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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex geometry\\[15mm]
\small lecture 16: Riemann-Hilbert correspondence}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
 November 14, 2020
}
\end{center}


\newpage


{\bf \blue Sheaves} 

\definition
{\bf\blue An open cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$.

\remark The definition of a sheaf below {\bf \purple is a
more abstract version of the notion of 
``sheaf of functions''} defined previously.

\definition
A {\bf\blue presheaf} on a
topological space $M$ is a collection of vector spaces (or abelian groups)
${\cal F}(U)$, for each open 
subset $U\subset M$,  together with {\bf \blue restriction maps}
$R_{UW}{\cal F}(U)\arrow {\cal F}(W)$ defined for each $W\subset U$,
such that for any three open sets $W\subset V\subset U$,
$R_{UW}=R_{UV}\circ R_{VW}$. Elements of ${\cal F}(U)$
are called {\bf \blue sections of ${\cal F}$ over $U$},
and the restriction map often denoted $f\restrict W$

\definition
A presheaf 
${\cal F}$ is called {\bf\blue a sheaf} 
if for any open set $U$ and any cover $U=\bigcup U_I$
the following two conditions are satisfied.\\
\phantom{xu} 1. Let $f\in {\cal F}(U)$ be a section of ${\cal F}$ on $U$
such that its restriction to each $U_i$ vanishes. {\bf \purple 
Then $f=0$.} \\
\phantom{xu}  2. Let $f_i\in{\cal F}(U_i)$ be a family of
sections compatible on the pairwise intersections:
$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$
for every pair of members of the cover. {\bf \purple Then there exists
$f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.}

\newpage


{\bf \blue Morphisms of sheaves}


\definition
Let ${\cal B}, {\cal B}'$ be sheaves on
$M$. {\bf\blue  A sheaf morphism} from ${\cal B}$ to ${\cal B}'$
is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$,
defined for each open subset $U\subset M$,
and compatible with the restriction maps:
\[
\begin{CD}
{\cal B}(U) @>>> {\cal B}'(U)\\
@VVV@VVV\\
{\cal B}(U_1) @>>> {\cal B}'(U_1)
\end{CD}
\]

\definition
{\bf\blue A sheaf isomorphism} is a homomorphism
$\Psi:\; {\cal F}_1 \arrow {\cal F}_2$, for which there exists
an homomorphism $\Phi:\; {\cal F}_2 \arrow {\cal F}_1$,
such that $\Phi\circ \Psi =\Id$ and 
$\Psi\circ \Phi =\Id$.

\newpage


{\bf \blue Sheaves of modules}

\remark
Let  $A:\; \phi \arrow B$ be a ring homomorphism, and
$V$ a  $B$-module. {\bf \purple Then $V$ is equipped with a natural
$A$-module structure: $a v:= \phi(a) v$.}


\definition
Let  ${\cal F}$ be a sheaf of rings on
a topological space $M$, and 
 ${\cal B}$ another sheaf.
It is called {\bf\blue a sheaf of  ${\cal F}$-modules}
if for all $U\subset M$ the space of sections
${\cal B}(U)$ is equipped with a structure of ${\cal
  F}(U)$-module, and for all  $U'\subset U$, 
the restriction map 
${\cal B}(U) \stackrel{\phi_{U,U'}}\arrow {\cal B}(U')$
is a homomorphism of ${\cal F}(U)$-modules
(use the remark above
to obtain a structure of ${\cal F}(U)$-module
on  ${\cal B}(U')$).

\definition
A {\bf \blue free sheaf of modules}  ${\cal F}^n$ 
over a ring sheaf  ${\cal F}$ maps an open set $U$ to 
the space ${\cal F}(U)^n$. 


\definition
{\bf\blue Locally free sheaf of modules}
over a sheaf of rings  ${\cal F}$ is a sheaf 
of modules ${\cal B}$ satisfying the following
condition. For each $x\in M$ there exists a neighbourhood $U\ni x$
such that the restriction ${\cal B}\restrict U$ is free.

\definition
{\bf\blue A vector bundle} on a 
smooth manifold $M$ is a locally free sheaf of $C^\infty M$-modules.

\example Clearly, {\bf \purple tangent bundle is a vector bundle}.


\newpage

{\bf \blue Locally constant sheaves}

\definition
Let ${\cal F}$ be a sheaf on $M$ which takes a connected
non-empty open subset $U\subset M$ to a vector space 
or abelian group ${\Bbb V}$.
Extend ${\cal F}$ to all open sets using the gluing axiom. 
Then ${\cal F}$ is called
{\bf \blue the constant sheaf}, denoted ${\Bbb V}_M$.

\exercise
Prove that {\bf \purple the constant sheaf ${\Bbb V}_M$ exists,
and is unique up to isomorphism.}

\exercise
Let $W$ be an open set in $M$, and $S_W$ 
its set of connected components. Prove that 
{\bf \purple ${\Bbb V}_M(W)={\Bbb V}^{|S_W|}$}.

\definition
A {\bf \blue locally constant sheaf} is a sheaf which
is locally isomorphic to a constant sheaf.

\example
Let $\pi:\; M' \arrow M$ be a covering. Given
$U\subset M$, let $S_U$ be the set of connected
components of $\pi^{-1}(U)$, and set ${\cal F}(U)={\Bbb V}^{|S_W|}$.
We are going to define the restriction map $r$ as follows.
For an open subset $W\subset U$, consider the map $S_W\arrow S_U$
induced by the natural embedding $\pi^{-1}(W)\stackrel j 
\hookrightarrow \pi^{-1}(U)$.
For each direct sum component ${\Bbb V}_u\subset {\Bbb V}^{|S_U|}$ corresponding
to $u \in \im j$, let $r_u:\; {\Bbb V}_u\arrow {\Bbb V}_{j(u)}$  be identity.
For a component ${\Bbb V}_u\subset {\Bbb V}^{|S_U|}$ 
corresponding to $u \notin \im j$, we set $r_u=0$.
Then $r:=\bigoplus_{u\in S_U} r_u:\;  \bigoplus_{u\in S_U}{\Bbb V}\arrow 
\bigoplus_{w\in S_W}{\Bbb V}$. {\bf \purple This defines a locally constant sheaf 
on $M$} {\bf \red (prove it).}

\newpage

{\bf \blue \'Etal\'e space of a sheaf}

\definition
Let ${\cal F}$ be a sheaf on $M$,
and $U, V\supset x$ be two open set containing $x\in M$.
Two sections $f\in {\cal F}(U)$, $g\in {\cal F}(V)$
are called {\bf \blue equivalent in $x$} if 
there exists an open set $W\ni x $ such that $W\subset U\cap V$
and $f\restrict W= g\restrict W$. 
%{\bf \blue A germ of ${\cal F}$
%in $x$} is the set of all sections on all $U\ni x$ up to
%this equivalence. 
%When $K=x$ is a point, 
%we say that {\bf \blue $f$ is equivalent to $g$ in a point $x$.}
{\bf \blue A germ of a sheaf ${\cal F}$ in $x$} is a class of equivalence
of sections of ${\cal F}$ in all open sets $U \ni x$ under
this equivalence relation. {\bf \blue The stalk}
of a sheaf ${\cal F}$ in $x$ is the space ${\cal F}_x$ of all germs
in $x$.

\definition
Let $E({\cal F})$ be the set of all 
stalks of a sheaf ${\cal F}$ in all points $x\in M$.
A germ $f \in {\cal F}_m$ is called {\bf \blue a limit of a sequence of germs 
$f_i \in {\cal F}_{m_i}$} if $\lim_i m_i =m$ and
there exists a section $\tilde f$ of ${\cal F}$ over $U\ni x$
such that almost all $f_i$ are germs of $\tilde f$.
The {\bf \blue \'etal\'e topology} on $E({\cal F})$ is
defined as follows:
a subset $K\subset E({\cal F})$ is {\bf \blue closed in \'etal\'e topology}
if it contains all its limit points.

\remark Usually $E({\cal F})$ {\bf \red is non-Hausdorff}.


\newpage

{\bf \blue \'Etal\'e space of a constant sheaf}

\claim
Let ${\cal F}= {\Bbb V}_M$ be a constant sheaf on a 
manifold, and $x\in M$ a connected subset. {\bf \red Then
the space of germs of ${\cal F}$ in $x$ is equal to 
${\Bbb V}$.}

\proof
Since ${\cal F}$ is constant, the set of its sections
on any connected open set is equal to ${\Bbb V}$.
This gives a natural map  $r_x:={\cal F}(U) \arrow {\Bbb V}$:
we restrict $f\in {\cal F}(U)$ to a connected component 
$U_1$ of $U$ containing $x$, and obtain an element of ${\Bbb V}$.
{\bf \purple Clearly, two sections $f, g$ are equivalent in $K$
if and only if $r_x(f)=r_x(g)$.} This identifies ${\Bbb V}$
with the set of equivalence classes of sections in $x$.
\endproof

{\bf \green Corollary 1:}
Let ${\cal F}= {\Bbb V}_M$ be a constant sheaf on a 
manifold. {\bf \red Then the \'etal\'e space $E({\cal F})$
  of ${\cal F}$ is identified
with ${\Bbb V}$ disconnected copies of $M$.}

\proof
Indeed, a sequence 
$f_i \in {\cal F}_{m_i}$ converges to $f$
if $\lim_i m_i =m$ and $r_{m_i}(f_i)= r_m(f)$ for almost
all $i$.
\endproof

\newpage

{\bf \blue Local systems}

\definition
{\bf \blue Category of coverings} of $M$ is 
category $\cac$ with $\Ob(\cac)$ all coverings
and morphisms continuous maps of coverings 
compatible with projections to $M$.

\definition
Let $\pi_1:\; M_1 \arrow M$,
 $\pi_2:\; M_2 \arrow M$ be continuous maps.
{\bf\blue Fibered product} $M_1 \times_M M_2$ is the
subset of $M_1 \times M_2$ defined as 
$M_1 \times_M M_2:= \{(x, y)\in M_1 \times M_2\ \ |\ \ \pi_1(x)=\pi_2(y)\}$,
with induced topology. 

\exercise
Prove that {\bf \purple a fibered product of coverings is
  a covering}.

\definition
{\bf \blue An abelian group structure on a covering $\pi_1:\; M_1 \arrow M$}
is a morphism of coverings $\mu:\; M_1 \times_M M_1\arrow M_1$
together with a morphism $e:\; M \arrow M_1$ from a
trivial covering to $M_1$ and $\in \Hom_M(M_1)$ such that $\mu$ defines
an additive structure of an abelian group on the set $\pi_1^{-1}(x)$
for each $x\in M$, with $e(x)$ a unit in this group and $a$ the inverse.

\remark If, in addition, we have a 
group homomorphism
$\R^* \arrow \Aut_M(M_1, M_1)$ which equips each $\pi_1^{-1}(x)$
with a structure of a vector space, we obtain
{\bf\blue a structure of a vector space on a covering}.

\definition
{\bf \blue A local system} is a 
covering with a structure of an abelian  group or a vector space.

\newpage

{\bf \blue \'Etal\'e space of a locally constant sheaf}

\theorem
Let ${\cal F}= {\Bbb V}_M$ be a locally constant sheaf on a 
manifold. {\bf \red Then its \'etal\'e space $E({\cal F})$ is a
covering of $M$.}

\proof Immediately follows from Corollary 1.
\endproof

\theorem
{\bf \red Category of locally constant sheaves is equivalent to
the category of local systems.}

\proof
Let ${\cal F}$ be a locally constant sheaf, and 
$E({\cal F})$ its etale space. Then $E({\cal F})$ is a
covering of $M$. The structure of vector space
on germs defines the structure of vector space on 
$E({\cal F})$. {\bf \purple This gives a functor from
locally constant sheaves to local systems.}

Conversely, let $\pi:\; M_1 \arrow M$ be a local system,
and ${\cal F}(U)$ be the space of the sections
of $\pi^{-1}(U)\stackrel \pi \arrow U$. Then
${\cal F}(U)$ is a vector space. The correspondence $U
\arrow {\cal F}(U)$ gives
a sheaf, which is clearly locally constant.
\endproof

\newpage

{\bf \blue Connections (reminder)}

{\bf \green Notation:}
Let $M$ be a smooth manifold, $TM$ its tangent bundle,
$\Lambda^iM$ the bundle of differential $i$-forms,
$C^\infty M$ the smooth functions. {\bf \purple The space of sections 
of a bundle $B$ is denoted by $B$.}


\definition
A {\bf\blue connection}  on a vector bundle $B$
is an operator $\nabla:\; B \arrow B \otimes \Lambda^1 M$
satisfying $\nabla(fb) = b \otimes df + f \nabla(b)$,
where $f \arrow df$ is de Rham differential.
When $X$ is a vector field, we denote by
$\nabla_X(b)\in B$ the term $\langle \nabla(b), X\rangle$.

\remark
When $M=[0,a]$ is an interval, any bunlde $B$ on $M$ is trivial.
Let $b_1, ..., b_n$ be a basis in $B$. Then $\nabla$ can
be written as 
\[ \nabla_{d/dt}\left(\sum f_i b_i\right)= 
\sum_i \frac{df_i}{dt} b_i + \sum f_i \nabla_{d/dt} b_i
\]
with the last term linear on $f$. Therefore, the equation
$\nabla_{d/dt}(b)=0$ is a first order ODE, and {\bf \purple
it has a unique solution for any initial value 
$b_0=b\restrict{\{0\}}$.}



\newpage

{\bf \blue Curvature}

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

\remark This operation is well defined, because
\begin{multline*} \nabla(\eta \otimes fb)= d\eta\otimes fb + 
(-1)^{\tilde \eta} \eta \wedge \nabla (f b)= \\ d\eta\otimes fb +
(-1)^{\tilde \eta} \eta \wedge df\otimes b + f \eta \wedge \nabla b
= d(f\eta) \otimes b + f \eta \wedge \nabla b= \nabla(f\eta\otimes b)
\end{multline*}

\remark Sometimes $\Lambda^{2}(M)\otimes B 
\stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes B$
is denoted $d_\nabla$.

\definition
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 $b\in B$.
{\bf \red This is the formula expressing the action of
$\nabla^2$ on $\Lambda^* M \otimes B$}.


\newpage

{\bf \blue Riemann-Hilbert correspondence}

\definition
A connection is {\bf \blue flat} if
its curvature vanishes.



\theorem
Let $M$ be a connected manifold, $\cac_1$ the category 
of representations of $\pi_1(M)$, and $\cac_2$ the
category of local systems. {\bf \red Then the categories
$\cac_1$ and $\cac_2$ are naturally equivalent.}

\proof Follows from the equivalence between
locally constant sheaves and local systems. \endproof

\theorem
The categories $\cac_1$ and $\cac_2$ {\bf \red are naturally
equivalent to the category of vector bundles on $M$
equipped with flat connection.}

{\bf \green We prove it later in this lecture.}

\newpage

{\bf \blue Curvature and commutators}

\claim
Let $X, Y\in TM$ be vector fields, 
$(B, \nabla)$ a bundle with connection, and $b\in B$ its section.
Consider the operator
\[ \Theta_B^* (X, Y, b):= \nabla_X\nabla_Yb-\nabla_Y\nabla_Xb-\nabla_{[X,Y]}b
\]
{\bf \red Then $\Theta_B^* (X, Y, b)$ is linear in all three arguments.}

\pstep
The term $\Theta_B^* (X, Y, fb)$ {\bf \purple has 3 components: one
which is $C^\infty$-linear in $f$, one which takes
first derivative and one which takes the second derivative.}
The first derivative part is 
\[ \Lie_Yf\nabla_X b+ \Lie_Xf\nabla_Y b-\Lie_Yf\nabla_X b-\Lie_Xf\nabla_Y b-
\Lie_{[X,Y]}f b=-\Lie_{[X,Y]}f b,\]
the second derivative part is
$\Lie_X\Lie_Y (f)b- \Lie_Y\Lie_X (f)b=\Lie_{[X,Y]}f$, they cancel.
Therefore, {\bf \purple $\Theta_B^* (X, Y, b)$ is $C^\infty$-linear in $b$.}

{\bf \green Step 2:}
Since $[X, fY]= \Lie_X fY + f[X,Y]$, we have
$\nabla_{[X,fY]}b=f\nabla_{[X,Y]}b + \Lie_X f\nabla_Y b$.

{\bf \green Step 4:} The term $\Theta_B^* (X, fY, b)$
has two components, $f$-linear and the component
with first derivatives in $f$. Step 2 implies
that the component with derivative of first order is
$\Lie_X f\nabla_Y b- \Lie_X f\nabla_Y b=0$.
\endproof


\newpage

{\bf \blue Curvature and commutators (2)}

\remark
\[ \Theta_B^* (X, Y, b):= \nabla_X\nabla_Yb-\nabla_Y\nabla_Xb-\nabla_{[X,Y]}b
\]
is another definition of the curvature. {\bf \red The following
theorem shows that it is equivalent to the usual definition}.

\theorem
Consider $\Theta_B^*:\; TM \otimes TM \otimes B \arrow B$
as a 2-form with coefficients in $\End(B)$. {\bf \red Then
$\Theta^*_B=\Theta_B$,} where $\Theta_B=\nabla^2$
is the usual curvature.

\pstep
Since $\Theta^*_B(X,Y)$, $\Theta_B(X,Y)$ are linear
in $X, Y$, it would suffice to prove this equality
for coordinate vector fields $X, Y$.

{\bf \green Step 2:} Consider the operator
$i_X:\; \Lambda^iM \otimes B \arrow \Lambda^{i-1}M \otimes B$ 
of convolution with a vector field $X$.
Writing $\nabla = d+ A$, where $A\in \Lambda^1 M \otimes \End B$,
we obtain $\nabla_X = \Lie_X + A(X)$, which gives
$[\nabla_X, i_Y]= [\Lie_X, i_Y]= 0$ when $X, Y$ are coordinate
vector fields.

{\bf \green Step 3:} 
\[
 \nabla^2(b)(X,Y)= (i_Xi_Y -i_Xi_Y) \nabla^2(b)=
 i_Y\nabla_X \nabla b- i_X\nabla_Y \nabla b=
 \nabla_X\nabla_Y b- \nabla_Y \nabla_X b.
\]
\endproof


\newpage

{\bf \blue Parallel transport along the connection (reminder)}

\remark
When $M=[0,a]$ is an interval, any bundle $B$ on $M$ is trivial.
Let $b_1, ..., b_n$ be a basis in $B$. Then $\nabla$ can
be written as 
\[ \nabla_{d/dt}\left(\sum f_i b_i\right)= 
\sum_i \frac{df_i}{dt} b_i + \sum f_i \nabla_{d/dt} b_i
\]
with the last term linear on $f$. 

\theorem
Let $B$ be a vector bundle with connection over $\R$.
Then for each $x\in \R$ and each vector $b_x \in B\restrict x$
{\bf \red there exists a unique section $b\in B$ such that
$\nabla b=0$, $b\restrict x= b_x$.}

\proof This is existence and uniqueness of solutions
of an ODE $\frac {db}{dt} + A(b)=0$. \endproof

\definition
Let $\gamma:\; [0, 1] \arrow M$ be a smooth path
in $M$ connecting $x$ and $y$, and $(B, \nabla)$ 
a vector bundle with connection. Restricting 
$(B, \nabla)$ to $\gamma([0,1])$, we obtain a 
bundle with connection on an interval.
Solve an equation  $\nabla(b)=0$
for $b\in B\restrict{\gamma([0,1])}$
and initial condition $b\restrict x= b_x$.
This process is called {\bf\blue parallel transport}
along the path via the connection.
The vector $b_y:= b\restrict y$
is called {\bf\blue vector obtained by parallel
transport of $b_x$ along $\gamma$}.

\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 Let $B_1= B^{\otimes n} \otimes (B^*)^{\otimes m}$
be a tensor power of $B$. The connection on $B$ gives
the connection on $B_1$. Since parallel transport
is compatible with the tensor product, 
{\bf \purple the holonomy representation, associated with $B_1$,
is the corresponding tensor power of $B\restrict x$.}

\definition
Let $B$ be a vector bundle, and $\Psi$ a section of its tensor power.
We say that {\bf \blue connection $\nabla$ preserves $\Psi$}
if $\nabla(\Psi)=0$. In this case we also say that the tensor $\Psi$
is {\bf \blue parallel} with respect to the connection.

\newpage

{\bf \blue Flat bundles}

\remark
$\nabla(\Psi)=0$ is equivalent to $\Psi$ being a solution of
$\nabla(\Psi)=0$ on each path $\gamma$. This means that 
{\bf \purple parallel transport preserves $\Psi$.} 

We obtained

\corollary
{\bf \red 
A section of the tensor power of $B$ 
is parallel if and only if it is holonomy invariant.}

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

The following theorem will be proven later today.

\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 Fiber of a locally free sheaf}


\definition
Recall that {\bf\blue a vector bundle}
is a locally free sheaf of modules over
 $C^\infty M$. A vector bundle is called
{\bf \blue trivial} if it is isomorphic to $(C^\infty M)^n$.



\definition
Let ${\cal B}$ be an $n$-dimensional locally free
sheaf of $C^\infty$-modules on $M$,
$x\in M$ a point,   ${\goth m}_x\subset C^\infty M$
an ideal of $x\in M$ in $C^\infty M$.
Define {\bf\blue the fiber} of
${\cal B}$ in $x$ as a quotient
${\cal B}(M)/{\goth m}{\cal B}$.
A fiber of ${\cal B}$ is denoted
${\cal B}\restrict x$.

\remark
{\bf \purple A  fiber of a vector bundle of rank $n$
is an $n$-dimensional vector space.}

\remark
Let  ${\cal B}= C^\infty M^n$, and  
$b\in {\cal B}\restrict x$ a point of a fiber, represented
by a germ $\phi \in {\cal B}_x=C^\infty_m M^n$,
$\phi=(f_1, ..., f_n)$. Consider a map $\Psi$ from the set
of all fibers ${\cal B}$ to $M \times \R^n$,
mapping $(x, \phi=(f_1, ..., f_n))$
to $(f_1(x), ..., f_n(x))$. {\bf \red Then $\Psi$ is bijective.}
Indeed, ${\cal B}\restrict x=\R^n$.

\newpage

{\bf \blue Total space of a vector bundle}


\definition
Let ${\cal B}$ be an $n$-dimensional locally free sheaf
of $C^\infty$-modules. Denote the set of all
vectors in all fibers of  ${\cal B}$ over all points of $M$
by $\Tot {\cal B}$. Let  $U\subset M$ be an open subset of $M$,
with  ${\cal B}\restrict U$ a trivial bundle.
Using the local bijection $\Tot {\cal B}(U)=U \times \R^n$
we consider topology on $\Tot {\cal B}$
induced by open subsets in 
$\Tot {\cal B}(U)=U \times \R^n$ for all
open subsets $U\subset M$ and all trivializations of
${\cal B}\restrict U$. Then
$\Tot {\cal B}$ is called {\bf \blue a total space of a
  vector bundle ${\cal B}$.}

\claim 
The space $\Tot {\cal B}$ with this topology {\bf \purple is
a locally trivial fibration over $M$, with fiber $\R^n$.}


\remark
Let $B$ be a vector bundle on $M$, and $\psi\in B^*$
a section of its dual. Then $\psi$ defines a function
$x\arrow \langle \psi, x\rangle$ on its total space
$\Tot(B)\stackrel \pi \arrow M,$ linear on fibers of $\pi$.
This gives a {\bf \purple bijective correspondence between
sections of $B^*$ and functions on $\Tot(B)$
linear on fibers.}

This gives the following claim

\claim
Let $B$ be a vector bundle and
$\Sym^* B^*$ the direct sum of all symmetric
tensor powers of $B^*$. {\bf \red Then
the ring of sections of $\Sym^* B^*$ is
identified with the ring of all smooth functions
on $\Tot B\stackrel \pi\arrow M$ which are polynomial
on fibers of $\pi$.} \endproof


\newpage

{\bf \blue Polynomial functions on $\Tot(B)$ }

\claim
Let $D$ be the space of derivations 
$\delta:\; \R[x_1, ..., x_n] \arrow \C^\infty \R^n$.
{\bf \red Then $D$ is the space of derivations of the ring
$\C^\infty \R^n$.}
\endproof


\exercise {\bf \purple Prove it.}

The same argument brings the following

{\bf \green CLAIM 1:}
Let $D$ be the space of derivations 
$\delta:\; \Sym^* B^* \arrow \C^\infty (\Tot B)$.
{\bf \red Then $D$ is the space of derivations of
the ring $\C^\infty (\Tot B)$.}

\proof
Indeed, any derivation which vanishes on fiberwise
polynomial functions vanishes everywhere on $\C^\infty (\Tot B)$.
\endproof

\newpage

{\bf \blue Vector fields on $\Tot(B)$ }

\theorem
Let $(B,\nabla)$ be a bundle on $M$ with connection, and $X\in TM$
a vector field. {\bf \red Then there exists a vector field $\tau_\nabla(X)$
on $\Tot(B)$ mapping a section  $u\in \Sym^* B^*$ to $\nabla_X u$.}

\proof
Let $u, v \in \Sym^* B^*$, and $uv\in \Sym^* B^*$ their product.
Then $\nabla_x(uv)= u \nabla_x v + v \nabla_x u$
because $\nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2)$.
Therefore, $\tau_\nabla(X)(u):=\nabla_x(u)$ is a derivation of the
ring of functions on $\Tot(B)$ which are polynomial on fibers.
By Claim 1, any such derivation can be uniquely extended
to a vector field on $\Tot(B)$.
\endproof

\definition
Let $(B,\nabla)$ be a bundle with connection on $M$.
The corresponding {\bf \blue Ehresmann connection} on $\Tot(B)$
is the distribution $E_\nabla\subset T \Tot(B)$ obtained as $\tau_\nabla(TM)$.

\newpage

{\bf \blue Vector fields on $\Tot(B)$ and parallel sections}

{\bf \green CLAIM 2:}
Let $(B,\nabla)$ be a bundle with connection, and
$\pi:\; \Tot(B)\arrow M$ the standard projection,
and $T_\pi\Tot(B)=\ker D\pi$ is the vertical tangent space
(Lecture 14). \\
\phantom{x}\ \ \ (i) {\bf \red Then $T\Tot B =  E_\nabla\oplus T_\pi\Tot(B)$,
where $E_\nabla$ is the Ehresmann connection. } \\
\phantom{x}\ \ \ (ii) Moreover, {\bf \red a section $f$ of $B$ is parallel if an
only if its image $f(M)\subset \Tot(B)$ is tangent to $E_\nabla$.}

\proof The second assertion is clear from the definition:
{\bf \purple a section $b$ is tangent to $E_\nabla$ if it is preserved by
all vector fields $a=\tau_\nabla(X)$ generating $E_\nabla$.} In this case
$\Lie_a(\tilde b)=0$, where $\tilde b$ is a function
on $\Tot(B^*)$ defined by $b$. However, $\Lie_a(\tilde b)=\widetilde{\nabla_X(b)}$
where $\widetilde{\nabla_X(b)}$ is a function on  $\Tot(B^*)$
associated with $\nabla_X(b)$. Therefore, $\Lie_a(\tilde b)=0$ $\Leftrightarrow$
$\nabla_X(b)=0$. 

To prove (i), we notice that $D\pi\restrict{E_\nabla}:\; E_\nabla \arrow TM$
is an isomorphism at every point of $\Tot B$. Indeed, these bundles have the same rank,
and for each $\tau_\nabla(X)\in E_\nabla$, this vector field
acts on functions pulled back from $M$ as $\Lie_X$, hence
$D\pi\restrict{E_\nabla}$ is injective.
\endproof


{\bf \blue Bundles with trivial holonomy }


\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}.\\
\proof
Let $B$ be a flat bundle on $M$, and $X, Y \in TM$ commuting vector fields.
Then $\nabla_X:\; B \arrow B$ commutes with $\nabla_Y$. Then 
the Ehresmann connection bundle $E_\nabla\subset T\Tot B$ is generated
by commuting vector fields $\tau_\nabla(X)$, $\tau_\nabla(Y)$, ..., 
hence it is involutive. By Frobenius theorem, every
point $b\in \Tot(B)$ is contained in a leaf $\cal E$ of 
the corresponding foliation, tangent to $E_\nabla$.
By Claim 2, such a leaf is a parallel section of $B$.
The projection from $\cal E$ to $M$ is a covering.
Since $M$ is simply connected, $\cal E=M$, and
$B$ is trivialized by parallel sections.

Conversely, assume that $B$ has trivial holonomy.
Then $\Tot(B)= M \times B\restrict x$ because
each point is contained in a unique parallel section,
hence the bundle $E_\nabla$ is involutive. Then $[\nabla_X, \nabla_Y]=0$
for any commuting $X, Y\in TM$, and the curvature vanishes.
\endproof

{\bf \green Corollary 1:} Let $B$ be a flat vector bundle
on a simply connected, connected manifold $M$.
{\bf \red Then for each $x\in M$ and each $b \in B\restrict x$,
there exists a unique parallel section of $B$ passing through $b$.}
\endproof

\newpage

{\bf \blue Riemann-Hilbert correspondence}

\theorem
The category of locally constant sheaves of vector spaces {\bf \red is naturally
equivalent to the category of vector bundles on $M$
equipped with flat connection.} \\
\pstep
Consider a constant sheaf $\R_M$ on $M$. This is a sheaf of rings, and
any locally constant sheaf is a sheaf of $\R_M$-modules.

Let ${\Bbb V}$ be a locally constant sheaf, and $B:= {\Bbb V}\otimes_{\R_M} \C^\infty M$.
Since ${\Bbb V}$ is locally constant, the sheaf $B$ is a locally free sheaf of $C^\infty$-modules,
that is, a vector bundle. Let $U\subset M$ be an
open set such that ${\Bbb V}\restrict U$ is constant.
If $v_1, ..., v_n$ is a basis in ${\Bbb V}(U)$, all sections of $B(U)$
have a form $\sum_{i=1}^n f_i v_i$, where $f_i \in C^\infty U$.
Define the connection $\nabla$ by 
$\nabla\left (\sum_{i=1}^n f_i v_i\right) = \sum df_i \otimes v_i$.
This connection is flat because $d^2=0$. It is independent from the choice of $v_i$
because $v_i$ is defined canonically up to a matrix with constant coefficients.
{\bf \purple We have constructed a functor from locally constant sheaves
to flat vector bundles}.

{\bf \green Step 2:}
Let now $(B, \nabla)$ be a flat bundle over $M$.
The functor to locally constant sheaves takes 
 $U\subset M$ and maps it to the space of parallel
sections of $B$ over $U$. This defines a sheaf
${\Bbb B}(U)$. For any simply connected $U$, and any $x\in M$,
the space ${\Bbb B}(U)$ is identified with a vector space
$B\restrict x$ (Corollary 1), hence ${\Bbb B}(U)$ is locally constant.
Clearly, $B= {\Bbb B}\otimes_{\R_M} \C^\infty M$, hence
{\bf \purple this construction gives an inverse functor to 
${\Bbb V}\mapsto {\Bbb V}\otimes_{\R_M} \C^\infty M$.}
\endproof


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