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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small lecture 19: Chern connections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, April 4, 2018 } 
\end{center}

\newpage

{\bf \blue Curvature}

\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 Flat bundles}

\definition
Let $(B, \nabla)$ be a bundle with connection.
{\bf\blue Holonomy group} of $\gamma$ is the group
of endomorphisms of the fiber $B_x$ obtained
from parallel transports along all paths 
starting and ending in $x\in M$

\definition
A bundle is {\bf \blue flat} if and only if
its curvature vanishes.

The following theorem easily follows from the
Frobenius theorem and interpretation of connection
on $B$ as of splitting of the tangent bundle of
the total space of the principal $GL(n)$-fibration associated with $B$.

\theorem
Let $(B, \nabla)$ be a vector bundle with connection
over a simply connected manifold. {\bf \red Then $B$ is flat if and only if
its holonomy group is trivial}.

\newpage

{\bf \blue Holomorphic bundles}

\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}

\remark
Let $M$ be a complex manifold. Then
{\bf \purple the operator \\ $\bar\6:\; C^\infty M \arrow \Lambda^{0,1}(M)$
is $\calo_M$-linear.}

\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 $B_{C^\infty}=B \otimes_{\calo_M} C^\infty M$.

\remark
The kernel of $\bar\6:\; B_{C^\infty}\arrow B_{C^\infty}\otimes \Lambda^{0,1}(M)$
{\bf \red coincides with the image of $B$} under the natural
sheaf embedding $B\hookrightarrow B_{C^\infty}$, with 
$b \arrow b \otimes 1$.

\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 $\eta \in \Lambda^{0,i}(M)$.

\newpage

{\bf \blue Complexification (reminder)}

\definition
Let $M_\R$ be a real analytic manifold, and
$M_\C$ a complex analytic manifold equipped with
an antiholomorphic involution, such that $M_\R$
is the set of its fixed points. Then $M_\C$ is called
{\bf \blue complexification} of $M_\R$.

\definition
A tensor on a real analytic manifold is called 
{\bf\blue real analytic} if it is expressed locally
by a sum (infinite, generally speaking)  of coordinate monomials with real
analytic coefficients.

\claim
Let $M_\R$ be a real analytic manifold, $M_\C$ its complexification,
and $\Phi$ a tensor on $M_\R$. {\bf \red Then $\Phi$ is real analytic
if and only if $\Phi$ can be extended to a holomorpic tensor
$\Phi_\C$ in some neighbourhood of $M_\R$ inside $M_\C$.}

\proof
The ``if'' part is clear, because every complex analytic 
tensor on $M_\C$ is by definition real analytic on $M_\R$.

Conversely, suppose that $\Phi$ is expressed by a sum of
coordinate monomials with real analytic coefficients $f_i$.
Let $\{U_i\}$ be a cover of $M$, and 
$\tilde U_i:=U_i\times B_\epsilon$ the corresponding cover
of a neighbourhood of $M_\R$ in $M_\C$ constructed above.
Chosing $\epsilon$ sufficiently small, we can assume that
the Taylor series giving coefficients of $\Phi$ 
converges on each $\tilde U_i$. {\bf \purple We define $\Phi_\C$
as the sum of these series.}
\endproof

\newpage

{\bf \blue Holomorphic structure operator}


\remark
{\bf \purple For any holomorphic bundle, one has $\bar\6^2=0$}.
Indeed, a holomorphic bundle admits a local trivialization.

\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  $V=B_{\C^\infty}$.}

\proof
We prove this theorem in additional assumption that $V$ and
$\bar \6$ is real analytic. This assumption can be justified
using the Newlander-Nirenberg theorem. 

Let $M_\C\subset M\times \bar M$ be a small
neighbourhood of diagonal, considered as a complexification
of $M$, and $V_\C$ a holomorphic
vector bundle on $M_\C$ obtained from the real
vector bundle $V$. We extend $\bar\6$ to a 
differential operator $\bar\6_\C$ on $V_\C$. Then 
$\bar\6_\C$ is a connection in the restriction of $V_\C$
to the fibers of
the natural projection $\pi:\; M_\C \arrow M$, and
the condition $\bar\6^2=0$ implies that it is flat.
{\bf \purple The sheaf $\ker \bar\6$ constant on fibers of $\pi$,}
hence $B=\pi_*(\ker \bar\6)$ is a holomorphic 
bundle of the same rank as $V$. \endproof


\newpage

{\bf \blue Connections and holomorphic structures}

\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.

\newpage

{\bf \blue Chern connection}

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

\pstep
Given a complex vector bundle $B$, define
{\bf \blue complex conjugate bundle} $\bar B$
as the same $\R$-bundle with complex
conjugate $\C$-action. Then
{\bf \purple a connection $\nabla$ on $B$
defines a connection $\bar\nabla$ on $\bar B$,
with $\bar\nabla^{1,0}=\overline{\nabla^{0,1}}$ and
$\bar\nabla^{0,1}=\overline{\nabla^{1,0}}$.}

{\bf \green Step 2:} Define {\bf \blue
 $\nabla^{1,0}$-operator} on a complex vector bundle
$B$ as a map  $B \stackrel 
{\nabla^{1,0}}\arrow \Lambda^{1,0}(M)\otimes B$,
satisfying  $\Lambda^{1,0}(fb) = \6(f)\otimes b + f\nabla^{1,0}(b)$
for any  $f\in C^\infty M, b\in B$.
{\bf \purple A $\bar\6$-operator on $B$ defines
an $\nabla^{1,0}$-operator on $\bar B$, and vice versa}.

{\bf \green Step 3:} Hermitian form defines an isomorphism
of complex vector bundles $B \stackrel g \arrow \bar B^*$.
Holomorphic structure on $B$ defines a
$\bar\6$-operator on $\bar B=B^*$, which is the
same as $\nabla^{1,0}$-operator $\nabla^{1,0}_g$  on $B$.
{\bf \purple This gives a connection operator
$\nabla:= \bar\6 + \nabla^{1,0}_g$ on $B$,}
which is Hermitian by construction.
\endproof



\end{document}

