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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Teichm\"uller spaces for geometric structures,\\
lecture 3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[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 Plan:}

0. Hyperk\"ahler geometry and Teichm\"uller spaces (reminder)

1. Period space and birational Teichm\"uller space

2. Global Torelli theorem

3. Subtwistor metrics and the proof of global Torelli

\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}

{\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 $M$.

\remark 
${\Bbb Per}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$

{\bf \green THEOREM:} {\bf \blue (Bogomolov)}
Let $M$ be a simple hyperk\"ahler manifold,
and $\Teich$ its Teichm\"uller space. Then
 {\bf \red The period map $P:\; \Teich \arrow {\Bbb Per}$ is etale.}\\

\remark Bogomolov's theorem implies that
{\bf \purple $\Teich$ is smooth.} It is {\bf \red non-Hausdorff}
even in the simplest examples.

\newpage


{\bf \blue Hausdorff reduction}

\remark {\bf \blue A non-Hausdorff manifold} is a topological
space locally diffeomorphic to $\R^n$.

\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 $M$.

{\bf \green Problems:} \\
1. {\bf \red $\sim$ is not always an equivalence relation.\\
2. Even if $\sim$ is equivalence, the $M/\sim$ is not always Hausdorff.}

\remark 
{\purple A quotient $M/\sim$ is Hausdorff, if $M \arrow M/\sim$ is open,
and the graph $\Gamma_\sim \in M\times M$ is closed.}

\theorem 
The Teichm\"uller space of a hyperk\"ahler manifold
{\bf \red admits a Hausdorff reduction}.

\newpage


{\bf \blue Global Torelli theorem}

\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 Proven later today.}

\newpage


{\bf \blue Period space as $\Gr_{++}(3, b_2-3)$}

\proposition
The period space 
\[ {\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ | \  q(l,l)=0, q(l, \bar l) >0\}\]
{\bf \red is identified with $\frac{SO(3,b_2-3)}{SO(2) \times SO(1,b_2-3)}$,} which
is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$.

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

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

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


\newpage

{\bf \blue Period space and hyperk\"ahler lines}

\definition
Let $(M,I,J,K)$ be a hyperk\"ahler manifold.
{\bf \blue A hyperk\"ahler 3-plane} in $H^2(M,\R)$ is a positive oriented
3-dimensional subspace $W$, generated by $\omega_I, \omega_J, \omega_K$.


\remark
The set of oriented 2-dimensional planes in $W$ is identified 
with $S^2 =\C P^1$. It is called {\bf \blue the twistor family}
of a hyperk\"ahler structure. A point in the twistor family
corresponds to a complex structure $aI + bJ + cK \in {\Bbb H}$,
with $a^2+b^2+c^2=1$. We call the corresponding $\C P^1\subset \Teich$
{\bf \blue the twistor curves}.

\remark
Let $I\in \Teich$ be a complex structure, and ${\cal K}(I)$
its K\"ahler cone. The set of twistor curves
passing through $I$ {\bf \red
is parametrized by ${\cal K}(I)$,} by Calabi-Yau theorem.
{\bf \purple The corresponding 3-dimensional subspaces are 
generated by $\Per(I)+\omega$, where $\omega\in {\cal K}(I)$.}

\newpage

{\bf \blue A K\"ahler cone for generic hyperk\"ahler manifolds}

\newcommand{\NS}{\operatorname{\sf NS}}
\newcommand{\Pos}{\operatorname{\sf Pos}}

\definition
{\bf \blue Neron-Severi lattice}, or {\bf\blue Hodge lattice}
of a manifold is $\NS(M):=H^{1,1}(M)\cap H^2(M,\Z)$

\theorem
Let $M$ be a hyperkaehler manifold with 
$\NS(M)$ of rank 0 or 1 and generated by $z$ with $z^2 > 0$. 
{\bf \red Then the K\"ahler cone is
equal to the positive cone $\Pos(M)$,} that is, one of two components of the set 
\[ \{ \nu \in H^{1,1}(M) \ | \ q(\nu, \nu)> 0\}.
\]
{\bf \green Proof:} Nakai-Moischezon-Huybrechts-Boucksom:
obstructions to K\"ahlerness of $\eta\in \Pos(M)$ are  curves
$C$ such that $\int_C \eta \leq 0$. For any non-zero $x, y \in \Pos(M)$,
such that $[C]=x^{\dim_\C M-1},$ one has $\int_Cy= \int_M x^{\dim_\C M-1}y>0$.
\endproof

\newpage

{\bf \blue Generic hyperk\"ahler lines}

\definition
Let $S\subset \Teich$ be a $\C P^1$ 
associated with a twistor family. It is called {\bf \blue
generic} if it passes through a point $I\in \Teich$
with $\NS(M,I)=0$. 

\remark 
For a generic point $I$ in such $S$, one has $\NS(M,I)=0$.
{\bf \purple This condition is equivalent to $l^\bot \cap H^2(M,\Z)=0$,}
where $l\in \Perspace$ is the corresponding 2-plane.

\remark 
A 3-plane $W\subset H^2(M,\R)$
corresponds to a generic twistor family {\bf \purple if and only if
its orthogonal complement $W^\bot\subset H^2(M,\R)$
does not contain rational vectors. }

\definition
A hyperk\"ahler 3-plane $W\subset H^2(M,\R)$
is called {\bf \blue generic} if $W^\bot\cap H^2(M,\Z)=0$.
The corresponding $\C P^1\subset \Perspace$ in the 
period space is called {\bf \blue a GHK line}.

\newpage

{\bf \blue A lifting property for GHK lines}

\remark
Consider a 3-plane  $W= \langle \omega_I,\omega_J,\omega_K\rangle$
associated with a hyperk\"ahler structure, and let $S$ be the
set of oriented 2-planes in $W$. 
Denote by $S_{ng}$ the set of $x\in S$ satisfying 
$x^\bot \cap H^2(M,\Z)\neq 0$. If $W$ is generic, then
$S_{ng}$ is countable.


\theorem {\bf \blue (A lifting property for GHK lines)}\\
Let $W\subset H^2(M,\R)$ be a generic 3-plane, and
$S\subset \Perspace$ the corresponding GHK line.
Consider the period map $P:\; \Teich \arrow \Perspace$.
{\bf \red Then $P^{-1}(S)$ is a union of a countable
set mapped to $S_{ng}$, and a disconnected
set of rational curves bijectively mapped to $S$.}

{\bf \green Proof. Step 1:} Let $x\notin S_{ng}$
We are going to prove that for all
$I\in P^{-1}(x)$, {\bf \purple $y$ is contained in 
a connected component of $P^{-1}(S)$, 
 bijectively mapped to $S$.} 

{\bf \green Step 2:} Notice that $NS(I)=x^\bot \cap H^2(M,\Z)=0$.
Therefore the K\"ahler cone of $(M,I)$ {\bf \purple is one of two components
of the set $\{\omega \in P(I)^\bot\ \ |\ \ q(\omega,\omega)>0\}$.}

{\bf \green Step 3:}
For each positive 3-plane $W\subset H^2(M,\R)$,
$W= \langle \omega_I,\omega_J,\omega_K\rangle$ for some
hyperk\"ahler structure $I,J,K$. {\bf \purple Then the twistor family
associated with $I,J,K$ is mapped to $S$.} \endproof


\newpage

{\bf \blue Subtwistor metric on the Teichm\"uller space}

\definition
Let $g_0$ be a Riemannian metric on $\Perspace$,
and $g$ its lift to $\Teich$.
Define {\bf \blue the subtwistor metric} $d$
as the distance function $d(x,y)$ given by 
infimum of the length (in $g$) for all
paths from $x$ to $y$ going through GHK curves
which intersect in points $z\in \Teich$ with $\NS(z)=0$.
Define the subtwistor metric on $\Perspace$ in the same way,
by using paths which go through $S^2$ obtained
from 3-dimensional subspaces $W\subset H^2(M,\R)$
containing no rational vectors.

\theorem
{\bf \red 
The subtwistor metric induces the standard topology} on 
any open subset $W\subset \Teich_b$.

{\bf \green Remark:} Its proof {\bf \purple follows from 
  Gleason-Palais-Montgomery}
classification of continuous groups.

\theorem
    {\bf \red The period map induces an isometry} from
    $\Teich_b$ to $\Perspace$
with respect to the subtwistor metric.

\proof Immediately follows from the GHK lifting property.
\endproof

\corollary
    {\bf \red The map $\Teich_b\stackrel \Per \arrow \Perspace$ is a homeomorphism.}

{\bf \green    This proves the Global Torelli theorem.}



\end{document}


