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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hyperkahler manifolds, \\[15mm]
\small lecture 2: Calabi-Yau theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, September 21, 2019 } \\ [15mm]
\url{http://bogomolov-lab.ru/KURSY/HK-2019/}
\end{center}

\newpage


{\bf \blue 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 $\nabla$.
The operator $\nabla:\Lambda^{i}(M)\otimes B 
\stackrel{\nabla}\arrow \Lambda^{i+1}(M)\otimes B$
{\bf\blue is often denoted $d_\nabla$.}


\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 $M$. }

\remark {\bf \blue (Bianchi identity)}\\
Clearly,
$[\nabla, \nabla^2]=[\nabla^2,\nabla]+
[\nabla, \nabla^2]=0$, hence $[\nabla, \nabla^2]=0$.
This gives {\bf \blue the Bianchi identity:}
$d_\nabla(\Theta_B)=0$, where $\Theta$ is considered as
a section of $\Lambda^2(M)\otimes \End(B)$, and
$d_\nabla:\; \Lambda^2(M)\otimes \End(B)\arrow\Lambda^3(M)\otimes \End(B)$
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 $\eta\in \Lambda^{p,q}(M)$ 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
Let $M$ be a Calabi-Yau manifold, $\omega$ its K\"ahler form,
$\Omega$ a non-degenerate section of the canonical bundle.
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 $f$ such that $-2\6\bar\6 f= \Theta_{K,\omega}$,
the curvature of the metric $h\arrow \frac {h\wedge \bar h}{\omega^n e^f}$ 
on $K_M$ is equal to $\Theta_{K,\omega}+2\6\bar\6 f=0$.

{\bf\green Proof. Step 2:} {\bf \red 
 $\omega_1$ is Ricci-flat if and only if
the induced metric on $K_M$ is flat,} which is equivalent
to $(\omega+dd^c\phi)^n = \omega^n e^f$.
\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 strictly 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 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.



\end{document}

