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          \scriptsize {\it \small Teichmuller space of geometric structures 4 \hfil
  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Teichm\"uller spaces for geometric structures,\\
lecture 4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf Conference - Teichm\"uller Theory in Higher 
Dimension and Mirror Symmetry \\
April 24-28, 2017}
\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 Hyperk\"ahler manifolds (reminder)}

\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 (reminder)}

{\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 The Bogomolov-Beauville-Fujiki form (reminder)}

{\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 The Teichm\"uller space and the mapping class group (reminder)}
\renewcommand{\Teich}{\operatorname{\sf Teich}}

{\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 $\widetilde \Teich$
the space of complex structures on $M$, and let
$\Teich:=\widetilde \Teich/\Diff_0(M)$. We call 
it {\bf \blue the Teichm\"uller space.}

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

{\bf \green Definition:} Let $\Diff(M)$ be the group of 
diffeomorphisms of $M$. We call $\Gamma:=\Diff(M)/\Diff_0(M)$ {\bf \blue the
mapping class group}.

\remark 
For hyperk\"ahler manifolds, we take for $\Teich$
{\bf \blue the space of all complex structures of hyperk\"ahler type}, that
is, {\bf \red holomorphically symplectic and K\"ahler type}. It is open
in the usual Teichm\" uller space. 

\newpage


{\bf \blue The period map (reminder)}

{\bf \green Remark:} For any $J\in \Teich$,
$(M,J)$ is also a simple hyperk\"ahler manifold, hence
$H^{2,0}(M,J)$ is one-dimensional. 

{\bf \green Definition:} Let 
$P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$
map $J$ to a line $H^{2,0}(M,J)\in {\Bbb P}H^2(M, \C)$.
The map $P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ is 
called {\bf\blue the period map}.

\remark 
{\purple $P$ maps $\Teich$ into an open subset of a 
quadric,} defined by
\[
{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0\}.
\]
It is called {\bf \blue the period space} of $M$.

\newpage


{\bf \blue Global Torelli theorem and mapping class group (reminder)}


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

{\bf \green THEOREM:} {\bf \blue (D. Huybrechts) }\\
If $I_1$, $I_2\in \Teich$ are non-separable points, 
then $P(I_1)=P(I_2)$, and $(M, I_1)$ {\bf \purple is birationally 
equivalent} to  $(M, I_2)$ 

\definition
Let $M$ be a topological space for which $M/\sim$ is Hausdorff.
Then $M/\sim$ is called {\bf \blue a Hausdorff reduction} of $M$.

\definition
The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the
birational Teichm\"uller space} of $M$.

\theorem {\bf \blue (``Global Torelli theorem'')}\\ {\bf \red The period map 
$\Teich_b\stackrel \Per \arrow \Perspace$ is a diffeomorphism,}
for each connected component of $\Teich_b$.


{\bf \green Theorem:} {\bf \blue (``Mapping class group is arithmetic'')}\\
Let $M$ be a simple hyperk\"ahler manifold,
and $\Gamma$ the mapping class group.  Then \\
(i)  $\Gamma\restrict{H^2(M,\Z)}$ {\bf \red is a finite index 
subgroup of $O(H^2(M, \Z), q)$.}\\
(ii) The map $\Gamma\arrow O(H^2(M, \Z), q)$
{\bf \red has finite kernel.}

\newpage


{\bf \blue Ergodic complex structures}

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

\claim {\bf \blue (ergodic actions have dense orbits)}\\
Let $M$ be a manifold, $\mu$ a Lebesgue measure, and
$G$ a group acting on $M$ ergodically. {\bf \red Then the 
set of non-dense orbits has measure 0.}

{\bf\green Proof. Step 1:}
Consider a non-empty open subset $U\subset M$. 
Then $\mu(U)>0$, hence $M':=G\cdot U$ satisfies 
$\mu(M\backslash M')=0$. For any orbit $G\cdot x$
not intersecting $U$, $x\in M\backslash M'$.
Therefore the set $Z_U$ of such orbits has measure 0.

{\bf\green Step 2:} Choose a countable base
$\{U_i\}$ of topology on $M$. Then the set of 
points in dense orbits is $M \backslash \bigcup_i Z_{U_i}$.
\endproof

\definition
Let $M$ be a complex manifold, $\Teich$ its Techm\"uller
space, and $\Gamma$ the mapping group acting on $\Teich$.
{\bf\blue An ergodic complex structure} is a complex
structure with dense $\Gamma$-orbit.

\claim
Let $(M,I)$ be a manifold with ergodic complex structure,
and $I'$ another complex structure.
{\bf \purple Then there exists a sequence of diffeomorphisms
$\nu_i$ such that $\nu_i^*(I)$ converges to $I'$.}
% This is another characterization of ergodic complex structures.

\newpage

{\bf \blue Ergodicity of the monodromy group action}

\definition
{\bf\blue A lattice} in a Lie group is a discrete
subgroup $\Gamma\subset G$ such that $G/\Gamma$ has finite
volume with respect to Haar measure.

\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 subgroup. {\bf \red Then the left action of $\Gamma$
on $G/H$ is ergodic.}


\theorem Let ${\Perspace}$ be a component of 
a birational Teichm\"uller space, and
$\Gamma$ its monodromy group. Let $\Perspace_e$ be
a set of all points $L\subset \Perspace$
such that the orbit $\Gamma\cdot L$ is dense
(such points are called {\bf \blue ergodic}). {\bf \red Then 
$Z:=\Perspace\backslash \Perspace_e$ has measure 0.}

{\bf \green Proof. Step 1:}
 Let $G=SO(b_2-3,3)$, $H=SO(2) \times SO(b_2-3,1)$.
{\bf \purple Then $\Gamma$-action on $G/H$ is ergodic,}
by Moore's theorem.

{\bf \green Step 2:}
Ergodic orbits are dense, non-ergodic orbits have measure
0.
\endproof


\remark 
Generic deformation of $M$ has no rational curves,
and no non-trivial birational models. Therefore,
{\bf \purple outside of a measure zero subset,
$\Teich=\Teich_b$.} This implies that 
{\bf \red almost all complex structures on $M$ are ergodic.}


\newpage

{\bf \blue Ratner's theorem}

\definition
Let $G$ be a connected Lie group equipped with
a Haar measure. {\bf \blue A lattice} $\Gamma\subset G$
is a discrete subgroup of finite covolume (that is,
$G/\Gamma$ has finite volume).

\example
By Borel and Harish-Chandra theorem, {\bf \purple any 
integer lattice in a simple Lie group has finite covolume.}

\definition
    {\bf \blue Unipotent element} in a Lie group is an exponent of a
    nilpotent element.

\theorem
Let $H\subset G$ be a Lie subroup generated by 
unipotents, and $\Gamma\subset G$ an arithmetic lattice.
Then {\bf \red the closure of any $\Gamma$-orbit in $G/H$
is an orbit of a Lie subgroup $S\subset G$,
such that $S\cap \Gamma\subset S$ is a lattice.}

\example
Let $V$ be a real vector space with a non-degenerate
bilinear symmetric form of signature $(3,k)$, $k>0$, 
$G:=SO^+(V)$ a connected component of the 
isometry group, $H\subset G$ a subgroup
acting trivially on a given positive 2-dimensional plane,
$H\cong SO^+(1,k)$, and $\Gamma\subset G$ an arithmetic
lattice. Consider the quotient $\Perspace:=G/H$. {\bf \purple Then
a closure of $\Gamma\cdot J$ 
in $G/H$ is an orbit of a closed connected
Lie subgroup $S\subset G$ containing $H$.}

\newpage

{\bf \blue Classification of $\Gamma$-orbits on $\Perspace$}


\claim Let $G=SO(3,k)$ be a group of oriented isometries of $V=\R^{3, k}$, and 
$H\cong SO(1,k)\subset G$. Denote by $\goth h$, $\goth g$ 
their ie algebras. Then {\bf \red any Lie algebra
$\goth s$ such that $\goth h \subsetneq \goth s \subsetneq \goth g$
is isomorphic to $\goth{so}(2, k)$.} This is the Lie algebra
of the Lie group $S=SO(2,k)$ fixing a positive vector $v\in V$.

\corollary
Let $J\in \Perspace=G/H$, and $\Gamma\subset G$ be
am arithmetic lattice. {\bf \red Then one of three things happens.}\\
\phantom{XX} (i) ether $J$ is ergodic, \\
\phantom{XX} (ii) or the closure of $\Gamma$-orbit of $J$ is an orbit of $S$ \\
or its connected component $S^+$,\\
\phantom{XX} (iii) or the orbit $\Gamma \cdot J$ is closed.

\newpage

{\bf \blue Characterization of ergodic complex structures}

\remark By Ratner's theorem, 
the $S^+$-orbit of $J$ in (ii) and the $H$-orbit of $J$ in (iii)
has finite volume in $G/\Gamma$  
Therefore, {\bf \purple its intersection with $\Gamma$ is a lattice in 
$H$.} This brings


\corollary
Consider the action of the mapping class group $\Gamma$
of a hyperk\"ahler manifold on its period space $\Perspace$.
Let $J\in \Perspace$ be a point such that its 
$\Gamma$-orbit is closed in $\Perspace$. Consider its stabilizer
$\St(J)\cong H \subset G$. {\bf \red Then $\St(J)\cap \Gamma$
is a lattice in $\St(J)$.}

\corollary 
Let $J$ be a complex structure with closed $\Gamma$-orbit
on a hyperk\"ahler manifold, $\Omega$ its holomorphic
symplectic form, and $W\subset H^2(M,\R)$ a plane generated
by $\Re\Omega, \Im \Omega$. {\bf \red Then $W$ is rational.}

Similarly, one has

\corollary
Let $J$ be a non-ergodic complex structure on a hyperk\"ahler manifold,
and $W\subset H^2(M,\R)$ be a plane generated
by $\Re\Omega, \Im \Omega$. {\bf \red Then $W$
  contains a rational vector.}





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