambda^{p,q}V$
is called {\bf \blue the Hodge decomposition of the Grassmann algebra}.

\remark The operator $I$ induces $U(1)$-action on $V$ by the formula
$\rho(t)(v) = \cos t\cdot v + \sin t \cdot I(v)$. We extend this
action on the tensor spaces by muptiplicativity.

\newpage


{\bf \blue K\"ahler manifolds}

{\bf\green DEFINITION:} A Riemannian metric $g$ on
a complex manifold $(M,I)$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.


{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

{\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$
is called {\bf \blue K\"ahler} if $d\omega=0$. 
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$, and
$\omega$ {\bf \blue the K\"ahler form}. 

\remark {\bf \purple This is equivalent to $\nabla\omega=0$,} where
$\nabla$ is Levi-Civita connection.

\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 $I,J,K$.

{\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
$I,J,K$).

\claim
A compact hyperk\"ahler manifold $M$ {\bf \purple has maximal holonomy of 
Levi-Civita connection $Sp(n)$} if and only if {\bf \purple
$\pi_1(M)=0$, $h^{2,0}(M)=1$.}

\theorem {\bf \blue (Bogomolov decomposition)}\\
{\bf \red Any compact hyperk\"ahler manifold has a finite covering isometric
to a product of a torus and several maximal holonomy hyperk\"ahler
manifolds.}


\newpage

{\bf \blue Holomorphically symplectic manifolds}


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

\remark In this talk, all  holomorphically symplectic manifolds
are assumed to be K\"ahler and compact.

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

\claim 
In these assumptions, $\omega_J + \1\omega_K$ is
holomorphic symplectic on $(M,I)$.

{\bf\green THEOREM:} {\bf \blue (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
of maximal holonomy.}


\newpage


{\bf \blue EXAMPLES.}

\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 two series:} Hilbert schemes of K3, and
generalized Kummer.


\newpage

{\bf \blue Main result}

\definition
A symplectic structure $\omega$ on a hyperk\"ahler
manifold is called {\bf 
\blue standard} if $\omega$ is a K\"ahler form for
some hyperk\"ahler structure.

\remark
Any known symplectic structure on a hyperk\"ahler manifold
or a torus is of this type. {\bf \purple It was conjectured that non-standard
symplectic structures don't exist.}


\theorem (E. Amerik, V.)
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$
{\bf \red is an open embedding on the set of all standard symplectic
structures}, and {\bf \red its image is the set of all cohomology classes
$v$ such that $q(\omega, \omega) >0$,} where $q$ is a quadratic form 
on cohomology defined below.

\newpage

{\bf \blue 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 positive
definite on $\langle \Omega, \bar \Omega, \omega\rangle$,
 where $\omega$ is a K\"ahler form. 

\newpage

{\bf \blue MBM classes}

\definition
{\bf \blue K\"ahler cone} of a K\"ahler manifold
is the set of all cohomology classes $\omega\in H^{1,1}(M)$

\definition
{\bf \blue Face} of a K\"ahler cone $K$ is  a subset 
$V\cap \6 K$ containing an open subset of $V$, for some hyperplane
$V\subset H^{1,1}(M)$.

\definition
Let $M$ be a hyperk\"ahler manifold.
A homology class
$z\in H_2(M,\Q)$ is called {\bf \blue an MBM class}
(monodromy birational minimal)
if for some complex structure in the same deformation
class, the annihilator $z^\bot$ contains a face of its
K\"ahler cone.

\definition
A cohomology class $z\in H^2(M,\Q)$ is called {\bf \blue MBM class}
if it becomes MBM after an identification $H^2(M,\Q)\cong H_2(M,\Q)$
provided by the Bogomolov-Beauville-Fujiki form.

\newpage

{\bf \blue Properties of MBM classes}
\newcommand{\Pos}{\operatorname{\sf Pos}}

\definition
{\bf \blue Negative class} on a hyperk\"ahler manifold
is $\eta\in H^2(M,\R)$ satisfying $q(\eta,\eta)<0$.

\theorem (E. Amerik, V.)
Let $(M,I)$ be a hyperk\"ahler manifold, $\rk H^{1,1}(M,\Z)=1$, and
$z\in H_{1,1}(M,I)$ a non-zero negative class. {\bf \red Then $z$ is
MBM if and only if 
$\pm z$ is $\Q$-effective,} that is, $\lambda z$ is represented
by a complex curve. \endproof

\definition
{\bf \blue Positive cone} $\Pos(M)$ on a K\"ahler surface
is the one of the two components of 
\[ \left \{v\in H^{1,1}(M,\R)\ \ |\ \  \int_M\eta\wedge\eta>0\right \}
\]
which contains a K\"ahler form.

\theorem (E. Amerik, V.)
Let $(M,I)$ be a hyperk\"ahler manifold, 
and $S\subset H_{1,1}(M,I)$ the set of all MBM classes in 
$H_{1,1}(M,I)$. Consider the corresponding set of hyperplanes
$S^\bot:=\{W=z^\bot\ \ |\ \ z\in S\}$ in $H^{1,1}(M,I)$.
{\bf \red Then the K\"ahler cone of $(M,I)$ is 
a connected component of $\Pos(M,I)\backslash \cup S^\bot$},
where $\Pos(M,I)$ is a positive cone of $(M,I)$.
\endproof



%\newpage
%
%{\bf \blue Equivalence of hyperk\"ahler structures}
%
%\definition
%Let $(M,I,J,K,g)$ and $(M,I',J',K',g')$ be two
%hyperk\"ahler structures. We say that these structures
%are {\bf\blue equivalent} if the corresponding quaternionic algebras
%coincide.
%
%\proposition\label{_hk_equiv_Proposition_}
%Let $M$ be a hyperk\"ahler manifold,
%and $(M,I,J,K,g)$ and $(M,I',J',K',g')$ be two
%hyperk\"ahler structures of maximal holonomy. {\bf \red Then the following
%conditions are equivalent.} \\
%\phantom{AAA} (i) $g$ is proportional to $g'$. \\
%\phantom{AAA} (ii) The hyperk\"ahler structure $(M,I,J,K,g)$ is equivalent to
%  $(M,I',J',K',g')$.
%
%{\bf\green Proof of (i) $\Rightarrow$ (ii):} If $g$ is proportional to $g'$,
%the corresponding Levi-Civita connections coincide.
%Its holonomy group is $Sp(n)$,
%and {\bf \purple holonomy stabilizer in $\End(TM)$ is a quaternionic
%algebra, generated by $I,J,K$ and by $I',J',K'$.}
%
%\newpage
%
%{\bf \blue Equivalence of hyperk\"ahler structures (2)}
%
%
%{\bf \green Proof of (ii) $\Rightarrow$ (i):} Conversely, assume
%that the algebras generated by $I,J,K$ and by $I',J',K'$
%coincide. Recall that a manifold is called {\bf\blue 
%hypercomplex} if it is equipped with a triple
%complex structures $I,J,K$ satisfying quaternionic
%relations. By Obata's theorem, there exists a
%unique torsion-free connection preserving $I,J,K$
%on any hypercomplex manifold; this connection
%is called {\bf\blue the Obata connection}. Since the
%Levi-Civita connection on $(M,I,J,K,g)$ satisfies
%this condition, it coincides with the Obata
%connection for $(M,I,J,K)$ and for $(M,I',J',K')$
%(the latter is true because the corresponding 
%hypercomplex manifolds are equivalent). However,
%$g$ and $g'$ are invariant with respect to the
%holonomy of Levi-Civita connection, which is equal
%$Sp(n)$. {\bf \purple The space of $Sp(n)$-invariant 
%symmetric tensors is 1-dimensional, hence
%$g$ and $g'$ are proportional.}
%\endproof

\newpage

{\bf \blue Teichm\"uller space of hyperk\"ahler structures }

\definition
Consider the space $\Hyp$ of all hyperk\"ahler structures 
$(I,J,K,g)$, let $\Teich_h:= \Hyp/\Diff_0$ be the corresponding
Teichm\"uller space, called {\bf \blue Teichm\"uller 
space of hyperk\"ahler structures}.


%\remark A quotient 
%$\Hyp/SU(2)$ {\bf \purple is naturally identified with the set
%of equivalence classes of hyperk\"ahler structures.}

%\remark
%As shown above, {\bf for manifolds
%with maximal holonomy the quotient $\Hyp_m:=\Hyp/SU(2)$  is
%also identified with the space of  all hyperk\"ahler
%metrics of fixed volume, say, volume 1.}
%
%\definition
%Define {\bf\blue the Teichm\"uller space of 
%hyperk\"ahler structures} as the quotient $\Hyp_m/\Diff_0$,
%where $\Diff_0$ is the connected component of the group
%of diffeomorphisms $\Diff$, and {\bf\blue the moduli
%of hyperk\"ahler structures} as $\Hyp_m/\Diff$. 
%
%\remark
%For most geometric structures, the Teichm\"uller spaces 
%and especially the moduli spaces are non-Hausdorff.
%However, for manifolds of maximal holonomy the moduli space of
%hyperk\"ahler structures is Hausdorff. This is because
%$\Hyp_m$ is identified with the space of all hyperk\"ahler
%metrics of fixed volume. However, there is a metric on the moduli 
%space of all metrics, known as {\bf \blue Gromov-Hausdorff
%metric,} and a metric space is necessarily Hausdorff.



\definition
 Consider the
space $\Perspace_h$ of all
triples $x,y,z\in H^2(M,\R)$ satisfying $x^2 = y^2=z^2 >0$.
Let  $\Per:\; \Teich_h \arrow\Perspace_h$ 
the map associating to a hyperk\"ahler structure
$(M,I,J,K,g)$ the triple $\omega_I, \omega_J,\omega_K$.
This map called {\bf\blue  the period map for the 
Teichm\"uller space of hyperk\"ahler structures},
and $\Perspace_h$ {\bf\blue the period space of  hyperk\"ahler
  structures}.

\theorem  (E. Amerik, V.)
Let $M$ be a hyperk\"ahler manifold of maximal holonomy,
and $\Per:\; \Teich_h \arrow\Perspace_h$ the period map
for the Teichm\"uller space of hyperk\"ahler structures.
Then {\bf \red 
$\Per$ is an open embedding
for each connected component.} Moreover, {\bf \red its
image is the set of all spaces $W\in \Perspace_h$
such that the orthogonal complement $W^\bot$
contains no MBM classes.} \\ \ \ \\
{\bf \green Ingredients of its proof:} 
Follows from Calabi-Yau theorem,
global Torelli theorem for complex structures of hyperk\"ahler type, and
the description of the K\"ahler cone in terms of the 
MBM classes. Main idea: {\bf \purple bijective correspondence between hyperk\"ahler
structures and pairs $(I, [\omega])$, where $I$ is a complex structure
of hyperk\"ahler type, 
and $[\omega]$ a K\"ahler class on $(M,I)$.} \endproof


\newpage

{\bf \blue Torelli theorem for symplectic structures}

\theorem
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$
{\bf \red is an open embedding on the set of all standard symplectic
structures}, and {\bf \red its image is the set of all cohomology classes
$v$ such that $q(v,v) > 0$.}

{\bf \green Proof. Step 1:}
Let  $P:\;  \Teich_h \arrow \Teich_s$ be the forgetful map
putting $\omega_I, \omega_J, \omega_K$
to $\omega_I$.
{\bf \purple Calabi-Yau implies that  $P$ is surjective.} Indeed, any
K\"ahler form can be deformed to a Ricci-flat K\"ahler form
in the same cohomology class.


{\bf \green Step 2:}
From Torelli theorem for hyperk\"ahler structures
it follows that {\bf \purple the fiber $P^{-1}(\omega)$ 
of $P$ is the space of pairs $x,y\in H^2(M)$
satisfying $x^2=y^2=\omega^2$, such that 
the space $\langle \omega, x, y\rangle^\bot$ contains no
MBM classes.}

{\bf \green Step 3:} Since the fibers of $P$ are complements
to subsets of codimension 2, they are connected. By Moser's theorem,
for each $(M,\omega_I, \omega_J, \omega_K) \in P^{-1}(\omega)$
{\bf \purple the symplectic forms $\omega_I$ are diffeomorphic. }

\newpage

{\bf \blue Torelli theorem for symplectic structures (2)}


\theorem
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$
{\bf \red is an open embedding on the set of all standard symplectic
structures}, and {\bf \red its image is the set of all cohomology classes
$v$ such that $q(v,v) > 0$.}


{\bf \green Step 4:} Consider the diagram
\[ \begin{CD}
 \Teich_h @>P>> \Teich_s \\
@VV{\Per_h}V @V{\Per_s}VV\\
{\small \begin{array}{c} \{x,y,z\in H^2(M)|x^2=y^2=z^2>0, \\ \text{$\langle x, y, z\rangle^\bot$ 
contains no MBM classes} \} \end{array}} @>{P'}>> \{x \in H^2(M)|x^2>0\} \\
\end{CD}
\]
The map $\Per_h$ is an isomorphism as shown,
and the fibers of $P$ are identified with fibers of $P'$
as follows from Moser's theorem and Step 3.
Therefore, $\Per_s$ is injective.
The rest of the arrows are surjective as shown, 
{\bf \purple hence $\Per_s$ is also surjective.}
\endproof

\newpage

{\bf \blue Ergodicity of mapping class group action}

\theorem (V., 2009)\\
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
{\bf \purple Then the image of the mapping class group 
$\Gamma$ in $O(H^2(M,\Z))$ has finite index.}

\corollary {\bf \red $\Gamma$ acts on $\Teich_s$ with dense orbits.}

{\bf \green Proof:} We use a theorem of Calvin Moore:

\theorem (Calvin C. Moore, 1966)
Let $\Gamma$ be a lattice in a non-compact 
simple Lie group $G$ with finite center, and $H\subset G$ a 
non-compact semisimple Lie subgroup. {\bf \red Then the left 
action of $\Gamma$ on $G/H$ is ergodic.}

Applying this theorem to $\Gamma$ inside $G=SO(H^2(M,\R), q)$
and $H$ the stabilizer of $\omega\in H^2(M,\R)$,
we obtain that the action of $\Gamma$ on $\Teich_s\subset H^2(M,\R)$
{\bf \purple is ergodic, hence has dense orbits.} \endproof

\question
The Teichm\"uller space of standard symplectic structures on K3
is Hausdorff, as shown above. {\bf \red 
Are there any non-Hausdorff non-standard symplectic
structures in the same connected component of $\Teich_s$?}



\end{document}


