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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Teichm\"uller spaces and moduli of geometric structures (1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[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. Set-up: Teichm\"uller space and the moduli space
of geometric structures.

2. Moser's theorem. Teichm\"uller space of symplectic structures on a torus.

3. Other geometric structures and their Teichm\"uller spaces.
$G_2$ structures, holomorphically symplectic structures.

\newpage



{\bf \blue Geometric structures}


\definition
{\bf \blue ``Geometric structure''} on a manifold $M$ is a reduction of its structure
group $GL(n,\R)$ to a subgroup $G\subset GL(n,\R)$. However, it is easier to 
define it by a collection of tensors $\Psi_1, ..., \Psi_n$ such that
the stabilizer $\St_{\langle \Psi_1, ..., \Psi_n\rangle}$ of 
$\Psi_1, ..., \Psi_n$ at each point  $x\in M$ is conjugate to
$G\subset GL(T_xM)$.

Let me give some examples.


{\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 $TM\otimes \C=T^{0,1}M\oplus T^{1,0}(M)$.

\definition
{\bf \blue Symplectic form} on a manifold
is a non-degenerate differential 2-form $\omega$ satisfying
$d\omega=0$.



\newpage

{\bf \blue Teichm\"uller space of geometric structures}

Let ${\cal C}$ be the set of all geometric structures of a given
type, say, complex, or symplectic. We put topology of
uniform convergence with all derivatives on ${\cal C}$.
Let $\Diff_0(M)$ be the connected component of its 
diffeomorphism group $\Diff(M)$ ({\bf\blue the group of isotopies}).

\definition
The quotient  ${\cal C}/\Diff_0$ is called
{\bf \blue Teichm\"uller space} of geometric strictures of this type.

\definition The group $\Gamma:=\Diff(M)/\Diff_0(M)$ is
called {\bf\blue the mapping class group} of $M$.
It acts on $\Teich$ by homeomorphisms.

\definition
The orbit space ${\cal C}/\Diff=\Teich/\Gamma$ 
is called {\bf \blue the moduli space} of 
geometric structure of this type.

Today I will describe $\Teich$ and $\Gamma$ in
some interesting cases and explain some important
concepts, such as {\bf \red ergodicity of $\Gamma$-action.}

\newpage

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


\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.}

\remark
In many cases $\Gamma$ acts on $\Teich_s$ with dense orbits,
hence {\bf \red the moduli space is not always well defined.}

\definition
Two symplectic structures are called {\bf \blue isotopic}
if they lie in the same orbit of $\Diff_0$, and {\bf \blue
diffeomorphic} is they lie in the same orbit of $\Diff$.

\newpage

{\bf \blue Moser's theorem}

\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 The proof of Moser's theorem}

\theorem {\bf \blue (Moser)}\\
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.}

{\bf \green Proof. Step 1:} 
We can locally find a section $S$ for the $\Diff_0$-action on $\Symp$,
producing a local decomposition $\Symp=O\times S$, where $O$ is a $\Diff_0$-orbit.
Here $O$ and $S$ are both Frechet manifolds.

{\bf \green  Step 2:} The period map $P:\; U\arrow H^2(M,\R)$ is
a smooth submersion. By Theorem 1, the fibers of $P$ are 0-dimensional.
Therefore, $P$ is locally a diffeomorphism. \endproof


\newpage

{\bf \blue Symplectic structures on a compact torus}


\definition
A symplectic structure $\omega$ on a torus is called {\bf 
\blue standard} if there exists a flat torsion-free connection
preserving $\omega$.

\remark Moser's theorem immediately implies that {\bf\purple the set 
$\Teich_{st}$ of standard symplectic
structures is open in the Teichm\"uller space.} Indeed, 
the period map from $\Teich_{st}$ to $H^2(M)$ is also locally a diffeomorphism.

\remark {\bf \red It is not known if any non-standard symplectic structures exist}
(even in dimension =4). 


\theorem
Let $\Lambda^2_{nd}(H_1(T))\subset H^2(T)$ be the space of symplectic
forms on $H_1(T)$, where $T$ is an even-dimensional torus.
Consider the period map
$\Per:\; \Teich_{st}\arrow \Lambda^2_{nd}(H_1(T))\subset H^2(T)$, 
where $\Teich_{st}$ is the  Teichm\"uller space of standard 
symplectic structures on $T$. 
{\bf \red Then $\Per$ is a diffeomorphism.}


\newpage


{\bf \blue The space of flat Hermitian metrics}


\theorem
Let $\Lambda^2_{nd}(H_1(T))\subset H^2(T)$ be the space of symplectic
forms on $H_1(T)$, where $T$ is an even-dimensional torus.
Consider the period map
$\Per:\; \Teich_{st}\arrow \Lambda^2_{nd}(H_1(T))\subset H^2(T)$, 
where $\Teich_{st}$ is the  Teichm\"uller space of standard 
symplectic structures on $T$. 
{\bf \red Then $\Per$ is a diffeomorphism.}

\pstep Let $\Teich_{h}$ be the Teichm\"uller space of flat Hermitian
metrics on $T$. Clearly, $\Teich_h= GL(2n, \R)/U(n)$. Moreover, 
{\bf \purple the natural forgetful map $\Teich_h\arrow\Teich_{st}$ is surjective}.

{\bf \green Step 2:} The fibers of the natural projection
$\Teich_{h}\arrow\Lambda^2_{nd}(H_1(T))$ are connected.
Using the diagram
\[\begin{CD}
\Teich_{h}@>>> \Lambda^2_{nd}(H_1(T))\\
@VVV @VV\Id V\\
\Teich_{st}@>>> \Lambda^2_{nd}(H_1(T))\\
\end{CD}\]
we obtain that
 $\Per:\; \Teich_{st}\arrow\Lambda^2_{nd}(H_1(T))$
has connected fibers. By Moser's theorem, this map is a diffeomorphism.
\endproof

\newpage

{\bf \blue $G_2$-structures}

\definition
Let $\rho\in \Lambda^2 \R^7$ be a 3-form on $\R^7$.
We say that $\rho$ is {\bf\blue non-degenerate} if the 
dimension of its stabilizer is maximal:
\[
\dim St_{GL(7)}\rho = \dim GL(7) - \dim \Lambda^3(\R^7) =
49-35 =14.
\]
In this case, $St(\rho)$ is one of two real forms 
of a 14-dimensional Lie group $G_2(\C)$. 
We say that $\rho$ is {\bf \blue non-split} if it satisfies
$St(\rho|_x)\cong G_2$, where $G_2$ denotes the
compact real form of $G_2(\C)$.
{\bf\blue A $G_2$-structure} on a 7-manifold
is a 3-form $\rho \in \Lambda^3(M)$, which is
non-degenerate and non-split at each point $x\in M$
(``stable'', in the sense of Hitchin).

\remark
A form $\rho$ defines
a $\Lambda^7 M$-valued metric on $M$:
\[
g(x,y) = (\rho\cntrct x)\wedge (\rho \cntrct y) \wedge \rho
\]
This defines a conformal structure on $M$.
The conformal factor is fixed if we want $|\rho|=1$.
Therefore, {\bf \red every $G_2$-manifold is equipped with a natural
Riemannian structure.}


\newpage

{\bf \blue Holonomy $G_2$-manifolds}



\definition 
An  $G_2$-manifold is called {\bf \blue a holonomy $G_2$-manifold}
if $\rho$ is preserved by the corresponding Levi-Civita connection.

\remark $(M,\rho)$ {\bf \purple is a holonomy $G_2$-manifold if and only if
the form $\rho$ and its Hodge dual $*\rho$ are closed.}

\theorem {\bf \blue (Joyce)}
Let $\Teich_{G_2}$ be the Teichm\"uller space of $G_2$-structures on 
a compact 7-manifold $M$, and $\Per \Teich_{G_2}\arrow H^3(M)$ the period
map associating to $\rho$ its cohomology class. {\bf \red Then $\Per$ is 
locally a diffeomorphism.}



\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 Holomorphic symplectic form}


\definition
{\bf\blue Holomorphic symplectic form} on an almost complex manifold $(M,I)$
is a non-degenerate differential 2-form $\omega\in \Lambda^2(M,\C)$ satisfying
$d\omega=0$ and $\Omega(Ix, y)=\1 \Omega(x,y)$.

\remark This is the same as to say that 
{\bf \purple $\Omega(X, \cdot)=0$ for all $X\in  T^{1,0}(M)$, and
$\Omega$ is non-degenerate on $T^{0,1}(M)$.}


\definition
Let $\Omega$ be a differential form on $M$. The {\bf \blue
  kernel}, or {\bf \blue the null-space}
$\ker(\Omega)\subset TM$ of $\Omega$ is 
the space of all vector fields $X\in TM$ such that
the contraction $\Omega\cntrct X$ vanishes.

\pstep
Let $X,X_1\in \ker(\Omega)$, and $X_2, ..., X_p$ any
vector fields. Cartan's formula
implies that $\Lie_X(\Omega) = d(i_X(\Omega))+
i_x(d\Omega)=0$, hence $\Lie_X(\Omega)=0$.

{\bf \green Step 2:} $\Lie_X(\Omega)(X_1, ..., X_p)=
\Lie_X(\Omega(X_1, ..., X_p))- \sum_{i=1}^p\Omega(X_1, ..., [X, X_i], ... X_p).
$
All terms of this sum, except $\Omega([X,X_1], X_2,
..., X_p)$, vanish, because $X_1 \in \ker(\Omega)$.
Since $\Lie_X(\Omega)=0$, we have $\Omega([X,X_1], X_2,
..., X_p)=0$ for all $X_2, ..., X_p$. Therefore,
$[X, X_1]\in \ker(\Omega)$.
\endproof

\corollary Let $(M,I)$ be an almost complex manifold
admitting a holomorphic symplectic form. {\bf \red Then $I$ is integrable.}



\newpage

{\bf \blue Teichm\"uller space for holomorphic symplectic structures}

\theorem {\bf \blue (Kaledin, V.)}
Let $(M,\Omega)$ be a holomorphic symplectic manifold,
$\Teich_\Omega$ the Teichm\"uller space of holomorphic
symplectic structures on $M$, and $\Per:\; \Teich_\Omega\arrow H^2(M,\C)$
the map associating to $(M,\Omega)$ the cohomology class of $\Omega$,
{\bf \red Then $\Per$ is locally an embedding.}




\end{document}


