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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Teichm\"uller spaces for geometric structures,\\
lecture 2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[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:}


1. Hyperk\"ahler manifolds (introduction)

2. Supersymmetry on K\"ahler and hyperk\"ahler manifolds

3. Automorphisms of cohomology

4. Computation of the mapping class group

\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 $I,J,K$.

{\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 (reminder)}


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

\remark In these lectures, 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 these lectures, 
{\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 The Bogomolov-Beauville-Fujiki form}

{\bf \green THEOREM:} (Fujiki). 
Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
hyperk\"ahler. {\bf \red 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 Global Torelli theorem.}


\theorem
Let $M$ be a simple hyperk\"ahler manifold,
and $\Gamma:= \Diff/\Diff_0$ its mapping class group.
Then $H^2(M,\Z)$ admits a $\Gamma$-invariant,
non-degenerate integer quadratic form $q$
such that the natural action of $\Gamma$ on $H^2(M,\Z)$
{\bf \red induces a homomorphism $\Gamma\arrow SO(H^2(M,\Z), q)$
with finite index and finite kernel.}

\remark Suppose that $\phi:\; M \arrow M'$
is a bimeromorphic map of Calabi-Yau manifolds.
Then the exceptional set of $\phi$ has codimension $\geq 2$,
{\bf \red hence $H^2(M)=H^2(M')$.}

\definition
Let $\Teich$ be the Teichm\"uller space of complex
structures of hyperk\"ahler type on $M$, and $\Teich_b$
the quotient of $\Teich$ by an equivalence relation 
induced by bimeromorphic maps $(M,I)\arrow (M,I')$
inducing identity on $H^2(M)$. Then $\Teich_b$ is called
{\bf \blue birational Teichm\"uller space} of $M$.
As we shall see, $\Teich_b$ is a smooth,
Hausdorff complex manifold.

\theorem
Consider the space
\[
\Perspace:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0\}
\]
and let the {\bf \blue period map} 
$\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$
map a complex structure $I$ to a 
line $H^{2,0}(M,I)\in {\Bbb P}H^2(M, \C)$.
{\bf \red Then $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$
induces a bijective map $\Teich_b\arrow \Perspace$}
on any connected component of $\Teich_b$.

\newpage

{\bf \blue Automorphisms of cohomology}

We are going to prove the following theorem.

{\bf \green Theorem 1:} The natural action of
$\Spin(H^2(M,\R), q)$
on $H^2(M)$
{\bf \red is extended to an action on the algebra $H^*(M)$ by automorphisms.}
Moreover, {\bf \red this action preserves the Chern classes.}

\claim
Let $M$ be a compact K\"ahler manifold.
Consider {\bf \blue the Weil operator} $W\in \End(H^*(M))$
acting on $H^{p,q}(M)$ as $W\restrict{H^{p,q}(M)}=\1(p-q)$,
and let $U(1)$ act on $H^*(M)$ as $e^{-\1 W}$. {\bf \red Then 
$U(1)$ acts by automorphisms.}


Theorem 1 is implied by the following result, proven later in this lecture.

\theorem
Consider the Lie subalgebra $\g_W\subset \Aut(H^*(M))$
generated by the Weil operators $W$ for all K\"ahler
structures of K\"ahler type. {\bf \red Then
  $\g_W$ is isomorphic to $\goth{so}(H^2(M,\R), q)$.}




\newpage

{\bf \blue Lie superalgebras}

\definition
A {\bf \blue graded vector space} is a space $V^* =\bigoplus_{i\in \Z} V^i$.

\remark If $V^*$ is graded, the endomorphisms space
$\End(V^*)=\bigoplus_{i\in \Z} \End^i(V^*)$ is also graded, with
$\End^i(V^*)= \bigoplus_{j\in \Z} \Hom(V^j, V^{i+j})$

\definition
An operator on a graded vector space is called {\bf \blue
even} ({\bf \blue odd}) if it shifts the grading by even 
(odd) number. The {\bf\blue parity} $\tilde a$ of an operator
$a$ is 0 if it is even, 1 if it is odd. We say that
an operator is {\bf \blue pure} if it is even or odd.


\definition
A {\bf \blue supercommutator}  of pure operators
on a graded vector space is defined by a formula
$\{a,b\}= ab - (-1)^{\tilde a \tilde b}ba$.

\definition
{\bf \blue A graded Lie algebra} (Lie superalgebra)
is a graded vector space $\g^*$
equipped with a bilinear graded map 
$\{\cdot,\cdot\}:\; \g^*\times \g^* \arrow \g^*$
which is graded anticommutative:
$\{a,b\} = - (-1)^{\tilde a \tilde b}\{b,a\}$
and satisfies {\bf \blue the super Jacobi identity}
$\{c, \{a,b\}\} = \{\{c, a\},b\}+ (-1)^{\tilde a \tilde c}\{a,\{c, b\}\}$

\newpage

{\bf \blue Supersymmetry in K\"ahler geometry }

Let $(M, I, g)$ be a Kaehler manifold, $\omega$ its Kaehler form.
{\bf \blue On $\Lambda^*(M)$, the following operators are defined.}

0. $d$, $d^*$, $\Delta$, because it is Riemannian.

1. $L(\alpha):= \omega\wedge \alpha$

2. $\Lambda(\alpha) := * L * \alpha$. 
It is easily seen that $\Lambda= L^*$.

3. The Weil operator $W\restrict{\Lambda^{p,q}(M)}=\1(p-q)$

\theorem
{\bf \red These operators generate a Lie superalgebra
$\goth a$ of dimension $(5|4)$,} 
acting on $\Lambda^*(M)$. Moreover, the Laplacian $\Delta$ is
central in $\goth a$, hence {\bf \purple $\goth a$ also acts on the
cohomology of $M$. }

\remark This is a convenient way to summarize 
the K\"ahler relations and the Lefschetz $\goth{sl}(2)$-action.

\newpage


{\bf \blue Supersymmetry in hyperk\"ahler geometry}

Let $(M, I, J,K, g)$ be a hyperkaehler manifold, $\omega_I$,
$\omega_J$, $\omega_K$ its Kaehler forms.
{\bf \blue On $\Lambda^*(M)$, the following operators are defined.}

0. $d$, $d^*$, $\Delta$, because it is Riemannian.

1. $L_I(\alpha):= \omega_I\wedge \alpha$

2. $\Lambda_I(\alpha) := * L_I * \alpha$. 
It is easily seen that $\Lambda_I= L^*_J$.

3. Three Weil operators
$W_{I}\restrict{\Lambda^{p,q}(M,I)}=\1(p-q)$,
$W_{ J}\restrict{\Lambda^{p,q}(M,J)}=\1(p-q)$,
$W_{K}\restrict{\Lambda^{p,q}(M,K)}=\1(p-q)$

\theorem
{\bf \red These operators generate a Lie superalgebra
$\goth a$} of dimension $(11|8)$,
 acting on $\Lambda^*(M)$. Moreover, the Laplacian $\Delta$ is
central in $\goth a$, hence {\bf \purple $\goth a$ also acts on the
cohomology of $M$. }

\remark The Weil operators form the Lie algebra
$\goth{su}(2)$ of unitary quaternions. This means that {\bf \blue 
the quaternionic action belongs to $\goth a$}. In particular,
$L_J, L_K, \Lambda_J$ and $\Lambda_K$.

\remark The twisted de Rham differentials 
$d_I, d_J, d_K$, associated to $I,J,K$ also belong to
$\goth a$: {\bf \purple $d_I= [ W_I, d]$, $d_J= [ W_J, d]$, 
$d_K= [ W_K, d]$}

\newpage

{\bf \blue $\goth{so}(4,1)$-action and the Hodge decomposition}

\remark 1.  $[L_I, \Lambda_J]=W_K$,
 $[L_J, \Lambda_K]=W_I$, $[L_I, \Lambda_K]=-W_J$.

2. The even part of $\goth a$
{\bf \red is isomorphic to $\goth{sp}(1,1, {\Bbb H})\oplus\R \cdot \Delta $.}

3. The odd part $\langle d, d_I, d_J, d_K, d,^* d_I^*, d_J^*, d_K^*\rangle$
{\bf \red generates the 9-dimensional odd Heisenberg algebra,} with the
only non-trivial supercommutators being 
$\{d, d^*\}=\{d_I, d^*_I\}=\{d_J, d^*_J\}=\{d_K, d^*_K\}=\Delta$

4. The action of $\goth a_{even}$ on $\goth a_{odd}$
{\bf \purple is the fundamental representation of $\goth{sp}(1,1, {\Bbb H})$ in 
${\Bbb H^2}$,} with the quaternionic Hermitian 
metric on $\goth a_{odd}$ provided
by the anticommutator.

\corollary
The weight decomposition of the $\goth{sp}(1,1, {\Bbb H})=
\goth{so}(4,1)$-action on $H^*(M)$ 
{\bf \purple coincides with the Hodge decomposition.}

%\newpage
%
%{\bf \blue Lefschetz-Frobenius algebras}
%
%\definition
%{\bf \blue A Frobenius algebra} is
%a graded commutative algebra $A =\bigoplus_{i=0}^{d} A^i$  equipped with 
%the Poincare-type non-degenerate product.
%
%\definition
%A {\bf\blue Lefschetz triple}
%in a Frobenius algebra $A =\bigoplus_{i=0}^{2n} A^i$ is a triple
%of operators $L_\eta, H, \Lambda_\eta$
%where $\eta \in A^2$ is a fixed element, 
%$L_\eta(x):= \eta \wedge x$, $H\restrict{A^i} = i-n$ and
%$\Lambda_\eta$ is an element such that
%$L_\eta, H, \Lambda_\eta$ is an $\goth{sl}(2)$-triple.
%A Frobenius algebra admitting a Lefschetz triple
%is called {\bf \blue a Lefschetz-Frobenius algebra}
%(Looijenga, Lunts).
%
%\remark
%Such $\Lambda_\eta$ {\bf \purple is uniquely
%determined by $H$ and $\eta$} (this statement is sometimes
%called ``Morozov's lemma'', and sometimes included
%in the statement of Jacobson-Morozov theorem).
%
%
%\remark
%Existence of $\Lambda_\eta$
%for given $\eta\in A^2$ is an open property in $A^2$,
%hence {\bf \purple a Lefschetz-Frobenius algebra admits 
%many $\goth{sl}(2)$-triples. }

\newpage

{\bf \blue Lia algebra $\g$ generated by $\goth{sl}(2)$-triples }

\theorem
Let $M$ be a hyperk\"ahler manifold of maximal holonomy, $A^*$ its
cohomology algebram and $\g:=\g(A)$ the Lie algebra 
generated by all Lefschetz $\goth{sl}(2)$-triples.
{\bf \red Then $\g$ is isomorphic to $\goth{so}(b_2 -2,4)$.}

{\bf \green Sketch of the proof. Step 1:}
Consider the action of $\g$ on the {\bf\blue Mukai extension}
$\hat H^2(M):= \R\cdot x \oplus H^2(M) \oplus \R\cdot y,$
where $x$ has grading 0, $y$ has grading 4, $H^2(M)$
has grading $2$. We equip $\hat H^2(M)$
 with {\bf\blue the Mukai form}  which is equal 
to BBF on $H^2(M)$, preserves grading, 
and satisfies $q_M(x, y)=1$ $q(x,x)=q(y,y)=0$, $x, y \bot H^2(M)$  and $(x,y)=1$. 
The action of $\g$ on $\hat H^2(M)$
is determined by the following properties: 
{\bf \purple 1. It is compatible with the grading. 
 2. For all $\alpha, \beta\in H^2(M)$, one has
$L_\alpha x = \alpha$, $L_\alpha \beta = q(\alpha, \beta)y$,
 where $q$ is the BBF form. 3. $\Lambda_\alpha y = \alpha$, $\Lambda_\alpha \beta = q(\alpha, \beta)x$.}


To see that this action is well-defined, we need to check that
commutator relations hold. This follows  from 
commutator relations in $\goth{so}(1,4)$ and Zariski density
of pairs $\alpha, \beta \in \langle \omega_I, \omega_J, \omega_K\rangle$
in the set of all pairs $\alpha, \beta\in H^2(M)$. 

{\bf\green Step 2:} The map $\g \arrow \goth{so}(\hat H^2(M))$
is surjective, which follows from the dimension argument
(dimensions are computed using the local Torelli theorem).
 Injectivity of $\g \arrow \goth{so}(\hat H^2(M))$
is clear, because $\goth{so}(\hat H^2(M))$ is given by generators
and relations which hold true in $\g$.
\endproof

\newpage

{\bf\blue Hodge structures and $\g$-action}


\remark 
The Lie algebra $\g=\goth{so}(b_1 -2,4)$ is equipped with a grading
$\g=\g_{-2}\oplus \g_0\oplus g_2$, induced by the grading on the
Mukai space: $\hat H^2(M):= H_0 \oplus H^2(M) \oplus H_4$,
with $H_0$ and $H_4$ 1-dimensional. Then 
$\g_0= \g_0'\oplus H$, where $H=[L_\omega,\Lambda_\omega]$
is the operator inducing the grading and commuting with 
the rest of $\g_0$, denoted by $\check \g_0$. 

\remark {\bf \purple The Lie algebra $\g_0':=\goth{so}(b_2 -3,3)$
is generated by the Weil maps $W_I$ for all complex
structures $I$ of hyperk\"ahler type.}
The corresponding Lie group $G_0'$ acts as $\Spin(b_2 -3,3)$
in odd-dimensional cohomology and $SO(b_2 -2,3)$ on 
even-dimensional ones.

\corollary The natural action of
$\g_0':=\goth{so}(b_2 -3,3)= \goth{so}(H^2(M,\R), q)$
on $H^2(M)$ {\bf \red is extended to an action on the algebra $H^*(M)$ by automorphisms.}
Moreover, {\bf \red this action preserves the Chern classes.}

{\bf \green This finishes the proof of Theorem 1.}

\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.


{\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 Yang-Mills bundle with $c_1(B)=0$.


{\bf \green Step 3:} ${\goth {so}}(H^2(M, \R), q)$
acts on $H^*(M, \R)$ by derivations preserving 
Pontryagin classes (Theorem 1). 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, which is positive definite.
\endproof 

\newpage

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


{\bf \green Theorem:} {\bf \blue (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.}

\proof (i) is clear from the $\Spin(H^2(M,\R))$-action on
$H^*(M)$, and (ii) follows because the 
kernel $K$ of the map
$\Aut(H^*(M)) \arrow G\restrict{H^2(M,\R)}$ is compact, hence
the discrete group $\Gamma_0\cap K$ is finite.
\endproof

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


\end{document}


