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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory} \\[15mm]
\small Lecture 23: Calabi-Yau theorem
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, May 16, 2018 } 
\end{center}

\newpage


{\bf \blue REMINDER: Holomorphic vector bundles}



\definition
{\bf\blue A $\bar\6$-operator} on a smooth bundle 
is a map $V \stackrel {\bar\6}\arrow \Lambda^{0,1}(M)\otimes V$,
satisfying $\bar\6(fb) = \bar\6(f)\otimes b + f\bar\6(b)$
for all $f\in C^\infty M, b\in V$.

\remark {\bf \purple A  $\bar\6$-operator on $B$ can be extended
to 
\[ \bar\6:\; \Lambda^{0,i}(M)\otimes V \arrow \Lambda^{0,i+1}(M)\otimes V,\]
} using
$\bar\6 (\eta \otimes b) = \bar\6(\eta)\otimes b + 
(-1)^{\tilde \eta}\eta\wedge\bar\6(b)$, 
where $b\in V$ and $\eta \in \Lambda^{0,i}(M)$.

\definition
{\bf \blue A holomorphic vector bundle}
on a complex manifold $(M,I)$ is a vector bundle
equipped with a $\bar \6$-operator which satisfies
$\bar\6^2=0$. In this case, $\bar\6$ is called
{\bf \blue a holomorphic structure operator}.

\exercise
Consider the Dolbeault differential 
$\bar\6:\; \Lambda^{p,0}(M)\arrow \Lambda^{p,1}(M)=
\Lambda^{p,0}(M)\otimes\Lambda^{0,1}(M)$.
{\bf \red Prove that it is a holomorphic structure
operator on $\Lambda^{p,0}(M)$.}

\definition
The corresponding holomorphic vector bundle
$(\Lambda^{p,0}(M),\bar\6)$ is called {\bf \blue
the bundle of holomorphic $p$-forms}, denoted by $\Omega^p(M)$.


\newpage

{\bf \blue REMINDER: Chern connection}

\definition
Let $(B, \nabla)$ be a smooth bundle with connection
and a holomorphic structure $\bar\6\; B \arrow \Lambda^{0,1}(M)\otimes B$. 
Consider a Hodge decomposition of $\nabla$,
$\nabla= \nabla^{0,1} + \nabla^{1,0}$,
\[
\nabla^{0,1}:\; V \arrow \Lambda^{0,1}(M)\otimes V, \ \ \ 
\nabla^{1,0}:\; V \arrow \Lambda^{1,0}(M)\otimes V.
\]
We say that $\nabla$ is {\bf \blue  compatible 
with the holomorphic structure} if $\nabla^{0,1}=\bar\6$.



\definition
{\bf \blue An Hermitian holomorphic vector bundle}
is a smooth complex vector bundle equipped with a Hermitian
metric and a holomorphic structure operator $\bar \6$.

\definition
{\bf\blue A Chern connection} on a
holomorphic Hermitian vector bundle is a connection
compatible with the holomorphic structure and preserving the metric.

\theorem
On any holomorphic Hermitian vector bundle, {\bf \red the
Chern connection exists, and is unique.}


\newpage

{\bf \blue REMINDER: Curvature of a connection}

\definition
Let  $\nabla:\; B \arrow B \otimes \Lambda^1 M$
be a connection on a smooth budnle. Extend it to an
operator on $B$-valued forms
\[
B \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes B
\stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes B 
\stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes B \stackrel{\nabla}\arrow ...
\]
using $\nabla(\eta \otimes b) = d\eta + (-1)^{\tilde \eta} \eta \wedge \nabla b$.
The operator $\nabla^2:\; B \arrow B\otimes \Lambda^{2}(M)$
is called {\bf \blue the curvature} of $\nabla$.


\remark The algebra of $\End(B)$-valued forms naturally acts
on $\Lambda^* M \otimes B$. The curvature
satisfies $\nabla^2(fb) = d^2 f b + df \wedge \nabla b - df \wedge
\nabla b + f \nabla^2 b= f \nabla^2 b$, hence it is
$C^\infty M$-linear. {\bf \purple We consider it as an $\End(B)$-valued
2-form on $M$. }

\proposition (Bianchi identity)
Clearly,
$[\nabla, \nabla^2]=[\nabla^2,\nabla]+
[\nabla, \nabla^2]=0$, hence $[\nabla, \nabla^2]=0$.
This gives {\bf \blue Bianchi identity:}
$\nabla(\Theta_B)=0$, where $\Theta$ is considered as
a section of $\Lambda^2(M)\otimes \End(B)$, and
$\nabla:\; \Lambda^2(M)\otimes \End(B)\arrow\Lambda^3(M)\otimes \End(B)$.
the operator defined above

\newpage

{\bf \blue REMINDER: Curvature of a holomorphic line bundle}


\remark If $B$ is a line bundle, $\End B$ is trivial,
and {\bf \red the curvature $\Theta_B$ of $B$ is a closed 2-form.}

\definition
Let $\nabla$ be a unitary connection in a line bundle.
The cohomology class
$c_1(B):=\frac{\1}{2\pi}[\Theta_B]\in H^2(M)$
is called {\bf \blue the real first Chern class of a line bunlde $B$}.

{\bf \green An exercise:} Check that $c_1(B)$
is independent from a choice of $\nabla$.


\remark When speaking of a {\bf \blue ``curvature of a holomorphic
bundle'',} one usually means the curvature of a Chern connection.

\remark 
Let $B$ be a holomorphic Hermitian line bundle, and $b$ 
its non-degenerate holomorphic section. Denote by $\eta$ a (1,0)-form
which satisfies $\nabla^{1,0} b=\eta\otimes b$.
Then $d|b|^2= \Re g(\nabla^{1,0} b, b) = \Re\eta|b|^2$.
{\bf \purple This gives $\nabla^{1,0} b= \frac{\6 |b|^2}{|b|^2}b=
   2\6\log|b| b.$}

\remark
Then $\Theta_B(b)= 2\bar\6\6\log|b| b$, {\bf \red  that is,
$\Theta_B = -2 \6\bar\6\log|b|$.}

\corollary
If $g' = e^{2f} g$ -- two metrics on a holomorphic line bundle,
$\Theta, \Theta'$ their curvatures, {\bf \purple one has
 $\Theta' - \Theta = -2 \6\bar\6 f$}


\newpage

{\bf \blue $\6\bar\6$-lemma}


\theorem {\bf \blue  (``$\6\bar\6$-lemma'')}\\
Let $M$ be a compact Kaehler manifold,
and $\eta \Lambda^{p,q}(M)$ an exact form.
Then $\eta =  \6\bar\6\alpha$, for some 
$\alpha \in \Lambda^{p-1,q-1}(M)$.

Its proof uses Hodge theory.

\corollary
Let $(L,h)$ be a holomorphic line bundle on a compact complex manifold,
$\Theta$ its curvature, and $\eta$ a (1,1)-form in the same cohomology 
class as $[\Theta]$. {\bf \red Then there exists a
 Hermitian metric $h'$ on $L$ such that its curvature
is equal to $\eta$}. 

{\bf \green Proof:}  Let $\Theta'$ be the curvature
of the Chern connection associated with $h'$. Then
$\Theta' - \Theta = -2\6\bar\6 f$, wgere $f = \log(h'h^{-1})$.
Then {\bf \purple $\Theta' - \Theta=\eta-
  \Theta=-2\6\bar\6 f$ has a solution 
$f$ by
$\6\bar\6$-lemma,} because $\eta- \Theta$ is exact. \endproof


\newpage

{\bf \blue Calabi-Yau manifolds}

\remark
Let $B$ be a line bundle on a manifold. Using 
the long exact sequence of cohomology associated with the
exponential sequence
\[ 
0 \arrow \Z_M \arrow C^\infty M \arrow (C^\infty M)^* \arrow 0,
\] 
{\bf \red we obtain $0 \arrow H^1(M, (C^\infty M)^*) \arrow H^2(M, \Z) \arrow 0$.}

\definition
Let $B$ be a complex line bundle, and $\xi_B$ its defining element
in $H^1(M, (C^\infty M)^*)$. Its image in $H^2(M, \Z)$ is called
{\bf\blue the integer first Chern class} of $B$, denoted by $c_1(B,\Z)$
or $c_1(B)$.

\remark
{\bf \purple A complex line bundle $B$ is (topologically)
trivial if  and only if
$c_1(B,\Z)=0$.}

\theorem (Gauss-Bonnet) 
A real Chern class of a vector bundle {\bf \purple is an image
of the integer Chern class $c_1(B,\Z)$} under the natural
homomorphism $H^2(M, \Z)\arrow H^2(M, \R)$. 

\definition 
{\bf\blue A first Chern class} of a complex $n$-manifold
is $c_1(\Lambda^{n,0}(M))$.

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


\newpage

{\bf \blue Ricci form of a K\"ahler manifold}

\theorem {\bf \blue(Bogomolov)} Let $M$ be a compact K\"ahler
$n$-manifold with  $c_1(M,\Z)=0$. {\bf \red Then the canonical bundle
$K_M:=\Omega^n(M)$ is trivial.}

\proof Follows from the Calabi-Yau theorem (later today). \endproof

In other words, a manifold is Calabi-Yau if and only if
its canonical bundle is trivial.

\definition
Let $(M,\omega)$ be a K\"ahler
manifold. The metric on $K_M$ can be written as
$|\Omega|^2 = \frac{\Omega\wedge \bar \Omega}{\omega^n}$.
The {\bf \blue Ricci form} on $M$ is the curvature of the
Chern connection on $K_M$. The manifold $M$ is
{\bf \blue Ricci-flat} if its Ricci form vanishes.

\remark Since a canonical bundle $K_M$ of a Calabi-Yau manifold
is trivial, it admits a metric with trivial connection.
Calabi conjectured that {\bf \purple this metric on $K_M$ is induced
by a K\"ahler metric $\omega$ on $M$} and proved that such a metric
is unique for any cohomology class $[\omega]\in H^{1,1}(M,\R)$.
Yau proved that it always exists.

\definition
A Ricci-flat K\"ahler metric is called {\bf \blue Calabi-Yau metric}.

\newpage

{\bf \blue Calabi-Yau theorem and Monge-Amp\`ere equation}

\remark
Let $(M, \omega)$ be a K\"ahler $n$-fold, and
$\Omega$ a non-degenerate section of $K(M)$,
Then $|\Omega|^2 = \frac{\Omega\wedge \bar \Omega}{\omega^n}$.
If $\omega_1$ is a new Kaehler metric on $(M,I)$, $h, h_1$ the
associated metrics on $K(M)$, then
$\frac {h} {h_1}= \frac {\omega_1^n}{\omega^n}$.

\remark 
For two metrics $\omega_1, \omega$ in the same K\"ahler class,
one has {\bf \purple $\omega_1-\omega=dd^c\phi$, for some function $\phi$}
($dd^c$-lemma).



\corollary
A metric $\omega_1= \omega+\6\bar\6\phi$ {\bf \red is Ricci-flat if and only if
$(\omega+dd^c\phi)^n = \omega^n e^f$,} where $-2\6\bar\6 f= \Theta_{K,\omega}$
{\bf \purple (such $f$ exists by $\6\bar\6$-lemma).}

{\bf\green Proof. Step 1:}
For such $f$, $\phi$, one has 
$\log\frac {h} {h_1}= -\log e^f=- f$. 
As shown above, the corresponding curvatures are related as
$\Theta_{K,\omega_1}-\Theta_{K,\omega}=-2\6\bar\6\log(h/h_1)$.
This gives
\[
 \Theta_{K,\omega_1}= \Theta_{K,\omega} -2\6\bar\6\log(h/h_1)
 = \Theta_{K,\omega}-2\6\bar\6 f.
\]
{\bf\green Proof. Step 2:} {\bf \red Therefore,
 $\omega_1$ is Ricci-flat if and only if
$\Theta_{K,\omega}-2\6\bar\6 f$.}
\endproof

To find a Ricci-flat metric {\bf \purple it remains to solve an
equation $(\omega+dd^c\phi)^n = \omega^n e^f$ for a given $f$.}

\newpage

{\bf \blue The complex Monge-Amp\`ere equation}


To find a Ricci-flat metric {\bf \purple it remains to solve an
equation $(\omega+dd^c\phi)^n = \omega^n e^f$ for a given $f$.}



\theorem
(Calabi-Yau) Let $(M, \omega)$ be a compact Kaehler $n$-manifold, and
$f$ any smooth function. {\bf \red Then there exists
a unique up to a constant function $\phi$} such that
$(\omega+ \1\6\bar\6 \phi)^n = A e^f \omega^n,$
where $A$ is a positive constant obtained from the
formula $\int_M A e^f \omega^n= \int_M \omega^n$.

\definition
\[
(\omega+ \1\6\bar\6 \phi)^n = A e^f \omega^n,
\]
is called {\bf\blue the Monge-Ampere equation.}


\newpage

{\bf \blue Uniqueness of solutions of
complex Monge-Ampere equation}

\proposition (Calabi)
{\bf \red A complex Monge-Ampere equation has at most one solution,}
up to a constant.

{\bf \green Proof. Step 1:}
Let $\omega_1, \omega_2$ be solutions of Monge-Ampere equation.
Then $\omega_1^n = \omega_2^n$. By construction, one has
$\omega_2= \omega_1 + \1\6\bar\6 \psi$. {\bf \purple We need to show $\psi=const$.}

{\bf \green  Step 2:}
$\omega_2= \omega_1 + \1\6\bar\6 \psi$ gives
\[
0 = (\omega_1 + \1\6\bar\6 \psi)^n - \omega_1^n= 
\1\6\bar\6 \psi\wedge \sum_{i=0}^{n-1}\omega_1^i \wedge\omega_2^{n-1-i}.
\]

{\bf \green  Step 3:} Let 
$P:=\sum_{i=0}^{n-1}\omega_1^i \wedge\omega_2^{n-1-i}$.
This is a positive $(n-1, n-1)$-form. {\bf \purple There exists
a Hermitian form $\omega_3$ on $M$ such that $\omega_3^{n-1}=P$.}

{\bf \green  Step 4:} Since $\1\6\bar\6 \psi\wedge P =0$,
this gives $\psi \6\bar\6 \psi\wedge P=0$. Stokes' formula implies
\[
0 = \int_M \psi \wedge\6 \bar\6\psi\wedge P=
- \int_M \6\psi \wedge\bar\6 \psi\wedge P = - \int_M  |\6\psi|_3^2\omega_3^n.
\]
where $|\cdot|_3$ is the metric associated to $\omega_3$.
{\bf \red Therefore $\bar\6 \psi=0$.}
\endproof

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

\theorem (Berger's theorem, 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

{\bf \blue Chern connection}


\definition
Let $B$ be a holomorphic vector bundle on a complex manifold, and
$\bar\6:\; B_{C^\infty}\arrow B_{C^\infty}\otimes \Lambda^{0,1}(M)$
an operator mapping $b \otimes f$ to $b\otimes \bar\6 f$,
where $b\in B$ is a holomorphic section, and $f$ a 
smooth function. This operator is called {\bf \blue a
holomorphic structure operator} on $B$. {\bf \red It is 
correctly defined, because $\bar\6$ is $\calo_M$-linear.}

\remark {\bf \purple A section $b\in B$ is holomorphic iff $\bar\6(b)=0$}



\definition
Let $(B, \nabla)$ be a smooth bundle with connection
and a holomorphic structure $\bar\6:\; B \arrow \Lambda^{0,1}(M)\otimes B$. 
Consider the Hodge decomposition of $\nabla$,
$\nabla= \nabla^{0,1} + \nabla^{1,0}$.
We say that $\nabla$ is {\bf \blue  compatible 
with the holomorphic structure} if $\nabla^{0,1}=\bar\6$.

\definition
{\bf \blue An Hermitian holomorphic vector bundle}
is a complex vector bundle equipped with a Hermitian
metric and a holomorphic structure.

\definition
{\bf\blue A Chern connection} on a
holomorphic Hermitian vector bundle is a connection
compatible with the holomorphic structure and preserving the metric.

\theorem
On any holomorphic Hermitian vector bundle, {\bf \red the
Chern connection exists, and is unique.}


\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, $K(M)= \Omega^n M$.
The natural Hermitian metric on $K(M)$ is written as
\[ (\alpha, \alpha') \arrow \frac{\alpha\wedge \bar \alpha'}{\omega^n}.\]
Denote by $\Theta_K$ the curvature
of the Chern connection on $K(M)$.
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 holonomy
in $Sp(n)$ 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 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 Harmonic forms}

Let $V$ be a vector space. {\bf \blue A metric $g$ on $V$ induces
a natural metric on each of its tensor spaces:}
$g(x_1\otimes x_2 \otimes ... \otimes x_k, x_1'\otimes x_2' \otimes ... \otimes x_k') = g(x_1, x'_1)g(x_2, x'_2) ... g(x_k, x'_k)$.

{\bf \purple This gives a natural positive definite scalar product
on differential forms over a Riemannian manifold $(M,g)$:}
$g(\alpha, \beta) := \int_M g(\alpha, \beta) \Vol_M$. The 
topology induced by this metric is 
called {\bf \blue $L^2$-topology.}

\definition 
Let $d$ be the de Rham differential and 
$d^*$ denote the adjoint operator. The {\bf \blue Laplace operator}
is defined as $\Delta:= dd^*+d^*d$.
A form is called {\bf \blue harmonic} if it lies in
$\ker \Delta$.

\theorem
{\bf \red 
The image of $\Delta$ is closed} in $L^2$-topology on differential
forms. 

\remark This is a very difficult theorem!


\remark 
On a compact manifold, the
 form $\eta$ is {\bf \purple harmonic iff $d\eta = d^*\eta =0$.}
Indeed, $(\Delta x, x)= (dx, dx) + (d^*x, d^*x)$. 

\corollary 
This defines a map ${\cal H}^i(M) \stackrel \tau 
\arrow H^i(M)$ from harmonic forms to cohomology.

\newpage

{\bf \blue Hodge theory}


\theorem (Hodge theory for Riemannian manifolds)\\
{\bf \purple On a compact Riemannian manifold,}  
the map ${\cal H}^i(M) \stackrel \tau 
\arrow H^i(M)$ to cohomology {\bf \red is
an isomorphism.}

{\bf \green Proof. Step 1:} 
$\ker d\; \bot\; \im d^*$ and $\im d \;\bot\; \ker d^*$.
Therefore, {\bf \purple a harmonic form is orthogonal to $\im d$.}
This implies that {\bf \red $\tau$ is injective}. 

{\bf \green Step 2:} {\bf \purple $\eta \bot \im \Delta$ if and only if
$\eta$ is harmonic.} Indeed, $(\eta, \Delta x)= (\Delta x, x)$.

{\bf \green Step 3:} Since $\im \Delta$ is closed, {\bf \red every
closed form $\eta$ is decomposed as $\eta = \eta_h + \eta'$,}
where $\eta_h$ is harmonic, and $\eta'= \Delta\alpha$.

{\bf \green Step 4:}
When $\eta$ is closed, $\eta'$ is also closed.
Then $0 =(d\eta,  d \alpha) = 
(\eta, d^* d \alpha)= (\Delta\alpha, d^* d \alpha)= 
(dd^* \alpha, d^* d \alpha) + (d^* d\alpha, d^* d \alpha)$.
The term $(dd^* \alpha, d^* d \alpha)$ vanishes, because
$d^2=0$, hence $(d^* d\alpha, d^* d \alpha)=0$. This gives
$d^* d\alpha=0$, and $(d^* d\alpha, \alpha) =(d\alpha, d\alpha)=0$.
We have shown that {\bf \purple for any closed $\eta$ 
decomposing as $\eta = \eta_h + \eta'$, with 
$\eta'= \Delta\alpha$, $\alpha$ is closed}

{\bf \green Step 5:}
This gives $\eta'= d d^*\alpha$, hence {\bf \red $\eta$ is a sum
of an exact form and a harmonic form.} \endproof

{\small \remark
This gives a way of obtaining the Poincare duality
via PDE.}

\newpage

{\bf \blue Hodge decomposition on cohomology}

\theorem
{\it (this theorem will be proven in the next lecture)}\\
On a compact Kaehler manifold $M$, {\bf \red the
Hodge decomposition is compatible with the 
Laplace operator. } This gives a decomposition of cohomology,
$H^i(M) = \bigoplus_{p+q=i}H^{p,q}(M)$, with
$\overline{H^{p,q}(M)} = H^{q,p}(M)$.

\corollary 
{\bf \red $H^p(M)$ is even-dimensional for odd $p$.}

{\bf \blue The Hodge diamond:}
{\small \[
\begin{array}{ccccccccc}
&&&&H^{n,n}&&&& \\[3mm]
&&&H^{n,n-1}&&H^{n-1,n}&&& \\[3mm]
\ \ \ \ &&H^{n,n-2}&&H^{n-1,n-1}&&H^{n-2,n}&& \ \ \ \ \\[3mm]
\vdots& &\vdots &&\vdots &&\vdots&&\vdots\\[3mm]
&&H^{2,0}&&H^{1,1}&&H^{0,2}&& \\[3mm]
&&&H^{1,0}&&H^{0,1}&& &\\[3mm]
&&&&H^{0,0}&&&& \\[3mm]
\end{array}
\]}

\remark {\bf \purple $H^{p,0}(M)$ is the space of holomorphic $p$-forms.}
Indeed, $dd^* + d^* d = 2 (\bar\6\bar\6^* + \bar\6^*\bar\6)$, 
hence {\bf \red a holomorphic form on 
a compact K\"ahler manifold is closed.}

\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 invariants theory imply:

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.

2. When $\Hol(M) = Sp(n)$,we have 
{\bf \red  $\dim H^{p,0}(M)= 1$ for even $p$
$0\leq p\leq 2n$,  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 $\Hol(M) = SU(2n+1)$,
$\pi_1(M)$ is finite. 



\end{document}

