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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Hyperbolic geometry and the proof of Morrison-Kawamata cone conjecture (1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\scriptsize Misha Verbitsky } 
\\[20mm]
  {\tiny\bf Complex Geometry: discussion meeting \\[3mm]
20 March 2017 to 25 March 2017\\[3mm]
Ramanujan Lecture Hall, ICTS, Bengaluru
  }
\end{center}

\newpage

{\bf \blue K\"ahler manifolds}

{\bf\green DEFINITION:} An Riemannian metric $g$ on
an almost complex manifiold $M$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.

{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

\remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}.

{\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$
is called {\bf \blue K\"ahler} if $d\omega=0$. 
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$, and
$\omega$ {\bf \blue the K\"ahler form}. The set of all K\"ahler
classes is called {\bf\blue K\"ahler cone}.

\newpage

{\bf \blue Hyperk\"ahler manifolds}


\definition {\bf \blue (E. Calabi, 1978)}\\ Let $(M, g)$ be a Riemannian
manifold equipped with three complex structure operators
$I, J, K:\; TM\arrow TM$, satisfying the quaternionic relation
$I^2=J^2=K^2=IJK=-\Id.$  Suppose that $I$, $J$, $K$ are
K\"ahler. Then $(M, I, J, K, g)$ is called {\bf \blue hyperk\"ahler}.

\claim A hyperk\"ahler manifold $(M,I,J,K)$
is {\bf \red  holomorphically symplectic} (equipped
with a holomorphic, non-degenerate 2-form). Then $M$ is
equipped with 3 symplectic forms $\omega_I$, $\omega_J$, 
$\omega_K$.

\lemma The form 
$\Omega:= \omega_J+\1\omega_K$ {\bf \purple is a holomorphic symplectic 2-form on
$(M,I)$.} \endproof

\theorem (Calabi-Yau, 1978)
Let $M$ be a compact, holomorphically symplectic K\"ahler
manifold. Then {\bf \red $M$ admits a hyperk\"ahler metric,} which is
uniquely determined by the cohomology class of its 
K\"ahler form $\omega_I$.



\newpage

\begin{center}
{\epsfig{file=Calabi.jpg,width=0.40\linewidth}} \\
{\it\green Eugenio Calabi, \\ born 11 May 1923}
\end{center}


\newpage

{\bf \blue Levi-Civita connection and K\"ahler geometry}


\definition
Let $(M, g)$ be a Riemannian manifold. A connection $\nabla$ 
is called {\bf \blue orthogonal} if $\nabla(g) =0$.
It is called {\bf \blue Levi-Civita} if it is torsion-free.

\theorem (``the main theorem of differential geometry'')\\
{\bf \red For any Riemannian manifold, the
Levi-Civita connection exists,\\ and it is unique}.

{\bf \green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian
manifold. {\bf \purple Then the following conditions are equivalent.}

(i) {\bf \red  $(M,I,g)$ is K\"ahler}

(ii) One has {\red $\nabla(I)=0$,} where $\nabla$ is the Levi-Civita connection.


\newpage


{\bf \blue Holonomy group}

\definition (Cartan, 1923)
Let $(B,\nabla)$ be a vector bundle with connection over $M$.
For each loop $\gamma$ based in $x\in M$, let 
$V_{\gamma, \nabla}:\; B\restrict x \arrow B\restrict x$
be the corresponding parallel transport along the connection.
The {\bf \blue holonomy group} of $(B,\nabla)$
is a group generated by $V_{\gamma, \nabla}$,
for all loops $\gamma$. If one takes all contractible
loops instead, $V_{\gamma, \nabla}$ generates
{\bf \blue the local holonomy}, or {\bf \blue
the restricted holonomy} group.

\remark A bundle is {\bf \blue flat} (has vanishing curvature)
{\bf\purple if and only if its restricted holonomy vanishes.}

\remark If $\nabla(\phi)=0$ for some tensor 
$\phi\in B^{\otimes i}\otimes (B^*)^{\otimes j}$,
{\bf \red the holonomy group preserves $\phi$.}

\definition {\bf \blue Holonomy of a Riemannian manifold}
is holonomy of its Levi-Civita connection.

\example Holonomy of a Riemannian manifold lies in
$O(T_x M, g\restrict x)=O(n)$.

\example  Holonomy of a K\"ahler manifold lies in
$U(T_x M, g\restrict x, I \restrict x)=U(n)$.

\remark The holonomy group {\bf \red does not depend
on the choice of a point $x\in M$.}


\newpage 

{\bf \blue The Berger's list}

\theorem {\bf \blue(de Rham)}\\ A complete, simply connected  
Riemannian manifold with non-irreducible holonomy 
{\bf \red splits as a Riemannian product.}

\theorem {\bf \blue (Berger's classification of holonomies, 1955)}\\
Let $G$ be an irreducible holonomy group of a
Riemannian manifold which is not locally symmetric. {\bf \red Then
$G$ belongs to the Berger's list:}

{
\begin{tabular}{|l|l|}
\hline
\multicolumn{2}{|c|}{\bf \color[rgb]{0,0,0.6}Berger's list}\\[1mm]
\hline
\it Holonomy  & \it Geometry\\[1mm]
\hline
$SO(n)$ acting on $\R^n$ & Riemannian manifolds\\[1mm]
\hline
$U(n)$ acting on $\R^{2n}$ & K\"ahler manifolds\\[1mm]
\hline
$SU(n)$ acting on $\R^{2n}$, $n>2$ & Calabi-Yau manifolds\\[1mm]
\hline
$Sp(n)$ acting on $\R^{4n}$ & hyperk\"ahler manifolds\\[1mm]
\hline
$Sp(n)\times Sp(1)/\{\pm 1\}$ & 
quaternionic-K\"ahler\\[1mm] acting on $\R^{4n}$, $n>1$ &  manifolds\\[1mm]
\hline
$G_2$ acting on $\R^7$ & $G_2$-manifolds \\[1mm]
\hline
$Spin(7)$ acting on $\R^8$ & $Spin(7)$-manifolds\\[1mm]
\hline
\end{tabular}
}

\newpage

\begin{center}
{\epsfig{file=berger.jpg,width=0.40\linewidth}} \\
{\it\green Marcel Berger, \\ 14 April 1927 -- 15 October 2016}
\end{center}


\newpage

{\bf \blue Subject of these lectures}

1. Determine the shape of the K\"ahler cone of a hyperk\"ahler manifold.
It turns out that it is determined by a quadratic inequality and
a set of linear inequaities associated with the rational curves.

2. Interpret various quantities associated with a hyperk\"aher
manifold, such as its automorphism group and its moduli space,
im terms of the shape of the K\"ahler cone.

3. Associate a hyperbolic manifold to each hyperk\"ahler manifold.
Interpret the statement about the shape of the K\"ahler cone
as a statement about this manifold.

4. Using ergodic theory and hyperbolic geometry, prove that
the group of holomorphic automorphisms of a hyperk\"ahler manifold
acts on the polyhedral faces of its K\"ahler cone with finitely
many orbits (``Morrison-Kawamata cone conjecture'').

{\bf \red The results are obtained in  
a serie of joint papers with Ekaterina Amerik.}

\newpage

{\bf \blue Calabi-Yau manifolds}

\definition\\
{\bf\blue A Calabi-Yau manifold} is a compact 
Kaehler manifold with $c_1(M,\Z)=0$.

\newcommand{\Ric}{\operatorname{Ric}}
\definition Let $(M,I, \omega)$ be a Kaehler $n$-manifold, and
$K(M):= \Lambda^{n,0}(M)$ its {\bf \blue canonical bundle.} We consider
$K(M)$ as a holomorphic line bundle, $K(M)= \Omega^n M$.
Denote by $\Theta_K$ the curvature
of the connection on $K(M)$ induced by Levi-Civita connection.
The {\bf\blue Ricci curvature} $\Ric$ of $M$ is a symmetric
2-form $\Ric(x,y)= \Theta_K(x, Iy)$.

\definition
A K\"ahler manifold is called {\bf \blue Ricci-flat}
if its Ricci curvature vanishes. 

\theorem
(Calabi-Yau) \\
Let $(M, I, g)$ be Calabi-Yau manifold. {\bf \red Then there exists
a unique Ricci-flat Kaehler metric in any given
Kaehler class.}

\remark 
Converse is also true: {\bf \purple any Ricci-flat K\"ahler manifold 
has a finite covering which is Calabi-Yau.} This is due to Bogomolov.

\newpage


{\bf \blue Bochner's vanishing}

\theorem
(Bochner vanishing theorem)
On a compact Ricci-flat Calabi-Yau manifold, {\bf \red any holomorphic
$p$-form $\eta$ is parallel} with respect to the Levi-Civita connection:
$\nabla(\eta)=0$.

\remark Its proof is based on spinors: $\eta$ gives a harmonic spinor,
and {\bf \purple on a Ricci-flat Riemannian spin manifold, any harmonic spinor
is parallel.}

\definition
A {\bf \blue holomorphic symplectic manifold} is a manifold
admitting a non-degenerate, holomorphic symplectic form.

\remark 
A holomorphic symplectic manifold is Calabi-Yau.
The top exterior power of a holomorphic symplectic form 
{\bf \purple is a non-degenerate section of canonical bundle.}


\newpage


{\bf \blue Hyperk\"ahler manifold}


\remark 
Due to Bochner's vanishing,  {\bf \red holonomy 
of Ricci-flat Calabi-Yau manifold
lies in $SU(n)$}, and {\bf \red holonomy of Ricci-flat 
holomorphically symplectic manifold  lies in $Sp(n)$}
(a group of complex unitary matrices preserving a 
complex-linear symplectic form).

\definition
A holomorphically symplectic K\"ahler manifold  
with a Calabi-Yau metric is called {\bf \blue hyperk\"ahler}.

\remark 
Since $Sp(n)=SU({\Bbb H}, n)$, a {\bf \purple hyperk\"ahler manifold admits
quaternionic action in its tangent bundle.}


\newpage


{\bf \blue EXAMPLES.}

\example An even-dimensional complex vector space.

\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 $\C^2/{\pm1}$.}

\remark Let $M$ be a 2-dimensional complex manifold with 
holomorphic symplectic form outside of singularities, which are
all of form $\C^2/{\pm1}$. Then {\bf \purple its resolution is also
holomorphically symplectic.}

\example Take a 2-dimensional complex torus $T$,
then all the singularities of $T/{\pm1}$ are of this form.
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 $\widetilde {T/{\pm1}}$.}

\newpage 

{\bf \blue K3 surfaces} 

\definition
{\bf \blue A K3-surface} is a deformation of a Kummer surface.

{\bf \red ``K3: Kummer, K\"ahler, Kodaira''} (a name is due to A. Weil).

\begin{center}\epsfig{file=Broad_Peak8051m.jpg,width=0.5\linewidth}

{\it\color{blue} ``Faichan Kangri (K3) is the 12th highest mountain on Earth.''}
\end{center}

\theorem Any complex compact surface with $c_1(M)=1$
and $H^1(M)=0$ {\bf \purple is isomorphic to K3.} Moreover, 
{\bf \blue it is hyperk\"ahler.}


\newpage 

{\bf \blue Hilbert schemes} 

\remark {\bf\blue A complex surface} is a 2-dimensional complex manifold.

\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}.


\example
Let $T$ is 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 $n>2$, it 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 of maximal holonomy are
these 2 and the three series:} tori, Hilbert schemes of K3, and
generalized Kummer.



\newpage

{\bf \blue Bogomolov's decomposition theorem}


\theorem 
{\bf \blue (Cheeger-Gromoll)} Let $M$ be a compact  
Ricci-flat Riemannian manifold with $\pi_1(M)$ infinite.
{\bf \red Then a universal covering of $M$ is a product 
of $\R$ and a Ricci-flat manifold.}

\corollary 
A fundamental group of a compact 
Ricci-flat Riemannian manifold is {\bf \blue
``virtually polycyclic'':} {\bf \purple it is projected
to a free abelian subgroup with finite kernel.} 

\remark This is equivalent to any compact Ricci-flat 
manifold having a finite covering which has free abelian
fundamental group.

\remark This statement contains the Bieberbach's 
solution of Hilbert's 18-th problem on classification 
of crystallographic groups.

\theorem 
{\bf \blue (Bogomolov's decomposition)}
Let $M$ be a compact, Ricci-flat Kaehler manifold.
{\bf \red Then there exists 
a finite covering $\tilde M$ of $M$ which is a product of 
Kaehler manifolds of the following form:}
\[
\tilde M = T \times M_1 \times ... \times M_i
\times K_1 \times ... \times K_j,
\]
with all $M_i$, $K_i$ simply connected, $T$ a torus,
and $\Hol(M_l) = Sp(n_l)$, $\Hol(K_l)=SU(m_l)$


\newpage

{\bf \blue Holomorphic Euler characteristic}

\definition
{\bf\blue A holomorphic Euler characteristic} $\chi(M)$ of a 
K\"ahler manifold is a sum $\sum(-1)^p\dim H^{p,0}(M)$.

\theorem (Riemann-Roch-Hirzebruch)
For an $n$-fold,  
{\bf \red $\chi(M)$ can be expressed as a polynomial expressions of
the Chern classes,} $\chi(M)=td_{n}$
where $td_n$ is an $n$-th component of the Todd polynomial,
{\small \[
td(M) = 
1 + \frac1 {2}c_1 + \frac{1}{12}(c_1^2+c_2) + \frac{1}{24}c_1c_2 + 
\frac1{720}(-c_1^4 + 4c_1^2c_2 + c_1c_3 + 3c_2^22 - c_4) + ...
\]}
\vspace{-10mm}

\remark 
The Chern classes are obtained as
polynomial expression of the curvature (Gauss-Bonnet).
Therefore {\bf \purple $\chi(\tilde M)= p\chi(M)$ for any
unramified $p$-fold covering $\tilde M \arrow M$.}

\remark Bochner's vanishing and the 
classical theory of invariants  imply: \\
\phantom{XXX} 1. When 
$\Hol(M) = SU(n)$, we have 
{\bf \red $\dim H^{p,0}(M)= 1$ for $p=1,n$, and 0 otherwise. }
In this case, $\chi(M)=2$ for even $n$ and
0 for odd. \\
\phantom{XXX}  2. When $\Hol(M) = Sp(n)$,we have 
{\bf \red  $\dim H^{p,0}(M)= 1$ for even $p$
with $0\leq p\leq 2n$,  and 0 otherwise.} 
In this case, $\chi(M)=n+1$.

\corollary
$\pi_1(M)=0$ if $\Hol(M) = Sp(n)$, or
$\Hol(M) = SU(2n)$. If $\Hol(M) = SU(2n+1)$,
{\bf \purple $\pi_1(M)$ is finite by Cheeger-Gromoll, but can be non-trivial. }




\end{document}


