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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Global Torelli theorem for hyperk\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf 
Master class: Around Torelli's theorem for K3 surfaces\\[2mm]
Arithmetic, Geometric and Dynamical aspects\\[2mm]
Institut de Recherche 
de Math\'ematique Avanc\'ee, Strasbourg\\ October 28-November 1 2013
}
\end{center}


\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 $H^1(M)=0$, $H^{2,0}(M)=\C$.

{\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\green Remark:} A simple hyperk\"ahler manifold
is always simply connected (Cheeger-Gromoll theorem). 

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

\newpage

{\bf \blue The Teichm\"uller space and the mapping class group}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}

{\bf \green Definition:} Let $M$ be a compact complex manifold, and 
$\Diff_0\subset \Diff$ 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
$\Teich:=\Comp/\Diff_0$. We call 
it {\bf \blue the Teichm\"uller space.}

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

{\bf \green Definition:} We call $\Gamma:=\Diff/\Diff_0$ {\bf \blue the
mapping class group}. The {\bf \blue moduli space of complex
structures on $M$} is a connected component of $\Teich/\Gamma$,
but this quotient is not always well-defined.

{\bf \green Remark:} This terminology is {\bf \purple standard for curves.}

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

\remark Two ingredients of global Torelli theorem:\\
{\bf \green 
(a) determine the mapping class group\\ (b) determine the Teichm\"uller space.}

\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  Lefschetz $\goth{sl}(2)$-action}

Let $(M, I, g)$ be a Kaehler manifold, $\omega$ its Kaehler form.
Consider the following operators.\\
1. $L(\alpha):= \omega\wedge \alpha$.\\
2. $\Lambda := L^*$ (Hermitian dual of $L$).\\
3. The Weil operator $W_I\restrict{\Lambda^{p,q}(M)}=\1(p-q)$

\claim
{\bf \red The triple $L, \Lambda, H$ 
satisfies the relations for the $\goth{sl}(2)$
Lie algebra:} 
$[L,\Lambda]=H$, $[H,L]=2L$, $[H,\Lambda]=2\Lambda$.

\definition
$L, \Lambda, H$ is 
called {\bf \blue the Lefschetz $\goth{sl}(2)$-triple}.

\theorem
The ${\goth sl}(2)$-action $\langle 
L, \Lambda, H\rangle$ and the action of Weil operator
commute with Laplacian, hence {\bf \red preserve the harmonic
forms on a K\"ahler manifold}.

\corollary 
Any cohomology class can be represented as
a sum of closed $(p,q)$-forms, {\bf \purple giving a decomposition
$H^i(M) = \bigoplus_{p+q=i}H^{p,q}(M)$, with
$\overline{H^{p,q}(M)} = H^{q,p}(M)$. }


\newpage

{\bf \blue Riemann-Hodge pairing}


\theorem 
Let $(M,\omega)$ be a K\"ahler $n$-manifold.
Consider the following pseudo-Hermitian form on $H^d(M)$:
\[ 
B(\alpha, \beta):= \int_M\alpha\wedge \bar \beta\wedge \omega^{n-d}
\]
{\bf \red 
Then $B$ is sign-definite on a space $H^{p,q}_k(M)$ of $(p,q)$-forms which
have weight $k$ with respect to Lefschetz $SL(2)$-action.}

\corollary 
Let $G$ be a group of automorphisms of the algebra $H^*(M,\R)$
preserving the $(p,q)$-decomposition and fixing a
K\"ahler class $\omega$. {\bf \red Then $G$ is compact.}

{\bf \green Proof:} Since $G$ fixes $\omega$, $G$ commutes
with the Lefschetz $SL(2)$-action, hence it fixes a sign-definite
form on each space $H^{p,q}_k(M)$. \endproof

\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.

{\small {\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 non-trivial stable 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 commutes with the Hodge decomposition and
 Lefschetz ${\goth sl}(2)$-action, hence preserves
the Riemann-Hodge form.
\endproof 

\newpage

{\bf \blue Sullivan's theorem}


{\bf \green Theorem:} (Sullivan) 
Let $M$ be a compact, simply connected 
K\"ahler manifold, $\dim_\C M\geq 3$. Denote by $\Gamma_0$ the group
of automorphisms of an algebra $H^*(M, \Z)$
preserving the Pontryagin classes $p_i(M)$. 
Then {\bf \red the natural map 
$\Diff(M)/\Diff_0\arrow \Gamma_0$ has finite kernel,
and its image has finite index in $\Gamma_0$.}

{\bf \green Theorem:}
Let $M$ be a simple hyperk\"ahler manifold,
and $\Gamma_0$ as above.  Then \\
(i)  $\Gamma_0\restrict{H^2(M,\Z)}$ {\bf \blue is a finite index 
subgroup of $O(H^2(M, \Z), q)$.}\\
(ii) The map $\Gamma_0\arrow O(H^2(M, \Z), q)$
{\bf \blue has finite kernel.}


{\bf \green Proof:} Follows from 
the computation of $G=\Aut(H^*(M, \R), p_1, ..., p_n)$
done earlier. Indeed, the kernel of $\Gamma_0\restrict{H^2(M,\Z)}$
is a set of integer points of a compact Lie group, hence finite.
The image of $\Gamma_0=G_\Z$ has finite index in 
$O(H^2(M, \Z), q)$, because the corresponding map of Lie groups
is surjective.
 \endproof

\newpage

{\bf \blue Computation of the mapping class group}

\corollary
The mapping class group {\bf \red $\Gamma$ is mapped to 
$O(H^2(M, \Z), q)$ with finite kernel and finite index.}

{\bf \green Proof:} By Sullivan, $\Gamma$ is mapped to $\Gamma_0$
with finite kernel and finite index, and $\Gamma_0\arrow O(H^2(M, \Z), q)$
has finite kernel and finite index, as shown above.
\endproof

\theorem (Kollar-Matsusaka, Huybrechts) 
{\bf \purple There are only finitely many connected components } of $\Teich$.

\corollary Let $\Gamma_I$ be the group of elements of mapping class 
group preserving a connected component of Teichm\"uller 
space containing $I\in \Teich$. {\red \bf Then $\Gamma_I$ 
has finite index in $\Gamma$}.

\remark $\Gamma_I$ is a group generated by monodromy of all
Gauss-Manin local systems for all deformations of $(M,I)$.
It is known as {\bf \blue the monodromy group} of $(M,I)$.

\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}.


\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}

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

\remark 
$\Perspace=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$

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

\remark Bogomolov's theorem implies that
{\bf \purple $\Teich$ is smooth.} It is {\bf \red usually non-Hausdorff}.

\newpage

{\bf \blue Birational equivalence and non-separable points}

\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:} (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  $(M, I_2)$.

\corollary
{\bf \red The set of all non-separable points in $\Teich$ belongs
to a union of countably many divisors.}

{\bf \green Proof. Step 1:} Let $z\in H^2(M,\Z)$ be a cohomology class,
and $\Teich_z$ the set of all $I\in \Teich$ such that
$z\in H^{1,1}(M,I)$. The relation $z\in H^{1,1}(M,I)$
is equivalent to $q(z,\Omega)=0$, hence 
\[
\Per(\Teich_z)\subset 
\{l\in {\Bbb P}(z^\bot)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0.
\]
{\bf \purple This shows that $\Per(\Teich_z)$ is a divisor.}

{\bf \green Step 2:} If $(M,I)$ is a holomorphic
symplectic manifold with non-trivial birational equivalence to $M'$,
it contains a rational curve $C$. {\bf \purple Therefore, $I\in \Teich_z$,
where $z=[C]$ is its homology class.} \endproof

\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 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.}

\newpage

{\bf \blue Weakly Hausdorff manifolds}

\definition A point $x\in X$ is called {\bf\blue Hausdorff}
if $x\not\sim y$ for any $y\neq x$.

\definition
Let $M$ be an $n$-dimensional 
real analytic manifold, not necessarily Hausdoff. 
Suppose that {\bf \purple the set $Z\subset M$ of non-Hausdorff points
is contained in a countable union of real analytic subvarieties}
of $\codim\geq 2$. Suppose, moreover, that

(S) For every $x\in M$, there is a 
closed neighbourhood $B\subset M$ of $x$ and a 
continuous surjective map 
$\Psi:\; B \arrow \R^n$ to a closed ball in $\R^n$,
{\bf \purple inducing a homeomorphism}  on an open neighbourhood of $x$.

Then $M$ is called {\bf\blue a weakly Hausdorff 
manifold}.

\remark {\bf \purple The period map satisfies (S)}. Also, 
the non-Hausdorff points of $\Teich$ {\bf \purple are contained 
in a countable union of divisors.}

\theorem A {\bf \red  weakly Hausdorff manifold $X$ admits a Hausdorff
reduction.} In other words, the quotient $X/\sim$ is a Hausdorff.
Moreover, $X \arrow X/\sim$ is locally a homeomorphism.

This theorem is proven using 1920-ies style point-set topology.

\newpage


{\bf \blue Birational Teichm\"uller moduli space}

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

\theorem {\bf \red The period map 
$\Teich_b\stackrel \Per \arrow \Perspace$ is an isomorphism,}
for each connected component of $\Teich_b$.

The proof is based on two results.

\proposition {\bf \blue (The Covering Criterion)}
Let $X\stackrel \phi \arrow Y$ be an etale map of smooth manifolds.
Suppose that each $y\in Y$ has a neighbourhood $B\ni y$ diffeomorphic
to a closed ball, such that  for each connected component
$B' \subset \phi^{-1}(B)$, $B'$ projects to $B$ surjectively.
{\bf \red Then $\phi$ is a covering.}

\proposition {\bf \red The period map satisfies the conditions of the
Covering Criterion.}





\newpage

{\bf \blue Global Torelli theorem}

\definition
Let $M$ be a hyperkaehler manifold,
$\Teich_b$ its birational Teichm\"uller space,
and $\Gamma$ the mapping class group.
The quotient $\Teich_b/\Gamma$ is called
{\bf \blue the birational moduli space} of $M$.

\remark 
The birational moduli space is obtained from the
usual moduli space {\bf \purple by gluing some (but not all) 
non-separable points.} {\bf \red It is still 
non-Hausdorff.}

\theorem 
Let $(M,I)$ be a hyperk\"ahler manifold, and $W$ 
a connected component of its birational
moduli space. {\bf \blue Then $W$ is isomorphic to ${\Perspace}/\Gamma_I$,
where ${\Perspace}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$
and $\Gamma_I$ is 
an arithmetic subgroup in $O(H^2(M, \R), q)$}.


{\bf \green A CAUTION:} Usually ``the global Torelli theorem''
is understood as a theorem about Hodge structures.
For K3 surfaces, {\bf \purple the Hodge structure on $H^2(M,\Z)$
determines the complex structure}. 
For $\dim_\C M >2$, {\bf \red it is false}.

\newpage

{\bf \blue The marked moduli space}

\definition
Let $\Gamma$ be the mapping class group, and
$K\subset \Gamma$ the kernel of the natural map
$\Gamma \arrow GL(H^2(M, \Z))$. {\bf \red It is finite,} as we have shown.
The quotient $\Teich/K$ is called {\bf\blue the marked
moduli space}.

\theorem
The natural map $\Teich \arrow \Teich/K$
{\bf \red is a homeomorphism on each connected component.}

{\bf \green Proof. Step 1:}\\
Let $I\in \Teich$ be a fixed point of a subgroup $K_I\subset K$.
By Bogomolov's theorem,
$T_I\Teich$ is naturally identified with $H^{1,1}_I(M)$.
Since the action of $K$ on $H^2(M)$ is trivial,
any $\alpha\in K_I$ acts trivially on $T_I\Teich$.
 Therefore, {\bf \purple $K_I$ acts as identity
on a connected component of $\Teich$ containing $I$}. 

{\bf \green Step 2:} 
From Step 1, obtain that {\bf \red the quotient map $\Teich\stackrel\Psi \arrow \Teich/K$
is a finite covering,} hence {\bf \purple it induces a finite covering
of the corresponding Hausdorff reductions.} However,
$\Psi$ induces an isomorphism on each connected
component of $\Teich_b$, because 
{\bf \red each component of $\Teich_b$ 
is isomorphic to $\Perspace$.} \endproof


\newpage

{\bf \blue The Hodge-theoretic Torelli theorem}

\remark The group $O(p,q)$ ($p, q >0$) has {\bf \blue 4 connected
components}, corresponding to the orientations of positive
$p$-dimensional and negative $q$-dimensional planes.


\definition
Let $M$ be a hyperkaehler manifold. One says that
{\bf \blue the Hodge-theoretic Torelli theorem holds
for $M$} if
\[ \Teich/\Gamma_I\arrow {\Perspace}/O^+(H^2(M, \Z), q),
\]
where $O^+(H^2(M, \Z), q)$ is a subgroup of $O(H^2(M, \Z), q)$
preserving orientation on positive 3-planes.
Equivalently, {\bf \purple it is true if $M$ is uniquely determined
by its Hodge structure}.

\remark
 The Hodge-theoretic Torelli theorem {\bf\purple is true
for K3 surfaces}. {\bf \red It is false} for all other known examples
of hyperkaehler manifolds.

{\bf \green Problems:} \\
1. The moduli space $\Teich/\Gamma$
{\bf \blue is not Hausdorff} (Debarre, 1984). Indeed,  bimeromorphically
equivalent hyperk\"ahler manifolds have isomorphic Hodge structures.\\
2. {\bf \red The covering $\Teich_b/\Gamma_I\arrow {\Perspace}/O^+(H^2(M, \Z), q)$
is non-trivial}, because the map $\Gamma_I\arrow O^+(H^2(M, \Z), q)$
is not surjective (Namikawa, 2002). 

\newpage

{\bf \blue The birational Hodge-theoretic Torelli theorem}


\definition
{\bf \blue The birational Hodge-theoretic Torelli theorem
is true for $M$} if $\Gamma_I$ (the stabilizer
of a Torelli component in the mapping class group) {\bf \purple is
isomorphic to $O^+(H^2(M, \Z), q)$.}

\remark 
If a birational Hodge-theoretic Torelli theorem 
holds for $M$, then any deformation of $M$ is 
up to a bimeromorphic equivalence {\bf \red determined by the
Hodge structure on $H^2(M)$.}

\theorem (Markman) The for $M=K3^{[n]}$, the group $\Gamma_I$
{\bf \blue is a subgroup of $O^+(H^2(M, \Z), q)$ generated by
oriented reflections.}

\theorem
Let $M=K3^{[n+1]}$ with $n$ a prime power.
Then the (usual) global Torelli theorem holds birationally:
{\bf \blue two deformations of a Hilbert scheme with isomorphic
Hodge structures are bimeromorphic}. {\bf \red For other $n$,
it is false} (Markman).



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