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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small Lecture 20: Curvature of line bundles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, April 7, 2018 } 
\end{center}

\newpage

{\bf \blue Curvature (reminder)}

\definition
Let $\nabla:\; B \arrow B \otimes \Lambda^1 M$ be a connection
on a vector bundle $B$. We extend $\nabla$ to an operator
\[
V \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes V
\stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes V 
\stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes V \stackrel{\nabla}\arrow ...
\]
using the Leibnitz identity
$\nabla(\eta \otimes b) = d\eta + (-1)^{\tilde \eta} \eta \wedge \nabla b$.
Then the operator $\nabla^2:\; B \arrow B\otimes \Lambda^{2}(M)$
is called {\bf \blue the curvature} of $\nabla$.

\remark
{\bf \purple The algebra of differential forms
with coefficients in $\End B$ acts on
$\Lambda^* M \otimes B$} via
$\eta \otimes a (\eta' \otimes b) = \eta \wedge \eta'
\otimes a(b)$, where
$a\in \End(B)$, $\eta, \eta'\in \Lambda^* M$, and $b\in B$.



\remark
$\nabla^2(fb) = d^2 f b + df \wedge \nabla b - df \wedge
\nabla b + f \nabla^2 b$, hence {\bf \purple
the curvature is a $C^\infty M$-linear operator.}
{\bf \red We shall consider the curvature $B$ as 
a 2-form with values in $\End B$}.
Then $\nabla^2 := \Theta_B \in \Lambda^2 M \otimes \End B$,
where  an $\End(B)$-valued form acts on $\Lambda^* M \otimes B$
as above.

\newpage

{\bf \blue Holomorphic bundles (reminder)}

\definition
{\bf \blue Holomorphic vector bundle} on a complex
manifold $M$ is a locally trivial sheaf of $\calo_M$-modules.

\definition
{\bf \blue The total space} $\Tot(B)$ of a holomorphic bundle $B$ over $M$ is
the space of all pairs $\{x\in M, b \in B_x/{\goth m}_x B\}$,
where $B_x$ is the stalk of $B$ in $x\in M$ and ${\goth m}_x$ the
maximal ideal of $x$. We equip $\Tot(B)$  with the natural topology
and holomorphic structure, in such a way that 
$\Tot(B)$ becomes a locally trivial holomorphic
fibration with fiber $\C^r$, $r=\rk B$.

\remark {\bf \purple The set of holomorphic sections of a map
$\Tot(B)\arrow M$ is naturally identified with the
set of sections of the sheaf $B$.}

\claim 
Ler $B$ be a holomorphic bundle. Consider the sheaf
$B_{C^\infty}:=B \otimes_{\calo_M} C^\infty M$.
{\bf \red Then $B_{C^\infty}$ is a locally trivial sheaf of
$C^\infty M$-modules}.

\definition
$B_{C^\infty}$ is called {\bf \blue smooth vector bundle
underlying the holomorphic vector bundle $B$}.

\remark The natural map
$\Tot(B)\arrow \Tot(B_{C^\infty})$ is a diffeomorphism.

\newpage

{\bf \blue $\bar\6$-operator on vector bundles (reminder)}

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

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

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


\theorem {\bf \blue (Malgrange)}
Let  $\bar\6:\; V \arrow \Lambda^{0,1}(M)\otimes V$
be a $\bar\6$-operator on a complex vector bundle,
satisfying  $\bar\6^2=0$, where $\bar\6$ is extended 
to \[
V \stackrel{\bar\6}\arrow \Lambda^{0,1}(M)\otimes V
\stackrel{\bar\6}\arrow \Lambda^{0,2}(M)\otimes V 
\stackrel{\bar\6}\arrow \Lambda^{0,3}(M)\otimes V \stackrel{\bar\6}\arrow ...
\]
as above.
{\bf \red Then
 $B:=\ker \bar\6\subset V$ is a holomorphic bundle of the
same rank, and  $V=B_{\C^\infty}$.}

\newpage

{\bf \blue Chern connection (reminder)}

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

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

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

\theorem
Every holomorphic Hermitian vector bundle
{\bf \red admits a Chern connection, which is unique.}

\remark
When people say about ``curvature of a holomorphic
Hermitian line bundle'', {\bf \red they speak about curvature
of a Hermitian connection.}

\newpage

{\bf\blue Bianchi identity}

\remark $[\nabla, \{\nabla, \nabla\}]=[\{\nabla, \nabla\},\nabla]+
[\nabla, \{\nabla, \nabla\}]=0$ by the super Jacobi identity.
This gives {\bf \blue the Bianchi identity:}
$\nabla(\Theta_B\wedge \eta) = \Theta_B \wedge \nabla(\eta)$.

\remark
When $B$ is a line bundle, $\End(B)$ is trivial,
and $\Theta_B$ is a 2-form.

\claim
{\bf \red A curvature of a line bundle is a closed 2-form.}

\proof
For any form $\theta\in \Lambda^{i}(M) \otimes \End(B)$,
Leibnitz identity gives
$\nabla(\theta \wedge \eta) = d\theta \wedge \eta +
(-1)^i\theta \wedge \nabla(\eta)$.
Bianchi identity gives
$\nabla(\Theta_B\wedge \eta) = \Theta_B \wedge \nabla(\eta)$.
Therefore $d\Theta_B=0$. \endproof

\remark The same argument can be used to show
that {\bf \red $\Tr_B \Theta_B^i$ is a closed $2i$-form},
where $\Tr_B$ denotes the trace in $\End(B)$, and
$\Theta_B^i$ is the $i$-th power of an $\End(B)$-valued form.

\definition
Cohomology classes of $\Tr_B \Theta_B^i$
are called {\bf \blue characteristic vlasses}
of a bundle $B$ (``Chern-Weil formula'').
When $B$ is a line bundle, the cohomology class
of $-\frac{\1}{\pi} \Theta_B$ is called
{\bf \blue first Chern class of $B$}, denoted
$c_1(B)$.

\newpage

{\bf \blue Luigi Bianchi}


\begin{center}{\epsfig{file=Bianchi.jpeg,width=0.43\linewidth}\\[10mm]
{\bf \small \blue Luigi Bianchi (1856 - 1928)}}
\end{center}



\newpage

{\bf \blue Exponential sequence}

\remark
Let $B$ be a line bundle on a manifold,
$\{U_\alpha\}$ its cover where $B$ is trivialized,
and $\phi_{\alpha\beta}$ the corresponding transition
functions defined on  $U_\alpha \cap U_\beta$.
On each intersection $U_\alpha \cap U_\beta\cap U_\gamma$ 
we have $\phi_{\alpha\beta}\phi_{\beta\gamma}=
\phi_{\alpha\gamma}$, hence {\bf \purple
a trivialization of $B$ on $\{U_\alpha\}$ 
defines a \v Cech 1-cocycle on $B$ with values in
$(C^\infty M)^*$.}

The following claim is clear from the definitions.

\claim
{\bf \red Isomorphism classes of vector bundles
are in bijective correspondence with
  $H^1(M, (C^\infty M)^*)$.}

\definition
{\bf \blue Exponential exact sequence} is the following exact sequence
of sheaves:
\[ 
0 \arrow \Z_M \arrow C^\infty M \arrow (C^\infty M)^* \arrow 0,
\] 
Since $H^i(C^\infty M)=0$ for $i>0$, 
{\bf \purple the corresponding long exact sequence gives
$0 \arrow H^1(M, (C^\infty M)^*) \tilde \arrow H^2(M, \Z) \arrow 0$.}

\newpage

{\bf \blue Line bundles on $\C P^\infty$}

\exercise
Let $B$ be a vector bundle on $M$.
{\bf \purple 
Prove that $B\oplus B'$ is trivial for some bundle $B'$.}

\claim For any complex line bundle $L$ on $M$,
{\bf \red  there exists a map $\phi:\; M \arrow \C P^n$
such that $L= \pi^* \calo(-1)$. }

\proof
Indeed, suppose that
$L\oplus B'$ is trivial, $B_1 =L \oplus B' = V \otimes_\C C^\infty M$.
Then each point $x\in M$ defines a line $\phi(x)\in {\Bbb P}V$
such that $L\restrict x = \phi(x) \subset B_1 \restrict x= V$.
In this situation, {\bf \purple 
$L$ is obtained as a pullback of the tautological 
vector bundle.}
\endproof


\newpage

{\bf \blue First Chern class}

\definition
The isomorphism $H^1(M, (C^\infty M)^*) \tilde \arrow H^2(M, \Z)$
maps a line bundle
$L\in H^1(M, (C^\infty M)^*)$
to its {\bf \blue integer Chern class} $c_1^\Z(B)\in H^2(M, \Z)$

\theorem {\bf \blue (Gauss-Bonnet)} \\
Let $L$ be a line bundle on $M$. Then
the natural map $H^2(M, \Z)\arrow H^2(M, \R)$
{\bf \red maps $c_1^\Z(L)$ to the class $c_1(L)=-\frac{\1}{\pi} [\Theta_L]$
defined above.}

\proof Let $\calo(-1)$ be the tautological bundle on $\C P^n$.
As shown above, any line bundle $L$ on $M$ can be obtained 
as $\phi^*(\calo(-1))$, hence it suffices to prove
$[\Theta_L]= \1\pi c_1^\Z(L)$ for $L=\calo(-1)$ on $\C P^n$.
Since a cohomology class in $H^2(\C P^n)$
is determined by its restriction to 
$\C P^1$, it would suffice to prove this
formula for $L=\calo(-1)$ on $\C P^1$.
In this case, $c_1^\Z(L)$ is the Euler characteristic
of $L$, and $[\Theta_L]= \1\pi c_1^\Z(L)$ is the usual
Gauss-Bonnet formula on a 2-sphere, which can be obtained
by computing the volume of $S^2$.
\endproof

\newpage

{\bf \blue Real structures on a complex vector space}

\definition
{\bf \blue Real structure on a complex vector space}
is anticomplex involution. 

\exercise For any complex vector space $V$ and
a real structure $\iota$, denote by $V_\R$ the
fixed point set of $\iota$. {\bf \purple Prove that
$V= V_\R \otimes_\R \C$.}

\example
Let $V$ be a Hermitian vector space, and$\End_\C V$
its endomorphism space. Consider the real structure
 $\phi \stackrel \iota \arrow -\phi^*$,
where $\phi^*$ denotes the Hermitian conjugate.
{\bf \purple Prove that the fixed point set of $\iota$
is the space of anti-Hermitian matrices $\goth u(V)$.}

\newpage

{\bf \blue Curvature of the Chern connection}

\proposition
{\bf \red Curvaure $\Theta_B$ of a Chern connection on $B$
is a (1,1)-form: $\Theta_B \in \Lambda^{1,1}(M)\otimes \End(B)$.}

\pstep
Let $B$ be a Hermitan bundle. Consider the
operator $\phi \stackrel \iota \arrow -\phi^*$
acting on $\End(B)$, where $\phi \arrow \phi^*$ 
denotes the Hermitian conjugation. Since
$\iota^2=\Id$, and this is an anticomplex
operator, it defines the real structure, and
its fixed point set is ${\goth u}_B$.

{\bf \green Step 2:} Since the Chern connection
preserves the Hermitian structure $g$,
one has $\nabla(g)=0$, which gives $\nabla^2(g)=0$.
This means that  $\Theta_B\in \Lambda^2 M \otimes {\goth
  u}_B$,
{\bf \purple and this for is real with respect to the
real structure defined by $\iota$}.

{\bf \green Step 3:} The $(0,2)$-part of the curvature
vanishes, because $\bar\6^2=0$. The $(2,0)$-part
of the curvature vanishes, because 
$\iota(\Theta_B) = \Theta_B$, and {\bf \purple any 
real structure on $\End(B)$ exchanges 
$\Lambda^{2,0}(M)\otimes \End(B)$ and 
$\Lambda^{0,2}(M)\otimes \End(B)$}.
\endproof

\corollary
{\bf \red For the Chern connection $\nabla=\bar\6+\nabla^{1,0}$
on $B$, one has $\Theta_B =\{\nabla^{1,0}, \bar\6\}$.}

\corollary 
{\bf \red A curvature of a holomorphic Hermitian line bundle
is a closed (1,1)-form.}


\newpage

{\bf \blue Curvature of the Chern  connection on a line bundle}


\remark
Let $B$ he a Hermitian holomorphic line bundle,
and $b\in \Gamma(B)$ a nowhere vanishing holomorphic
section. Then 
\[ d|b|^2= (\nabla^{1,0} b, b)+ (b, \nabla^{1,0} b)=
   2 \Re (\nabla^{1,0} b, b),
\]
which gives 
\[ \nabla^{1,0} b= \frac{\6 |b|^2}{|b|^2}b=
   2\6\log|b| b.
\] 
{\bf \purple We obtaim
that $\Theta_B(b)= 2\bar\6\6\log|b| b$, hence
$\Theta_B = -2 \6\bar\6\log|b|$.}

{\bf \green Corollary 1:}
Let $g' = e^{2f} g$ be Hermitian metrics on a holomorphic
line bundle, and  $\Theta, \Theta'$ the corresponding 
curvatures. {\bf \red Then  $\Theta' - \Theta = -2 \6\bar\6 f$.}
\endproof

\definition Tensor multiplication {\bf \purple defines the
structure of abelian group on the set $\Pic(M)$
of equivalence classes of holomorphic line bundles on 
a complex manifod $M$}. This group is called
{\bf \blue the Picard group} of $M$.
\newpage

{\bf \blue $dd^c$-lemma (reminder)}

\theorem
Let $\eta$ be a form on a compact K\"ahler manifold,
satisfying one of the following conditions.\\
(1). $\eta$ is an exact $(p,q)$-form. (2). 
$\eta$ is $d$-exact, $d^c$-closed. \\ 
(3). $\eta$ is $\6$-exact, $\bar\6$-closed. \\
{\bf \red Then $\eta\in \im dd^c=\im \6\bar\6$.}

\proof
Notice immediately that in all three cases
{\bf \purple $\eta$ is closed and orthogonal to
the kernel of $\Delta$}, hence its cohomology class vanishes.

Since $\eta$ is exact, it lies in the image of $\Delta$.
Operator $G_\Delta:=\Delta^{-1}$ is defined on $\im \Delta= \ker \Delta^\bot$
and commutes with $d, d^c$. 

In case (1), $\eta$ is $d$-exact, and $I(\eta)= \bar\eta$ is $d$-closed,
hence $\eta$ is $d$-exact, $d^c$-closed like in (2).

Then $\eta= d \alpha$, where $\alpha:=G_\Delta d^*\eta$.
Since $G_\Delta$ and $d^*$ commute with $d^c$,
the form $\alpha$ is $d^c$-closed;
since it belongs to $\im \Delta=\im G_\Delta$, 
it is $d^c$-exact, $\alpha=d^c \beta$ 
which gives $\eta = dd^c \beta$.

In case (3), we have $\eta= \6 \alpha$, where $\alpha:=G_\Delta \6^*\eta$.
Since $G_\Delta$ and $\6^*$ commute with $\bar\6$, 
the form $\alpha$ is $\bar\6$-closed;
since it belongs to $\im \Delta$, it is $\bar\6$-exact, $\alpha=\bar\6 \beta$ 
which gives $\eta = \6\bar\6 \beta$.
\endproof

\newpage

{\bf \blue Forms realized as a curvature of a line bundle}

\corollary 
Let $\omega$ be an integer (1,1)-form with integer
cohomology class on a compact K\"ahler manifold.
{\bf \red Then $\omega$ is a curvature of a holomorphic
line bundle}.

\pstep Exponential exact sequence 
 $0 \arrow \Z_M \arrow \calo_M \arrow \calo^*_M\arrow 0$
gives
\[
H^1(\calo^*_M) \stackrel c \arrow H^2(M, \Z) \stackrel p \arrow H^2(M, \calo_M),
\]
where $H^1(\calo^*_M)= \Pic(M)$ is the group of 
holomorphic line bundles, $c$ maps a bundle to its first
Chern class, and $p$ projects $H^2(M)$ to its Hodge
component $H^2(M, \calo_M)= H^{0,2}(M)$. Then
{\bf \purple for any integer class $[\omega]\in
  H^{1,1}(M)\cap H^2(M, \Z)$, there exists
a line bundle $L$ such that $[\omega]= c_1(L)$.}

{\bf \green Step 2:}
Take any metric $h$ on $L$. Its curvature
$\omega_h$ is a closed (1,1)-form, cohomologous to
$\omega$. By $dd^c$-lemma, $\omega_h -\omega = -2 \6\bar\6
f$ for some $f\in C^\infty M$. {\bf \purple By Corollary 1,
curvature of $h':=e^{2f}h$ satisfies 
$\omega_h -\omega_{h'}= -2 \6\bar \6 f$,}
giving  $\omega_{h'}= \omega$. \endproof





\end{document}

