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\title{Riemann Surfaces}
\author{JWR}
\date{Friday December 14, 2001, 8:42 AM}

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\begin{document}

\maketitle



%\input{algebraicCurves}

%\input{uniformization}


\section{The Classifying Map}


A reference for the material in this section is~\cite{GH}.

\para\label{grass}
Let $V$ be a vector space over $\C$. We denote by $\G_k(V)$ the
\jdef{Grassmannian} of $k$-dimensional subspaces of $V$ and by
$$
   \P(V)=\G_1(V)
$$
the projective space of $V$.
Two vector bundles over the  Grassmannian $\G_k(V)$ are
the \jdef{tautological bundle}
$$
   T\to\G_k(V), \qquad T_\lambda :=\lambda\subset V
$$
and the \jdef{co-tautological bundle}
$$
   H\to\G_k(V), \qquad H_\lambda := T_\lambda^*=V^*/\lambda^\perp
$$
where $V^*$ is the dual space to $V$ and $\lambda^\perp=\{\alpha\in V^*:\alpha|\lambda=0\}$.
In case $k=1$ the bundle $H\to\P(V)$ is also called the \jdef{hyperplane bundle}.
Note the canonical isomorphism
$$
  \G_{N-k}(V)\to\G_k(V^*):\lambda\mapsto\lambda^\perp
$$
where $N=\dim_\C(V)$.
For any holomorphic bundle $E\to X$ we denote by $\cO(X,E)$
the vector space of  its holomorphic sections.
\arap

\begin{theorem} \label{thm:cotaut}
Each functional $\alpha\in V^*$ determines a section $s_\alpha$ of the cotautological bundle
$H\to\G_k(V)$ via
$$
       s_\alpha(\lambda)=\alpha|\lambda.
$$
The map
$$
      V^*\to\cO(\G_k(V),H):\alpha\mapsto s_\alpha
$$
is an isomorphism.
\end{theorem}

\begin{proof} It is clear that the map is injective; to show it is surjective
we choose $s\in\cO(X,H)$; we must find $\alpha\in V^*$ with $s=s_\alpha$.
This can be found
via a power series argument using the standard affine coordinates on the
Grassmannian as follows.  \ETC
\end{proof}

\para Let $E\to X$ be a holomorphic vector bundle of rank $k$ over a
compact complex manifold $X$.
The vector space $\cO(X,E)$ of holomorphic sections of $E$ is finite dimensional by
elliptic theory.
A \jdef{base point} of $E$ is a point $p\in X$ where the space $\{s(p):s\in\cO(X,E)\}$
is a proper subspace of the fiber $E_p$;
the bundle is called \jdef{base point free} iff it has no base points.
If $p$ is not a base point of $E$ we have
$$
\{s\in\cO(X,E):s(p)=0\}\in\G_{N-k}(\cO(X,E))
$$
here $N:=\dim_\C(\cO(X,E))$
and hence, via the canonical isomorphism of paragraph~\ref{grass},
$$
f(p):=\{s\in\cO(X,E):s(p)=0\}^\perp\in\G_k(\cO(X,E)^*).
$$
For  a base point free bundle $E\to X$ this defines a map
$$
   f:X\to\G_k(\cO(X,E)^*)
$$
called the \jdef{classifying map} of $E$.
\arap

\begin{theorem}\label{thm:noBasePoint}
Let $E\to X$ a base point free holomorphic bundle
and let $H\to\G_k(\cO(X,E)^*)$ be the cotautological bundle.
Then the pull back of $H$ by the classifying map  is $E$.
\end{theorem}

\begin{proof}
For each $p\in X$ we have a linear isomorphism
$$
        E_p\to H_{f(p)}:=\{s\in\cO(X,E):s(p)=0\}^\perp: v\mapsto \eta_v
$$
where $\eta_v(\alpha)=\alpha(s)$ for
$\alpha\in\{s\in\cO(X,E):s(p)=0\}^\perp\subset\cO(X,E)^*$
and
$s\in\cO(X,E)$ with $s(p)=v$.
\end{proof}


\begin{theorem} A compact Riemann surface $X$ can be
(1)~embedded in $\P^3$
and
(2)~immersed in $\P^2$ so that its image has only transverse double points.
\end{theorem}

\begin{proof}[Sketch of Proof] Theorem~\ref{thm:noBasePoint} says that
vector bundles without base point correspond to maps to the Grassmannian.
In particular, line bundles without base point correspond to maps to projective space.
If $X$ is a Riemann surface,
Theorem~\ref{thm:kodaira4RS} below says that a holomorphic line bundle of sufficiently
high degree  has no base point and that the classifying map is injective
and the derivative of the classifying map  is never zero.
Thus shows that any Riemann surface can be embedded in projective space $\P^N$.
Suppose $\P^{N-k}$ and $\P^k$ are transverse projective  subspaces of $\P^N$
of the indicated dimensions.
By transversality theory, for $k\ge 2$, a generic $\P^{N-k}$ misses the image of $X$.
Then projection onto $\P^k$ along $\P^{N-k}$ gives a map from $X$ into $\P^k$.
By transversality theory this projection is generically an embedding of $X$ for $k=3$
and generically  an immersion with at worst transverse double points for $k=2$.
\end{proof}



\section{Degree}

\para Let $E\to X$ be a smooth fiber oriented vector bundle over a compact smooth
oriented manifold $X$. Assume that the  \jdef{rank} (=fiber dimension) of $E$
is the same as the dimension $n$ of $X$. For an isolated zero $p\in X$
of a smooth section $s$ of $E$ define the \jdef{local degree} $\deg_p(s)$
of $s$ at $p$ by
$$
 \deg_p(s) = \mathrm{degree}\left(S_p\to S(E_p):q\mapsto \frac{s(q)}{|s(q)|}\right)
$$
where $S_p$ is the boundary of a small disk $D$ in $X$  centered at $p$ and
$S(E_p)$ is the boundary of the unit disk of the fiber $E_p$ in some trivialization
of $E$ over $D$. Here the disk $D$ is small in the sense that the only zero of
$s$ in its closure is the point $p$. Because the degree of a map between spheres
of the same dimension is a homotopy invariant, the local degree of a smooth section $s$
at an isolated zero is independent of the
choice of the small disk $D\subset X$ about $p$
and of the choice of the local trivialization of $E|D$ used in the definition.
\arap

\dfn By  transversality theory (see Milnor,
{\em Topology from the differentiable viewpoint}) the number
$$
    \deg(E)=\sum_{s(p)=0}\deg_p(s)
$$
is independent of the choice of the smooth section with isolated zeros
used to defined it. This number is called the \jdef{degree} or \jdef{Euler number}
of the bundle $E\to X$. The cohomology class $e(E)\in H^n(X)$
defined by
$$
\inner{e(E)}{[X]}=\deg(E)
$$
is called the \jdef{Euler class} of the bundle $E\to X$;
here $[X]\in H_n(X)$ is the fundamental class.
In the case $X$ is a surface (i.e. a smooth manifold of real dimension two) and $L\to X$
is a complex line bundle the Euler class of $L$ is called the \jdef{Chern class}
and denoted by $c_1(L)$.
\nfd

\begin{theorem} The Euler number of the cotangent bundle $T^*X\to X$ (and hence also
of the tangent bundle $TX\to X$) is the Euler characteristic $\chi(X)$.
\end{theorem}

\begin{proof} Let $f:X\to\R$ be a Morse function. Then the section $df$ of $T^*X$
has isolated zeros. At a critical point $p$ the Morse lemma tells us that there are
coordinates $x_1,\ldots,x_n$ such that
$$
   f(q) = -x_(q)^2-\cdots-x_k(q)^2+x_{k+1}(q)^2+\cdots+x_n(q)^2
$$
so
$$
   \frac{df(q)}{|df(q)|}= -x_1\,dx-\cdots-x_k\,dx_k+x_{k+1}\,dx_{k+1}+\cdots+x_n\,dx_n.
$$
Hence $\deg_p(df)=(-1)^k$. The result now follows by Morse theory.
\end{proof}




\section{Line Bundles}


\para\label{deg-Ord}
Assume that $X$ is a Riemann surface
and $E=L\to X$ is a holomorphic line bundle over $X$.
Let $s$ be a  meromorphic section of $L$ not identically zero.
Then near a singularity (i.e. zero or pole) of $s$ we may choose a local trivialization
of $L$ and a holomorphic coordinate $z=re^{i\theta}$ such that
$$
    s(q)= z(q)^k= r^ke^{ik\theta}.
$$
The integer $k$ is independent of the choice
of the local trivialization and local coordinate and is
called the \jdef{order} of $s$ at $p$ and denoted $\Ord_p(s)$.
The order is the degree of the map $q\mapsto|s(q)|^{1}s(q)$
from a small circle about the singularity to the unit circle of the fiber.
Hence
$$
    \deg_p(s)=\Ord_p(s).
$$
The formula
$$
  \deg(L)=\sum_p \deg_p(s)
$$
holds for meromorphic sections since we may modify $s$ near each pole so as to
produce a smooth $s$ with a zero of the same degree  via the formula
$\tilde{s}(q) = \phi(r)e^{ik\theta}$ where $\phi(r)=r^k$ for $r$ near the boundary
of the  domain of $z$, $\phi(r)>0$ for $r>0$,  and $\phi(0)=0$.
\arap

\dfn The \jdef{canonical bundle} over a Riemann surface $X$ is
the bundle $K\to X$ whose fiber $K_p$ over a point $p\in X$
is the vector space
$$
    K_p=L_\C(T_pX,\C)
$$
of $\C$-linear maps from the tangent space $T_pX$ to $\C$.
This bundle should be distinguished from the cotangent bundle
$T^*X\to X$ whose fiber is the real  dual space
$$
    T_p^*X=L_\R(T_pX,\R)
$$
for $p\in X$.
Each holomorphic coordinate
$z$ gives a nonzero local section $dz$ of $K$ and on the overlap of the domains
of two holomorphic coordinates $z$ and $w$ we have
$$
    dw=\phi'\,dz
$$
where where $\phi$ is the holomorphic function such that $w(q)=\phi(z(q))$.
A meromorphic section of $K$ is called a \jdef{meromorphic  differential}.
A holomorphic section of $K$ is called a \jdef{holomorphic  differential}
or (in some books) an \jdef{abelian  differential}.
\nfd

\begin{theorem}\label{thm:deg-K}
Let $X$ be a compact Riemann surface. Then the degree of the canonical bundle
over $X$ is
$$
   \deg(K)=-\chi(X)
$$
where $\chi(X)$ is the Euler characteristic of $X$.
\end{theorem}

\begin{proof}
Let $\omega$ be a meromorphic differential on $X$.
The real valued form $\xi=\Re(\omega)$ is a section of
the cotangent bundle.
Near a singularity $\omega= z^k\,dz$ in a suitable holomorphic
coordinate. Now $z^k=r^k(\cos k\theta+i\sin k\theta)$  and
$dz=dx+idy$ so
$
  \xi=r^k\cos(k\theta)\,dx-r^k\sin(k\theta)\,dy
$
and hence
$
 \deg(K)=\deg_p(\omega)= -\deg_p(\xi)=-\chi(X).
$
\end{proof}


\section{Divisors}

\para\label{divisorNotation}
A \jdef{divisor} on a
Riemann surface $X$ is a $\Z$ valued function $D$ on $X$ with discrete support.
The divisors on $X$ form a group under pointwise addition denoted $\Div(X)$.
When $X$ is compact the support is finite and we usually represent
the divisor  as a formal finite sum
$$
  D= \sum_{k=1}^m n_k p_k
$$
where $n_k=D(p_k)$.
A meromorphic section $s$ of a holomorphic line
bundle $L\to X$ (in particular a meromorphic function) determines a
divisor
$$
      (s)=\sum_{p\in X} \deg_p(s) p
$$
whose support is the set of all singularities (zeros and poles) of $s$.
The \jdef{degree} of the divisor $D$ of finite support is the integer
$$
\deg(D)=\sum_{k=1}^m n_k;
$$
thus
$$
\deg((s))=\deg(L)
$$
for a meromorphic section $s$ of a holomorphic line bundle $L\to X$.
A \jdef{principal divisor} is one  of form $(f)$ where $f$ is a  meromorphic function.
two divisors are called \jdef{linearly equivalent} iff they differ by a principal divisor.
The notation $D\ge 0$ means that $D$ takes only nonnegative values.
A divisor $D$ is called \jdef{positive} or \jdef{effective} iff $D\ge 0$.
For any divisor $D$ we define the complex vector space
$$
   \cL(D):=\{f\in \cM(X): f=0 \mbox{ or } (f)+D\ge 0 \}
$$
and
$$
   \ell(D):=\dim_\C(\cL(D)).
$$
Here  $\cM(X)$ is the function field of $X$,
i.e. the field of meromorphic functions on $X$.
\arap

\begin{theorem}\label{thm:div=bundle}
 A divisor and a meromorphic section of a holomorphic line bundle
are essentially the same thing. More precisely
\begin{description}
\item[(i)] Every holomorphic line $L\to X$ admits a meromorphic section $s$.
\item[(ii)] A meromorphic section $s$ is holomorphic if and only if its divisor $(s)$
is effective.
\item[(iii)] For every divisor $D$ there is a holomorphic line bundle $L_D\to X$
and a meromorphic section $s_D$ of $L_D$ with $D=(s_D)$.
\item[(iv)] Assume that $s$ and $s'$ are meromorphic sections
of holomorphic line bundles $L$ and $L'$ respectively.
Then $(s)=(s')$ if and only if there
is an isomorphism $L\to L'$ of holomorphic line bundles
which carries $s$ to $s'$.
\item[(v)] Two divisors are linearly equivalent if and only if the
corresponding holomorphic line bundles are isomorphic.
\item[(vi)] Let $D$ be the divisor of a meromorphic section $s$
of a holomorphic line bundle $L\to X$. Then the map
$$
     \cL(D)\to\cO(X,L): f\mapsto fs
$$
is an isomorphism from the vector space $\cL(D)$ onto the vector space
$\cO(X,L)$ of holomorphic sections of $L$.
\end{description}
\end{theorem}


\begin{proof} For the proof of~(i) see Theorem~\ref{thm:meromorphicSection} below.
Part~(ii) is obvious; a meromorphic section is holomorphic if and only if it has no poles.
Given a divisor $D= \sum_{k=1}^m n_k p_k$ we will construct a line bundle $L_D$
and a meromorphic section $s_D$.
Choose a cover $\cU=\{U_k\}_{0\le k\le m}$ of $X$
such that $U_k$ is the domain of a holomorphic coordinate $z_k$ centered at $p_k$
for $k=1,2,\ldots,m$ and $U_0=X\setminus\{p_1,p_2,\ldots,p_m\}$. Define
a meromorphic function $s_k$ on $U_k$ by $s_k=z_k^{n_k}$ for $k=1,2,\ldots,m$ and
$s_0=1$. The holomorphic functions $g_{jk}:U_j\cap U_k\to\C^*$ defined by
$$
    g_{jk} = \frac{s_j}{s_k}
$$
satisfy $g_{ij}g_{jk}g_{ki}=1$ and thus form the transition functions for a line bundle
$L=L_D$. The formula $s_j=g_{jk}s_k$ says that the functions $s_k$ fit together to form
a (meromorphic) section $s_D$ of $L_D$ and by construction $(s_D)=D$. This proves~(iii).

For part~(iv) note that trivializations of $L$ and $L'$ over a common open set
$U$ determine a unique function $\phi$ on $U$ with $s'=\phi s$ and $\phi$
is holomorphic and nowhere zero on $U$ since $(s)=(s')$. For part~(v)
assume that $s$ and $s'$ are meromorphic sections of $L$. Then the unique function
$\psi$ on $X$ such that $s'=\psi s$ is meromorphic and satisfies
$(s')=(\psi)+(s)$. Conversely assume that $s$ and $s'$ are meromorphic sections of
$L$ and $L'$ respectively and that $(s')=(\psi)+(s)$ for some meromorphic function $\psi$
on $X$. Choose an open  cover $\{U_k\}_k$ such that both $L|U_k$ and $L'|U_k$
are trivial and let $s_k=s|U_k$, $s'_k=s'|U_k$, and $\psi_k=\psi|U_k$.
Then $s'_k$ and $\psi_k s_k$
have the same divisor in $U_k$ so $s'_k=\phi_k\psi_k s_k$ where $\phi_k:U_k\to\C^*$.
Thus
$$
    g_{jk} := \frac{s_j}{s_k}= \frac{s'_j}{s'_k}=:g'_{jk}
$$
which shows that the corresponding line bundles $L$ and $L'$ are isomorphic.

For part~(vi) first note that the condition $(f)+(s)\ge 0$ implies that
$fs$ is holomorphic so the map $f\mapsto fs$ carries $\cL(D)$ to $\cO(X,L)$.
This map is clearly injective.
Let $g_{jk}=s_j/s_k$ be the transition functions of part~(iii).
Then a holomorphic section of $L$ is a collection of holomorphic functions $\sigma_k$ such that
$\sigma_j=g_{jk}\sigma_k$. It follows that
$\sigma_j/s_j=\sigma_k/s_k$ on $U_j\cap U_k$, i.e. there is a meromorphic function
$f$ defined on $X$ with $f|U_k=\sigma_k/s_k$, i.e. $\sigma=f s$. Since
$\sigma$ is holomorphic we have $(f)+(s)=(\sigma)\ge 0$, i.e. $f\in\cL(D)$.
This shows that the map $f\mapsto fs$ is surjective.
\end{proof}

\para\label{def:picard}
The isomorphism classes of holomorphic line bundles over a Riemann surface $X$
form an abelian group called the \jdef{Picard group}
and denoted by $\Pic(X)$.
The trivial line bundle is the identity element
of $\Pic(X)$, the group operation is tensor product, and the inverse of
a line bundle is its dual bundle.
Theorem~\ref{thm:div=bundle} defines an isomorphism
$$
  \Pic(X)= \Div(X)/\sim
$$
where $\Div(X)$ is the group of divisors on $X$
and $\sim$ denotes linear equivalence of divisors.
This isomorphism is an isomorphism of groups since
$$
(s_1\otimes s_2)=(s_1)+(s_2)
$$
for meromorphic sections $s_1$ and $s_2$ of $L_1$ and $L_2$ respectively.
The degree of a holomorphic bundle defines a homomorphism of groups
$$
  \Pic(X)\to\Z: L\mapsto\deg(L).
$$
\arap

\rmk Some authors mean by the term {\em Picard group} the subgroup
$\Pic_0(X)= \Div_0(X)/\sim$ of line  bundles of degree zero.
\kmr

\section{Sheaves}

In this section $\cC$ denotes the category of abelian groups
but much of the theory described here works for any category $\cC$
which admits inverse limits.

\para Let $X$ be a topological space.
A \jdef{presheaf} on $X$ is contravariant functor $\cS$ from the category
of open subsets of $X$ and inclusions to the category $\cC$. Usually the
value $\cS(U)$ of $\cS$ on an open subset $U$ is a space of functions defined on $U$
(or more generally a space of sections of a bundle over $U$)
so we use the notation
$$
   s|V= \cS(\iota_{VU})(s)
$$
for open sets $V\subset U\subset X$, $s\in\cS(U)$, and where $\iota_{VU}:V\to U$ is the
inclusion; we call $s|V$ the \jdef{restriction} of $s$ to $V$.
A \jdef{sheaf} is a presheaf $\cS$ which satisfies the following
\jdef{completeness axiom}: {\em For every open set $U\subset X$ and every indexed open cover
$\{U_i\}_{i\in I}$ of $U$ the map
$$
   \cS(U)\to \left\{s\in\prod_{i\in I} \cS(U_i): s|U_i\cap U_j=s_j|U_i\cap U_j\right\}: s\mapsto (s|U_i)_{i\in I}
$$
is a bijection.}
\arap


\para Let $\cS$ be a presheaf over $X$. Consider triples
$(s,U,x)$ where $U\subset X$ is open, $s\in\cS(U)$, and $x\in U$.
Define an equivalence relation
on these triples by $[s_1,U_1,x_1]=[s_2,U_2,x_2]$ iff $x_1=x_2$
and there exists an open set $U\subset X$ with
$x\in U\subset U_1\cap U_2$ and $s_1|U_1=s_2|U_2$.
The equivalence class $[s,U,x]$ is called the \jdef{germ} of $s$ at $x$;
the \jdef{stalk} $\cS_x$ is the set of germs $[s,U,x]$ at $x$.
The natural projections
$$
  \cS(U)\to\cS_x:s\mapsto[s,U,x]
$$
commute with restrictions so each stalk $\cS_x$ is an object of $\cC$.
\arap

\rmk To any presheaf $\cS$ over $X$ we can associate the disjoint union
$S := \coprod_{x\in X} \cS_x$. Denote by
$\pi:S\to X$ the obvious projection $\pi([s,U,x])=x$. Each
$s\in\cS(U)$ determines a section $\gamma_s$ of $\pi$ over $U$  via the formula
$\gamma_s(x)=[s,U,x]$.
Equip $S$ with the topology that makes each of these sections
a homeomorphism onto its image. Then $\pi$ is a local homeomorphism.
For each open set $U\subset X$ let
$\cO(X,U)$ denote the set
of continuous sections of $\pi$ over $U$, i.e. the set of continuous maps
$\gamma:U\to \pi^{-1}(U)$ such that $\pi\circ\gamma(x)=x$ for $x\in U$.
It is not hard to prove that
\begin{itemize}
\item the functor $U\mapsto\cO(X,U)$ is a sheaf,
and
\item a presheaf $\cS$ is a sheaf if and only if the map
$
       \cS(U)\to\cO(X,U):s\mapsto\gamma_s
$
is a bijection for every open set $U\subset X$.
\end{itemize}
A local homeomorphism is called an \jdef{etale space}. Some authors
(e.g.\cite{GUNNING})
use the term {\em sheaf} to signify an  etale space where each $\pi^{-1}(x)$
is an abelian group, and the term
{\em complete presheaf}, for what we have called a sheaf.
The topology on $S$ is confusing (it is often not Hausdorff)
and we will avoid it.
\kmr

\para Let $\cS$ and $\cS'$ be sheaves over $X$.
A \jdef{morphism} from the sheaf $\cS$ to the sheaf $\cS'$
is a natural transformation $T:\cS\to\cS'$, i.e. an operation which
assigns to each open set $U\subset X$ a morphism
$T:\cS(U)\to \cS'(U)$ which intertwines the restriction morphisms.
The morphism induces a morphism $T_x:\cS_x\to\cS'_x$ of stalks
for each $x\in X$.
\arap

\para A sequence of sheaves and morphisms between them is called \jdef{exact}
iff induces an exact sequence on stalks for each $x\in X$.
A sheaf $\cS$ is a \jdef{subsheaf} of the sheaf $\cT$ iff
$\cS(U)$ is a subgroup of $\cT(U)$ for every open set $U\subset X$.
It is easy to prove that when $\cS$ is a subsheaf of  $\cT$ there
is an exact sequence
$$
   0\to\cS\to\cT\to\cT/\cS\to 0
$$
where the morphism $\cS\to\cT$ is the inclusion; the sheaf $\cT/\cS$ is unique
up to isomorphism. WARNING: It can happen that $(\cT/\cS)(U)\ne\cT(U)/\cS(U)$.
For example, this happens for the exponential sequence (see below)
$$
  0\to\Z\to\cE\mapright{\exp}\cE^*\to 0
$$
when $U$ is not simply connected.
\arap



\para \label{standardSheaves} Here is a list of some important sheaves.
\medskip

\begin{tabular}{ll}
 $\cE$ &    smooth functions; \\
 $\cE^p$ &    smooth $p$-forms ($\cE=\cE^0$); \\
 $\cE^*$ &     nowhere zero smooth functions;  \\
 $\cE^{p,q}$ &    smooth $(p,q)$-forms; \\
 $\cO$ &    holomorphic functions; \\
 $\cO^*$ &    nowhere zero holomorphic functions; \\
 $\Omega^p$ &    holomorphic $p$-forms ($\cO=\Omega^0$); \\
 $\cM$ &    meromorphic functions; \\
 $\cM^*$ &    not identically zero  meromorphic functions; \\
 $\cP$ & principal parts, $\cP=\cM/\cO$;\\
 $\cD$ & divisors, $\cD=\cM^*/\cO^*$.
\end{tabular}

\medskipnoindent The sheaves $\cE$, $\cE^p$, and $\cE^*$ are defined on
any smooth manifold $X$ and the sheaf $\cM^*$ is defined on an almost complex manifold;
the remaining sheaves are defined only on complex manifolds.
Unless otherwise specified the functions are complex valued.
The group operation in all these is addition of functions except for $\cE^*$, $\cO^*$,
and $\cM^*$ where the operation is multiplication.
When $E\to X$ is a vector bundle and $\cS$ is one of the above sheaves
we denote by $\cS(E)$ the corresponding sheaf of sections of $E$ and we
use the abbreviation
$$
    \cS(U,E):=\cS(E)(U)
$$
so that the elements of $\cS(X,E)$ are sections of $E\to X$.
For example, $\cE(E)$ denotes the sheaf which assigns to the open set
$U$ the space of smooth sections of $E|U$ and
$\cE^{p,q}(E)$ denotes the sheaf which assigns to the open set
$U$ the space of smooth $E$ valued forms of type $(p,q)$ defined on  $U$.
These sheaves are defined for smooth complex vector bundles;
the subsheaves $\cO(E)\subset\cE(E)$ and $\Omega^p(E)\subset\cE^{p,0}(E)$ are defined only
for holomorphic bundles.
\arap

\rmk \label{cM(L)}
For any holomorphic line bundle the sheaf $\cM$
of meromorphic functions and the sheaf $\cM(L)$ of meromorphic sections
of $L$ are isomorphic. Each meromorphic section $s\in\cM(X,L)$ determines
an isomorphism via the formula
$$
     \cM(U)\to\cM(U,L): f\mapsto fs.
$$
\kmr

\para
For each abelian group $G$ we denote the corresponding \jdef{constant sheaf}
by the same symbol, i.e. $G(U)=G$. For each point $p$ we define
the \jdef{skyscraper sheaf} $G_p$ by $G_p(U)=G$ if $p\in U$
and $G_p(U)=0$ if $p\notin U$.
\arap

\para \label{standardSequences} Here are some important exact sequences of sheaves.

\medskipnoindent the smooth exponential sequence:
$$
  0\to\Z\to\cE\mapright{\exp}\cE^*\to 0
$$
the holomorphic exponential sequence:
$$
  0\to\Z\to\cO\mapright{\exp}\cO^*\to 0
$$
the principal part sequence:
$$
  0\to\cO\to\cM\to\cP\to 0
$$
the divisor sequence:
$$
  0\to\cO^*\to\cM^*\to\cD\to 0
$$
the de Rham complex:
$$
  0 \to \R\to\cE^0\mapright{d}\cE^1\mapright{d}\ldots.
$$
the $p$th Dolbeault complex:
$$
  0 \to \Omega^p\to\cE^{p,0}\mapright{\pbar}\cE^{p,1}\mapright{\pbar}\ldots.
$$
\arap


\section{Sheaf Cohomology}

A good reference for the material in this section is~\cite{GUNNING}.

\para Let $X$ be a topological space. An \jdef{open cover} of  $X$
is an indexed collection $\cU=\{U_i\}_{i\in I}$ of open subsets of $X$
such that $X=\bigcup_{i\in I} U_i$.
A \jdef{refinement} of the open cover
$\cU$ is open cover  $\cV=\{V_j\}_{j\in J}$  such that there is a map
and $\tau:J\to I$ is a map such that $V_j\subset U_{\tau(j)}$ for $j\in J$;
the map $\tau$ is called a \jdef{refining map} from $\cV$ to $\cU$.
The \jdef{nerve} of the open cover $\cU$ is the set
$$
    \mathrm{Nerve}(\cU)=\bigcup_{k\ge 0} N^k(\cU)
$$
where $N^k(\cU)$ consists of all finite subsequences
$$
\sigma=( i_0,i_1,\ldots,i_k)\in I^{k+1}
$$
such that the open set
$$
    U_\sigma:=\bigcup_{r=0}^k U_{i_r}
$$
is nonempty.
\arap

\para Let $\cS$ be a sheaf over $X$ and $\cU=\{U_i\}_{i\in I}$ be an open cover
of $X$. We define a chain complex
$$
  0 \mapright{\delta} C^0(\cU,\cS)\mapright{\delta} C^1(\cU,\cS)\mapright{\delta} C^2(\cU,\cS)\mapright{\delta}\cdots
$$
by
$$
      C^k(\cU,\cS):= \bigoplus_{\sigma\in N^k(\cU)} \cS(U_\sigma)
$$
and
$$
    (\delta \mu)_\sigma = \sum_{r=0}^k (-1)^r \mu_{\sigma(r)}|U_\sigma
$$
where $\sigma=(i_0,i_1,\ldots,i_k)$ and $\sigma(r)=(i_0,i_1,\ldots,i_{r-1},i_{r+1},\ldots i_k)$.
The $k$th \jdef{cohomology group} of the sheaf $\cS$ for the cover $\cU$ is
$$
 H^k(\cU,\cS): = \frac{\mathrm{Kernel}(\delta: C^k(\cU,\cS)\to C^{k+1}(\cU,\cS))}{\mathrm{Image}(\delta: C^{k-1}(\cU,\cS)\to C^k(\cU,\cS))}
$$
\arap

\rmk
Our use of the term {\rm nerve} is not quite standard;
the standard definition takes sets $\{i_0,i_1,\ldots,i_k\}$
of cardinality $k+1$ rather than  sequences. With this definition
one must linearly order $I$ to define $\delta$; changing the linear order
changes the sign of $\delta$. The kernel and image and
(hence also the cohomology) are independent of the choice
and the two notions of nerve lead to the same cohomology.
\kmr

\rmk If $\delta \mu=0$ then $\mu_\sigma$ is a skew symmetric function
of the indices.
\kmr

\para Let $\cS$ be a sheaf, $\cU$ be an open cover, and $\cV$ be a refinement
of $\cU$. Each refining map $\tau$ induces a chain map
$\tau_*:C^*(\cV,\cS)\to C^*(\cU,\cS)$ and two refining maps induce
chain homotopic chain maps. Hence there is a
well defined map $H^*(\cV,\cS)\to H^*(cU,\cS)$. The open covers of $X$
form a directed set under refinement. The $k$th cohomology group of $\cS$
is defined by
$$
    H^k(\cS):=\lim_{\cU\to\infty} H^k(\cU,\cS).
$$
When (as is usually the case) the sheaf $\cS$
takes values in the category of complex vector spaces,
we denote the dimension  of the cohomology group by
$$
   h^k(\cS):=\dim_\C H^k(\cS).
$$
\arap

\begin{theorem} \label{thm:coveringDimension}
For any sheaf $\cS$ over a topological space $X$
we have that $H^k(\cS)=0$ for $k>n$ where $n$ is the covering dimension
of $X$.
\end{theorem}

\begin{proof} By the definition of covering dimension any open
cover has a refinement $\cU$ with $N^k(\cU)=\emptyset$ for $k>n$.
\end{proof}

\para Let $\cS$ be a sheaf.
A \jdef{partition of unity} on $\cS$
is a collection $\{\eta_i\}_{i\in I}$ of sheaf homomorphisms
$\eta_i:\cS\to\cS$ which is locally finite
(i.e. for every point of $X$ has a neighborhood
which intersects the supports of only finitely many $\eta_i$)
and satisfies $\sum_{i\in I}\eta_i=1$ the identity. The partition of unity
is \jdef{subordinate} to the cover $\cU=\{U_i\}_{i\in I}$ if the support
of $\eta_i$ is a subset if $U_i$.
A sheaf $\cS$ is called a \jdef{fine sheaf} iff for any open
cover $\cU$ there is partition of unity subordinate to it.
\arap

\begin{theorem} Let $\cS$ be a  fine sheaf. Then $H^k(\cS)=0$ for $k>0$.
\end{theorem}

\begin{corollary} \label{cor:skyscraper}
Let $G_p$ be a skyscraper sheaf. Then $H^0(G_p)=G$
and $H^k(G_p)=0$ for $k>0$.
\end{corollary}

\begin{proof} Let $\cU=\{U_i\}_{i\in I}$ be an open cover. Choose
$i_0$ with $p\in U_{i_0}$ and take $\eta_{i_0}$ the identity and
$\eta_i=0$ for $i\ne i_0$.
\end{proof}


\para Let
$$
   0\to\cA\to\cB\to\cC\to 0
$$
be a short exact sequence of sheaves. Then  there is a long exact sequence
$$
\begin{array}{r}
0 \to H^0(\cA)\to H^0(\cB)\to H^0(\cC)\mapright{\delta}\phantom{\cdots}\\ \\
  H^1(\cA)\to H^1(\cB)\to H^1(\cC)\mapright{\delta}\phantom{\cdots}\\ \\
  H^2(\cA)\to H^2(\cB)\to H^2(\cC)\mapright{\delta}\cdots
\end{array}
$$
(When $X$ is a Riemann surface the $\cdots$ may be replaced by $0$
according to Theorem~\ref{thm:coveringDimension}.)
\arap


\para\label{picard}
Let $L\to X$ be line bundle, $\cU=\{U_i\}_{i\in I}$ be an open cover
such that $L|U_i$ is trivial and $s_i\in\cO^*(U_i,L)$ be a nowhere
zero holomorphic section of $L$ over $U_i$.
Then the \jdef{transition functions}
$g_{ij}\in\cO^*(U_i\cap U_j)$ defined by $s_i=g_{ij}s_j$  form a cocycle in $C^1(\cU,\cO^*)$
and hence a cohomology class in $H^1(\cO^*)$ where $\cO^*$ is the sheaf of nowhere
zero holomorphic functions.
It is not hard to see that this defines a group isomorphism
$$
   \Pic(X)= H^1(\cO^*)
$$
where $\Pic(X)$ is the \jdef{Picard group}, i.e. the multiplicative group
(under tensor product) of isomorphism classes of line bundles over $X$.
\arap

\para\label{2ExactSequences}
The sheaf $\cO^*$ fits into two exact sequences, namely the
\jdef{divisor exact sequence}
$$
   0\to\cO^*\to\cM^*\to\cD\to 0
$$
and the \jdef{exponential exact sequence}
$$
  0\to\Z\to\cO\to\cO^*\to 0
$$
where the map $\cO(U)\to\cO^*(U)$ is $s\mapsto\exp(2\pi i s)$.
A common notation for the global divisors on $X$ is
$$
\Div(X):= H^0(\cD)=\cD(X).
$$
When $X$ is compact, it is not hard to see
a global section is a divisor in the sense of paragraph~\ref{divisorNotation}
and that where $L=L_D$ is the line bundle constructed
in part~(iii) of Theorem~\ref{thm:div=bundle}  we have
$L_D=\delta(D)$ where $\delta:H^0(\cD)\to H^1(\cO^*)$ is the boundary operator
in the long exact sequence associated to the divisor sequence.
(See paragraph~\ref{D-P} below.)
\arap

\para A \jdef{double complex} consists of a collection $\{C^{p,q}\}_{p,q\ge 0}$ of
abelian groups indexed by the nonnegative integers together with homomorphisms
$d:C^{p,q}\to C^{p,q+1}$ and $\delta:C^{p,q}\to C^{p+1,q}$ such that
$d^2=0$, $\delta^2=0$, and $d\delta=\delta d$.
For notational convenience define $C^{-1,q}=C^{p,-1}=0$ and define $d$ and $\delta$
as the inclusion of zero. Define cohomology groups
$$
 H_d^{p,q} = \frac{\mathrm{Kernel}(d:C^{p,q}\to C^{p,q+1})}{\mathrm{Image}(d:C^{p,q-1}\to C^{p,q})}
$$
and
$$
 H_\delta^{p,q} = \frac{\mathrm{Kernel}(\delta:C^{p,q}\to C^{p+1,q})}{\mathrm{Image}(\delta:C^{p-1,q}\to C^{p,q})}
$$
for $p,q\ge 0$.
\arap

\para We may represent the double complex as an array as follows
$${\renewcommand{\arraystretch}{1.4}
\begin{array}{ccccccccc}
 &   &\vdots   &                 &\vdots   &                 &\vdots   &            &\\
 &   &\mapup{d}&                 &\mapup{d}&                 &\mapup{d}&            &\\
0&\to&C^{0,2}  &\mapright{\delta}&C^{1,2}  &\mapright{\delta}&C^{2,2}  &\mapright{\delta}&\ldots\\
 &   &\mapup{d}&                 &\mapup{d}&                 &\mapup{d}&            &\\
0&\to&C^{0,1}  &\mapright{\delta}&C^{1,1}  &\mapright{\delta}&C^{2,1}  &\mapright{\delta}&\ldots\\
 &   &\mapup{d}&                 &\mapup{d}&                 &\mapup{d}&            &\\
0&\to&C^{0,0}  &\mapright{\delta}&C^{1,0}  &\mapright{\delta}&C^{2,0}  &\mapright{\delta}&\ldots\\
 &   &\uparrow &                 &\uparrow &                  &\uparrow&            &\\
 &   &0        &                 &0        &                  &0       &            &
\end{array}
}
$$
We will say that an $y\in C^{0,k}$ with $dy=0$ and an element $x\in C^{k,0}$ with $\delta x=0$
are \jdef{zig zag related} iff there exist
$z_j\in C^{j,k-j}$ with $z_0=y$, $z_k=x$, and $dz_j=\delta z_{j+1}$ for
$j=0, 1,\ldots, k-1$. When $k=2$ this may be represented by
$$
\begin{array}{ccccccc}
 0  \\
 \mapup{d}\\
 z_0=y  &\mapright{\delta}&\delta z_0=dz_1\\
        &                 &\mapup{d}\\
        &                 &z_1      &\mapright{\delta}&\delta z_1=dz_2\\
        &                 &         &                 &\mapup{d}\\
        &                 &         &                 &z_2=x  &\mapright{\delta}&0\\
\end{array}
$$
\arap


\begin{theorem}
Assume that the rows and columns of the double complex are exact
except for the leftmost column and the bottom row,
i.e. that $H_d^{p,q}=0$ for $p>0$ and $q\ge 0$
and that $H_\delta^{p,q}=0$ for $q>0$ and $p\ge 0$.
Then the groups $H_d^{0,k}$ and $H_\delta^{k,0}$ are isomorphic. More precisely
there is an isomorphism $T:H_d^{0,k}\to H_\delta^{k,0}$ such that
for $y\in C^{0,k}$ with $dy=0$ and $x\in C^{k,0}$ with $\delta x=0$
we have $T[y]=[x]$ if and only if $y$ and $x$ are zig zag related.

\end{theorem}

\begin{corollary}
Suppose that
$$
  0 \to\cS^0\to\cS^1\to\cS^2\to\cdots
$$
is a \jdef{fine resolution} of $\cS_0$, i.e. an exact sequence of  sheaves
with $\cS^k$ a fine sheaf for $k>0$.
Then there is an isomorphism
$$
    H^p(X,\cS_0) =
 \frac{\mathrm{Kernel}(\cS^p(X)\to\cS^{p+1}(X))}{\mathrm{Image}(\cS^{p-1}(X)\to\cS^p(X))}
$$
\end{corollary}

\begin{proof} Choose  an open cover  $\cU$ of  $X$. Let
$$
     C^{p,q}:= C^p(\cU,\cS^q)
$$
denote
the space of C\^ech $p$-cochains on the cover $\cU$ with values in $\cS^q$
and apply the Zig Zag Theorem.
\end{proof}

\begin{corollary}[De Rham] Let $X$ be a compact smooth manifold.
Then there is an isomorphism
$$
H_{DR}^*(X,\C) = \hat H^*(X,\C)
$$
between the de Rham cohomology  and the C\^ech cohomology.
\end{corollary}

\begin{proof} Let $\cE^p$ be the sheaf of smooth $p$-forms on $X$.
Then we have a fine resolution
$$
  0 \to \C\to\cE^0\mapright{d}\cE^1\mapright{d}\ldots.
$$
That the sequence is exact is the Poincar\'e Lemma.
\end{proof}

\begin{corollary}[Dobeault]\label{cor:dolbeault}
Let $E\to X$ be a holomorphic vector bundle over a  complex manifold $X$ and
$\Omega^p(E)$ be the sheaf of holomorphic $E$ valued $p$-forms on $X$.
Then there is an isomorphism
$$
H_\pbar^{p,q}(X,E) = H^q(X,\Omega^p(E))
$$
between the Dolbeault cohomology of $E$-valued forms of type
$(p,q)$  and the sheaf cohomology of $\Omega^p(E)$.
\end{corollary}


\begin{proof}
Let $\cE^{p,q}(E)$ be the sheaf of smooth $E$-valued forms
of type $(p,q)$.
Then we have a fine resolution
$$
  0 \to \Omega^p(E)\to\cE^{p,0}(E)\mapright{\pbar}\cE^{p,1}(E)\mapright{\pbar}\ldots.
$$
That the sequence is exact is the $\pbar$-Poincar\'e Lemma.
\end{proof}


\section{Serre Duality}

\para
Let $X$ be a compact complex manifold of complex dimension $n$,
$E\to X$ be a holomorphic vector bundle, and $E^*\to X$ denote the
dual bundle.
For $p,q,=0,1,\ldots,n$ define a pairing
$$
  \cE^{n-p,n-q}(X,E^*)\times \cE^{p,q}(X,E) \to\C: (\alpha,\beta)\mapsto \int_X\alpha\wedge\beta.
$$
By integration by parts
$$
   \int_X \pbar\gamma\wedge \beta=(-1)^p\int_X \gamma\wedge \pbar\beta
$$
for $\alpha\in\cE^{n-p,n-q-1}(X,E^*)$ and $\beta\in\cE^{p,q}(X,E)$
so
$$
\pbar^*:=(-1)^p\pbar:\cE^{n-p,n-q-1}(X,E^*)\to\cE^{n-p,n-q}(X,E^*)
$$
is a formal adjoint to the $\pbar$ operator
$$
\pbar:\cE^{p,q}(X,E)\to\cE^{p,q+1}(X,E).
$$
This defines a map
$$
 H_\pbar^{n-p,n-q}(X,E^*)\to H_\pbar^{p,q}(X,E)^* \eqno(*)
$$
between the Dolbeault cohomology groups.
Using the Dolbeault isomorphism  of Corollary~\ref{cor:dolbeault}
this induces a map
$$
     H^{n-q}(\Omega^{n-p}(E^*))\to H^q(\Omega^p(E))^*
$$
between the sheaf cohomology groups.
\arap

\begin{theorem}[Serre Duality]
The pairing is nondegenerate, i.e. the map~$(*)$ is an isomorphism.
\end{theorem}

\begin{proof}[Sketch of proof] Let $\cE^{p,q}(X,E)^*$ denote the dual space
of the Frech\'et space $\cE^{p,q}(X,E)$. The above pairing gives an inclusion
$$
\cE^{n-p,n-q}(X,E^*)\to \cE^{p,q}(X,E)^*.
$$
If, in the statement of Serre duality,
the cohomology group $H_\pbar^{n-p,n-q}(X,E^*)$ of the formal dual  complex
$$
\cdots\to  \cE^{n-p,n-(q+1)}(X,E^*)\mapright{\pbar^*}
\cE^{n-p,n-q}(X,E^*)\mapright{\pbar^*}
\cE^{n-p,n-(q-1)}(X,E^*)\to\cdots
$$
is replaced by the cohomology group of the dual complex
$$
\cdots\to  \cE^{p,q+1}(X,E^*)^*\mapright{\pbar^*}
\cE^{p,q}(X,E^*)^*\mapright{\pbar^*}
\cE^{p,q-1}(X,E^*)^*\to\cdots
$$
the Serre theorem becomes an exercise in linear algebra.
The replacement is justified by elliptic regularity;
i.e. a linear functional in the kernel of $\pbar^*$
is given by integration against smooth form.
For the details see~\cite{GUNNING}.
\end{proof}


\para
In case $X$ is a Riemann surface, its   complex dimension is $n=1$
and simpler notations are used: the sheaf of germs of holomorphic one forms is
denoted by $\Omega$ rather than $\Omega^1$ and the sheaf of germs of holomorphic
aero forms (functions) is denoted by $\cO$ rather than $\Omega^0$. In this
case
$$
   H^1(\cO(E))^*=H_\pbar^{0,1}(X,E)^*=H_\pbar^{1,0}(X,E^*)=H^0(\Omega(E^*))
$$
where the middle isomorphism is  Serre duality and the outer ones are the Dolbeault
isomorphisms. Taking dimensions we get
$$
           h^1(\cO(E))=h^0(\Omega(E^*))=h^0(\cO(K\otimes E^*))
$$
where $K$ is the canonical bundle.
\arap

\rmk\label{rmk:vanishing} A line bundle $L$ of negative degree can have no holomorphic sections
so
$$
    \deg(L)<0\implies h^0(\cO(L))=0.
$$
Since $\deg(K\otimes L^*)=\deg(K)-\deg(L)= 2g-2-\deg(L)$ where $g$ is the genus of $X$ we have
$$
    \deg(L)>2g-2\implies  h^1(\cO(L))=h^0(\cO(K\otimes L^*))=0.
$$
\kmr

\section{Existence Theorem}

In this section we will prove part~(i) of Theorem~\ref{thm:div=bundle}.
We follow~\cite{GUNNING}. We derive the Riemann Roch Theorem for
holomorphic line bundles as a corollary.
Throughout $X$ is a compact Riemann surface.


\para For any line bundle $L$ define an integer
$$
    \nu(L):= h^0(\cO(L))-h^1(\cO(L))-\deg(L).
$$
The Riemann Roch Theorem (Theorem~\ref{thm:RR-I} below) is that
$\nu(L)=1-g$ where $g$ is the genus of $X$,
\arap


\begin{lemma}\label{lem:nu} Let $L_D$ be the line bundle associated with the divisor $D$
as in Theorem~\ref{thm:div=bundle}. Then
$$
\nu(L\otimes L_D)=\nu(L)
$$
for any divisor $D$ and any line bundle $L$.
\end{lemma}

\begin{proof} By induction and the fact that $L_{D+p}=L_D\otimes L_p$ it suffices
to prove this in case $D=p$. There is a short exact sequence
$$
   0\to \cO(L)\to \cO(L\otimes L_p)\to \C_p\to 0;
$$
the map $\cO(L)\to\cO(L\otimes L_p) $ is given by $s\mapsto s\otimes s_p$.
Hence (by~\ref{cor:skyscraper}) there is a long exact sequence
$${\renewcommand{\arraystretch}{1.5}
\begin{array}{l}
0 \to H^0(\cO(L))\to H^0(\cO(L\otimes L_p))\to\C\mapright{\delta} \\
\phantom{0\to} H^1(\cO(L))\to H^1(\cO(L\otimes L_p))\to  0.
\end{array}
}$$
Now use the fact that the alternating sums of the dimensions
in an exact sequence is zero and the fact that
$\deg(L\otimes L_p)=\deg(L)+1$.
\end{proof}

\begin{theorem}\label{thm:meromorphicSection}
 A holomorphic line bundle admits a meromorphic section.
\end{theorem}

\begin{proof} Let $L$ be a holomorphic line bundle.
Since the line bundle $L_D$ has a holomorphic section,
to show that $L$ has a meromorphic section it suffices to show
that $L\otimes L_D$ has a holomorphic section for some divisor $D$.
If not, then $h^0(\cO(L\otimes L_D))=0$ for all $D$ and so
by Lemma~\ref{lem:nu} we have
$\nu(L)=\nu(L\otimes L_D)=-h^1(\cO(L\otimes L_D))-\deg(L\otimes L_D)$.
By Serre Duality this gives
$$
\nu(L)=-h^0(\cO(K\otimes L^*\otimes L_{-D}))-\deg(L\otimes L_D).
$$
Now $\deg(K\otimes L^*\otimes L_{-D})=\deg(K)-\deg(L)-\deg(L_D)$
which is negative for $D$ large.
If the degree of a holomorphic bundle is negative it has no holomorphic sections
so $h^0(\cO(K\otimes L^*\otimes L_{-D}))=0$ for $D$ large and hence
$\nu(L)=-\deg(L\otimes L_D)=-\deg(L)- \deg(L_D)$ for all
$D$ which is clearly absurd.
\end{proof}

\begin{corollary}\label{cor:L=LD}
Every line bundle is the line bundle of a divisor.
\end{corollary}

\begin{proof} Let $s$ be a meromorphic section of $L$ and $D=(s)$.
Then $L$ is isomorphic to $L_D$.
\end{proof}

\begin{theorem}[Riemann Roch for Line Bundles] \label{thm:RR-I} For any holomorphic line bundle
$L\to X$ over a Riemann surface $X$ of genus $g$ we have
$$
    h^0(\cO(L))-h^0(\Omega(L))=\deg(L)+1-g.
$$
In particular, $h^0(\Omega)=g$.
\end{theorem}

\begin{proof} In other words, we must show $\nu(L)=1-g$ where $\nu(L)$ was defined
in Lemma~\ref{lem:nu}. By Corollary~\ref{cor:L=LD} it is enough to show that $\nu(L_D)=1-g$
for every divisor $D$.
By Lemma~\ref{lem:nu} $\nu(L_D)=\nu(L_0)$ where $L_0$ denotes the trivial bundle;
so it is enough to show $\nu(L_0)=1-g$.
Now  $h^0(\cO(L_0))=1$ (a holomorphic function is constant) and $\deg(L_0)=0$ so
$$
\nu(L_0)=h^0(\cO(L_0))-h^0(\cO(K\otimes L_0^*))-\deg(L_0)
        =1-h^0(\cO(K)).
$$
Also $K\otimes K^*=L_0$ and by Theorem~\ref{thm:deg-K} $\deg(K)=2g-2$   so
$$
\nu(K)= h^0(\cO(K))-h^0(\cO(K\otimes K^*))-\deg(K)=h^0(\cO(K))-1-(2g-2).
$$
By Lemma~\ref{lem:nu}  $\nu(L_0)=\nu(K)$ so $h^0(\cO(K))=g$
so $\nu(L_0)=1-h^0(\cO(K))=1-g$ as required.
\end{proof}

\section{Riemann Roch for Vector Bundles}

\para Let $E\to X$ be a holomorphic vector bundle over a compact Riemann surface.
By the definition of cohomology the sequence
$$
    0\to H_\pbar^0(X,E)\to\cE^0(X,E)\mapright{\pbar}\cE^{0,1}(X,E)\to H_\pbar^{0,1}(X,E)\to 0
$$
is exact, i.e. $H_\pbar^0(X,E)$ is the kernel of $\pbar$ and
$H_\pbar^{0,1}(X,E)$ is the cokernel.
The operator $\pbar$ is elliptic
so these spaces are finite dimensional. The number
$$
   \ind(\pbar):=\dim_\C H^0(X,E)-\dim_\C H^{0,1}(X,E)
$$
is  called the \jdef{index} of the $\pbar$ operator of $E$.
By Serre duality the exact sequence my be written
$$
    0\to \cO(X,E)\to\cE^0(X,E)\mapright{\pbar}\cE^{0,1}(X,E)\to \cO(X,K\otimes E^*)^*\to 0
$$
where
$K\to X$ is the canonical bundle and so the index may be written as
$$
   \ind(\pbar) =h^0(\cO(E))-h^0(\Omega(E^*)).
$$
\arap


\para By a transversality argument,
any complex vector bundle $E$ of rank great than one over a Riemann surface
$X$ has a smooth nowhere vanishing section.
Hence $E$ has  decomposition $E=L_1\oplus\cdots\oplus L_n$
as a direct sum of line bundles.
(Warning: It is not true that a holomorphic vector bundle is a direct
sum of holomorphic line bundles.)
The number
$$
   c(E):=\sum_{k=1}^n \deg(L_k)
$$
is called the \jdef{Chern number} of the vector bundle $E$.
The Chern number is the
value of the  first Chern class  $c_1(E)\in H^2(X)$ on the fundamental
cycle $[X]\in  H_2(X)$  so it is independent of the decomposition
used to define it.
\arap

\begin{theorem}[Riemann Roch]\label{thm:RiemannRoch}
The index of the $\pbar$ operator of $E$
of  a holomorphic vector bundle $E\to X$
over a compact Riemann surface $X$  of genus $g$
is given by
$$
   \ind(\pbar) = n(1-g)+c(E).
$$
\end{theorem}

\begin{proof} \ETC
\end{proof}


\rmk
By Theorem~\ref{thm:deg-K} the degree of the canonical
bundle $K\to X$ is
$$
 \deg(K)= -\chi(X)= 2g-2
$$
Since $c(E^*)=-c(E)$ and the Chern number is additive and agrees with
the degree for line bundles,
the Riemann Roch theorem has  the suggestive form
$$
    \dim_\R\Gamma(E)-\dim_\R\Gamma(K\otimes E)=c(E)-c(K\otimes E^*).
$$
\kmr

\begin{theorem}[Kodaira Embedding for Riemann Surfaces]\label{thm:kodaira4RS}
Let $L\to X$ be a holomorphic line bundle
over a Riemann surface $X$ of genus $g$ and assume that $\deg(L)>2g+1$. Then
\begin{description}
\item[(i)] For $p,q\in X$ with $p\ne q$ there is a holomorphic section $s\in\cO(X,L)$
with $s(p)=0$ and $s(q)\ne 0$; and
\item[(ii)] For $p\in X$ there is a holomorphic section $s\in\cO(X,L)$ with
$s(p)=0$ and $ds(p)\ne0$. Here $ds(p):T_pX\to L_p$ is the canonical derivative.
\end{description}
Hence the corresponding map $X\to\P^N=\P(\cO(X,L))^*)$,
$N=h^0(\cO(L))-1$ is an embedding.
\end{theorem}

\begin{proof} \ETC See~\cite{FORSTER} page~144.
\end{proof}


\section{The Canonical Map}


\begin{lemma}  If $X$ has positive genus, then
the canonical bundle $K\to X$ has no base  point.
\end{lemma}

\begin{proof}
Suppose every $\omega\in\cO(X,K)=\Omega(X)$ vanishes at some point $p\in X$;
we will show that $X$ has genus zero.
Let $L_p$ denote the line bundle
corresponding to the divisor $D=p$
as in Theorem~\ref{thm:div=bundle}.
Then  $\cO(X,K\otimes L_p^*)$ and $\cO(X,K)$
are isomorphic so $h_0(\cO(K)=h_0(\cO(K\otimes L_p^*)$.
Hence by Riemann Roch,
$h_0(\cO(L_p))=h_0(\cO(K\otimes L_p^*)+\deg(L_p)+1-g=2$.
Hence there is a meromorphic function with a simple pole
at $p$ and no other pole so $X$ is isomorphic to $\P$
and hence has genus zero.
\end{proof}


\dfn
The classifying map $X\to\P^{g-1}=\P(\Omega(X))$
for the canonical bundle $K\to X$ of a Riemann surface is
called the \jdef{canonical map} of $X$. The automorphism group $\Aut(X)$ of $X$ acts on $K$
and hence on the target projective space of the canonical map; the canonical map
is equivariant.
\nfd



\section{Divisors and Principal Parts}

\para\label{div-pp}
Throughout  this section  $X$ is a Riemann surface, not necessarily compact.
Let
$\cM$ be  the sheaf of germs  of meromorphic functions,
$\cO$ the subsheaf of germs of holomorphic functions,
and $\cP$ the quotient sheaf.
We call $\cP$ the \jdef{sheaf of principal parts} in $X$ and the exact sequence
$$
      0\to\cO \to\cM\to\cP\to 0
$$
the \jdef{principal part exact sequence}.
Let
$\cM^*$ be  the sheaf of germs  of  nonzero meromorphic functions,
$\cO^*$ the subsheaf of germs of nowhere zero holomorphic functions,
and $\cD$ the quotient sheaf.
We call $\cP$ the \jdef{sheaf of divisors} on $X$  and the exact sequence
$$
      0\to\cO^*\to\cM^*\to\cD\to 0
$$
the \jdef{divisor exact sequence}.
If $U\subset X$ is connected and open then
$\cM^*(U)$ is the multiplicative group the field $\cM(U)$
and
$\cO^*(U)$ is the multiplicative group of units of the integral domain $\cO(U)$.
The group operation in $\cM^*$ is multiplication but we write the group operation
in $\cD$ additively.
\arap

\para\label{D-P}
The sheaf $\cD$ assigns to each open set $U$ the space $\cD(U)$ of
all formal sums
$$
  D = \sum_{p\in U} n_p p
$$
where $n_p$ is an integer and the support (i.e. the closure in $U$ of the
set of points $p$ where $n_p\ne 0$) is discrete.
Thus in the notation of paragraph~\ref{divisorNotation},
$\cD(U)=\Div(U)$ for any open subset $U$ of $X$ and in particular
$$
   \cD(X)=H^0(X,\cD)=\Div(X)
$$
for any Riemann surface $X$.
The map $H^0(\cM^*)\to H^0(\cD)$ in the cohomology exact sequence of
the divisor exact sequence
assigns to each global meromorphic function
$f\in\cM^*(X)$ its divisor
$$
   (f) = \sum_{p\in X} \Ord_p(f) p.
$$
The sheaf $\cP$ assigns to each open set $U$ the space $\cP(U)$ of
all formal sums
$$
  P = \sum_{p\in U} h_p p
$$
with discrete support (as for divisors) and
where $h_p$ a finite Laurent series
$$
     h_p= \sum_{k=1}^{n_p} a_k(p) z^{-k}.
$$
(Here $z$ is a holomorphic coordinate centered at $p$ so the values
$a_k(p)$ depend on the choice of $Z$.)
The map $H^0(\cM)\to H^0(\cP)$ in the cohomology exact sequence of the
principal part exact sequence assigns to each meromorphic function $f$
the formal sum $P$, supported at the poles of $f$ where
$h_p$ is the sum of the negative terms in the Laurent expansion for $f$ at $p$.
We next interpret the boundary operators
$H^0(\cD)\to H^1(\cO^*)$ and $H^0(\cP)\to H^1(\cO)$
in a parallel fashion.
\arap

\para
A \jdef{Weierstrass distribution} is a pair $(f,\cU)$
where $\cU=\{U_i\}_{i\in I}$ is an open cover of $X$,
$f_i\in\cM^*(U_i)$, and $f_i/f_j\in\cO^*(U_i\cap U_j)$;
a solution to $(f,\cU)$ is a
meromorphic function $f\in\cM^*(X)$ with $f_i=f|U_i$.
For a Weierstrass distribution the number $\Ord_p(f_i)$
is independent of the choice if $i$ with $p\in U_i$; this
gives an cocycle $D(f,\cU)\in C^0(\cU,\cD)$ and every
cocycle arises as a $D(f,\cD)$.
The boundary operator
$\delta:H^0(\cD)\to H^1(\cO^*)$ assigns to the cocycle $D$
the cocycle $f_i/f_j$ where $D=D(f,\cU)$.
Hence by the cohomology exact sequence
a Weierstrass distribution $(f,\cU)$ has a solution
$\iff$
$\delta[D(f,\cU)]=0$ in $H^1(\cO^*)$.
Note that under the identification
$\Pic(X)= H^1(\cO^*)$ of paragraph~\ref{picard} we have
$$
   \delta(D)=L_D
$$
where $L_D$ is the line bundle associated with the divisor $D$ as in Theorem~\ref{thm:div=bundle}.
\arap

\para
A \jdef{Mittag Leffler distribution} is a pair $(f,\cU)$
where
$f_i\in\cM(U_i)$  and $f_i-f_j\in\cO(U_i\cap U_j)$;
a solution to $(f,\cU)$ is a
meromorphic function $f\in\cM(X)$ with $f_i=f|U_i$.
A Mittag Leffler distribution determines a cocycle $P(f,\cU)\in C^0(\cU,\cP)$ and every
cocycle arises as a $P(f,\cU)$.
The boundary operator
$\delta:H^0(\cP)\to H^1(\cO)$ assigns to the cocycle $P$
the cocycle $f_i-f_j$ where $P=P(f,\cU)$.
Hence by the cohomology exact sequence
a Mittag Lefler distribution  $(f,\cU)$ has a solution
$\iff$
$\delta[P(f,\cU)]=0$ in $H^1(\cO)$.
\arap


\rmk In the theory of several complex variables
a Mittag Lefler distribution is called a {\em Cousin-I} distribution and a
Weierstrass distribution is called a {\em Cousin-II} distribution.
The problem of finding $f$ with $f|U_i=f_i$ is called the \jdef{Cousin problem}.
\kmr



\begin{lemma} \label{lem:divisors}
The sheaves $\cD$ and $\cP$ are fine. Hence
$H^k(\cD)=H^k(\cP)=0$ for $k>0$.
\end{lemma}

\begin{proof} Choose any open cover $\{U_i\}_i$.
Then write $X$ as a disjoint union
of sets $X_i$ with $X_i\subset U_i$; the sets $X_i$ are neither
closed nor open. Define $\eta_i:\cD\to\cD$ by
$$
  \eta_i(D)=\sum_{p\in X_i\cap U} n_p p
$$
for $U\subset X$ open and $D\in\cD(U)$ a local section of $\cD$. Then $\sum\eta_i=1$.
\end{proof}


\begin{theorem}  $H^2(\cO)=H^2(\cO^*)=H^2(\cM^*)=H^2(\cM)=0$.
\end{theorem}

\begin{proof}
By Dolbeault $H^2(\cO)=0$.
In the exponential cohomology exact sequence
$H^2(\cO^*)$ lies between $H^2(\cO)$ and $H^3(\Z)$ so $H^2(\cO^*)=0$.
In the divisor cohomology exact sequence $H^2(\cM^*)$
lies between $H^2(\cO^*)$ and $H^2(\cD)$ so $H^2(\cM^*)=0$.
In the principal part cohomology exact sequence $H^2(\cM)$
lies between $H^2(\cO)$ and $H^2(\cP)$ so $H^2(\cM)=0$.
\end{proof}

\begin{theorem}
If $X$  noncompact, $H^1(\cO)=H^{0,1}(X,\C)=0$.
\end{theorem}

\begin{proof}
The Dolbealt isomorphism $H^1(\cO)=H^{0,1}(X,\C)$ follows from the exactness of
the sheaf sequence
$$
  0\to\cO\to\cE^0\mapright{\pbar}\cE^{0,1}\to0
$$
and holds whether or not $X$ is compact. To show that
$H^{0,1}(X,\C)=0$ we must show that
for every $\omega\in\cE^{0,1}(X)$ there is an $f\in\cE^0(X)$ with $\omega=\pbar f$.
The proof uses Runge approximation and an exhaustion argument. See~\cite{FORSTER} page~200.
\end{proof}

\begin{corollary}
If $X$  noncompact, $H^1(\cO^*)=0$, i.e.
every holomorphic line bundle over $X$ is trivial.
\end{corollary}

\begin{proof}
In the exponential exact sequence
$H^1(\cO^*)$ is between $H^1(\cO)=0$ and $H^2(X,\Z)=0$.
\end{proof}


\begin{corollary}[Weierstrass] Assume that $X$ is not compact.
Then every divisor is the divisor of a meromorphic function.
Hence every Weierstrass distribution $(f,\cU)$ has a solution.
\end{corollary}

\begin{proof}
By the  sequence $H^0(\cO^*)\to H^0(\cM^*)\to H^0(\cD)\to H^1(\cO^*)=0$
is exact so $H^0(\cM^*)\to H^0(\cD)$ is onto and $\delta F(f,\cU)=0$.
\end{proof}


\begin{corollary}[Mittag Lefler] Assume that $X$ is not compact.
Then every principal part is the principal part of a meromorphic function.
Hence every Mittag Lefler distribution has a solution.
\end{corollary}

\begin{proof} The  sequence $H^0(\cO)\to H^0(\cM)\to H^0(\cP)\to H^1(\cO)=0$.
is exact so $H^0(\cM)\to H^0(\cP)$ is surjective and $\delta P(f,\cU)=0$.
\end{proof}

\begin{theorem} $H^1(\cM^*)=H^1(\cM)=0$.
\end{theorem}

\begin{proof} In the noncompact case the group this follows immediately
from the corresponding cohomology exact sequences and the results already proved:
$H^1(\cM^*)$ lies between $H^1(\cO^*)=0$ and $H^1(\cD)=0$
and $H^1(\cM)$ lies between $H^1(\cO)=0$ and $H^1(\cP)=0$.
Assume the compact case. By Corollary~\ref{cor:L=LD}
the first map in the sequence
$
 H^0(\cM^*)\to H^1(\cO^*)\to H^1(\cM^*)\to H^1(\cD)=0
$
is surjective so second map is zero and hence $H^1(\cM^*)=0$.
Next assume $f_{ij}$ represents an element
of $H^1(\cM)$. Refine the cover if necessary so that
the number of poles of the $f_{ij}$ is finite. Choose a divisor $D$ of
degree $>2g-2$ such that $f_{ij}$ is a cycle in $C^1(\cU,\cO(L_D))$.
By Remark~\ref{rmk:vanishing}  $H^1(\cO(L_D))=0$.
Hence the cycle $f_{ij}$ is
cohomologous to zero in $C^*(\cU,\cO(L_D)))$ and thus certainly in $C^*(\cU,\cM)$.
\end{proof}


\rmk A complex manifold (of any dimension) which is isomorphic to a closed submanifold
of $\C^N$ for some $N$ is
called a \jdef{Stein manifold}. One can prove that a noncompact Riemann surface
is a Stein manifold. The various cohomology vanishing theorems proved in this
section are special cases of more general theorems for Stein manifolds.
If $X$ is a submanifold of projective space then
$X$ has a finite open cover
$\cU=\{U_i\}_{i\in I}$ where each $U_\sigma$ for $\sigma\in\mathrm{Nerve}(\cU)$
is Stein. Thus for the various sheaves
$\cS=\cO,\Omega^p,\ldots$ the homologies $H^k(U_\sigma,\cS)$ vanish for $k>1$, i.e.
$\cU$ is what is called a \jdef{Leray cover} for the sheaf $\cS$.
A theorem of Leray (see~\cite{GUNNING} page~44) implies that $H^k(\cS)=H^k(\cU,\cS)$,
i.e. in computing the sheaf cohomology we don't need to take the limit over all
covers.
\kmr


\section{Serre Duality and Divisors}

\para Now suppose that $L_D$ is the line bundle associated to a divisor $D$.
Then Theorem~\ref{thm:div=bundle} gives isomorphisms
$$
   \cO_D:=\cO(X,L_D)=\cL(D), \qquad \Omega_{-D}:=\Omega(X,L_D^*)=\cL(K-D)
$$
where, as is traditional,  we have used the same letter $K$ for the canonical line bundle
$K\to X$  and the divisor of a meromorphic section of $K$, i.e. a meromorphic
one form.
Serre duality takes the form
$$
   H^1(\cO_D)=H^0(\Omega_{-D})^*.
$$
Our aim is to find a formula for the pairing
$$
   H^1(\cO_D)\times H^0(\Omega_{-D})\to\C.
$$
We will use the principal part exact sequence
$$
  0\to\cO(K)\to \cM(K)\to \cP(K)\to 0
$$
for one forms.
\arap

\para
Where $D=\sum_p n_p p$ and $U\subset X$ is open,
an  element  of $\cO_D(U)$ is a  meromorphic function $f\in\cM(U)$
such that for $p\in U$ with $n_p<0$ the pole of $f$ has order
$\le -n_p$ at $p$; similarly  an  element of $\Omega_{-D}(U)$ is a
meromorphic one form $\omega\in\cM(U,K)$ with zero of order $\ge -n_p$
at each $p$ where $n_p<0$. Thus for each open set $U$ there is a bilinear map
$$
     \cO_D(U)\times\Omega_{-D}(U)\to \Omega(U):(f,\omega)\mapsto f\omega
$$
defined by pointwise multiplication. In particular this induces a pairing
$$
    H^0(\cU,\cO_D)\times H^1(\cU,\Omega_{-D})\to H^1(\cU,\Omega):
((f_i)_i,(\omega_{ij})_{ij})\mapsto (f_i\omega_{ij})_{ij}.
$$
for each open cover $\cU$ and hence a pairing
$$
   H^0(\cO_D)\times H^1(\Omega_{-D})\to H^1(\Omega).
$$
\arap

\para The sum of the residues of a meromorphic one form in $H^0(\cM(K))$ is zero but there
is a map
$$
    \Res:H^1(\Omega)=H^1(\cO(K))\to\C
$$
defined as follows. Choose $\mu\in H^1(\Omega)$. Since $H^1(\cM(K))=0$
the boundary map $\delta:H^0(\cP)\to H^1(\cO(K))=H^1(\Omega)$ is surjective.
Therefore there is a Mittag Lefler distribution $(\nu,\cU)$ representing $\mu$,
i.e. $\nu_i\in\cM(U_i,K)$ and the cocycle $\nu_i-\nu_j\in \cO(U_i\cap U_j,K)$
represents $\mu$. Since the $\nu_i-\nu_j$ is holomorphic on $U_i\cap U_j$ we have that
$\res_p(\mu):=\res_p(\nu_i)$
is independent of the choice of $i$ with $p\in U_i$ used to define it. Define
$$
    \Res(\mu)=\sum_{p\in X} \res_p(\mu).
$$
\arap

\begin{theorem} The composition
$$
H^0(\cO_D)\times H^1(\Omega_{-D})\to H^1(\Omega)\mapright{\Res}\C
$$
is the Serre Duality pairing.
\end{theorem}

\begin{proof} \ETC
\end{proof}


\begin{theorem} Let $f\in C^0(\cU,\cM)$ be a Mittag Lefler distribution
of meromorphic functions on a compact Riemann surface $X$. Then $f$ has a solution
if and only if
$$
    \Res(f\omega)=0\qquad \mbox{for all $\omega\in\Omega(X)$}.
$$
\end{theorem}

\begin{proof} \ETC See~\cite{FORSTER} page~147,
\end{proof}

\begin{theorem}[Abel]\label{thm:abel}
 Let $D\in \Div_0(X)$ be a divisor of degree zero
on a compact Riemann surface $X$. Then $D=(f)$  for some $f\in\cM(X)$ if and only
if there is a singular one chain $c$ on $X$ with $\p c=D$ and
$$
   \int_c\omega=0\qquad \mbox{for all $\omega\in\Omega(X)$}.
$$
\end{theorem}

\begin{proof} \ETC See~\cite{FORSTER} page~163.
\end{proof}



\section{Abel Jacobi}

\para Let $X$ be a compact Riemann surface,
$\cM^*(X)$ be the multiplicative group of nonzero meromorphic functions on $X$,
$\Div(X)$  be the group of divisors on $X$,
$\Div_0(X)$  be the subgroup of divisors of degree zero,
$\Pic(X)$ be the Picard group of holomorphic line bundles on $X$,
$\Pic_0(X)$ be the subgroup line bundles of degree zero,
$\cM(X,K)$ be the space of meromorphic differentials in $X$,
$\Omega(X)=\cO(X,K)$ be the  space of holomorphic differentials on $X$, and
$\Omega(X)^*$ be its dual as a complex vector space.
Each $\gamma\in H_1(X,\Z)$ determines a functional $I_\gamma\in\Omega(X)^*$
via the formula
$$
    I_\gamma(\omega)=\int_\gamma\omega.
$$
Define $\Lambda\subset \Omega(X)^*$ by
$$
    \Lambda=\{I_\gamma:\gamma\in H_1(X,\Z)\}.
$$
It is easy to see that the map $\gamma\to I_\gamma$ is a homomorphism of groups
so that $\Lambda$ is a subgroup of the additive group of the vector space $\Omega(X)^*$.
The \jdef{Jacobian} of $X$ is the quotient group
$$
    \Jac(X):=\Omega(X)^*/\Lambda.
$$
\arap

\begin{lemma} Fix $o\in X$ and let $D=\sum_k n_k p_k\in\Div(X)$.
Then  the class  $u(D)\in\Jac(X)$ of the linear functional  defined by
$$
 \Omega(X)\to\C:\omega\mapsto   \sum_k n_k\int_o^{p_k}\omega
$$
is independent of the choice  of the arcs
from $o$ to $p_k$ used to define the integrals.
If $D\in\Div_0(X)$ then $u(D)(\omega)$ is also independent of the choice of $o\in X$.
The  map
$u:\Div_0(X)\to\Jac(X)$ thus defined  is called \jdef{Abel Jacobi map} of $X$.
\end{lemma}

\begin{theorem}[Abel-Jacobi] \label{thm:abel-jacobi}The sequence
$$
   \cM^*(X)\to\Div_0(X)\mapright{u}\Jac(X)\to 0
$$
is exact. Here the map $\cM^*(X)\to\Div_0(X)$ assigns to each meromorphic function $f$
its divisor $(f)$. Moreover, $\Lambda$ is a lattice so $\Jac(X)$ is a (compact) torus.
\end{theorem}

\rmk
The cohomology exact sequence of the divisor exact sequence restricts to
$$
  \cM^*(X)\to\Div_0(X)\to\Pic_0(X)\to 0
$$
so as a corollary we have that that the Abel Jacobi map induces
an isomorphism
$$
\Pic_0(X)=\Jac(X)
$$
also denoted by $u$.
The assertion that the map $u$ is injective is {\em Abel's Theorem};
the assertion that it is surjective is the {\em Jacobi Inversion Theorem}.
\kmr

\para Let $P\subset\C$ be a fundamental polygon for $X$, i.e. the sides of $P$ are
given by
$$
  \p P=\alpha_1\cup\beta_1\cup\alpha_1'\cup\beta_1'\cup\cdots\cup\beta_g'
$$
(in the indicated order)
where $g$ is the genus of $X$
and there is a holomorphic map from a neighborhood of $P$ onto $X$
which is injective in the interior of $P$
and identifies $\alpha_i'$ and $\alpha_i^{-1}$
and also identifies $\beta_i'$ and $\beta_i^{-1}$.
Define
$$
   \alpha_{g+i}=\beta_i, \qquad \beta_{g+1}=\alpha_i^{-1}.
$$
Fix a point $o$ in the interior of $P$.
\arap

\begin{lemma} Let $\omega$ be a meromorphic differential in $X$
having no pole on $\p P$, $\phi$ be a holomorphic  differential on $P$,
and $f:P\to\C$ be holomorphic function defined by
$$
     f(p)=\int_o^p\phi.
$$
Then
$$
  2\pi i\sum_{p\in X} \res_p(f\omega) = -\sum_{i=1}^{2g}w_ib_i, \quad\mbox{where}\quad
  w_i=\int_{\alpha_i}\omega,\quad b_i=\int_{\beta_i}\phi.
$$
\end{lemma}

\begin{proof} By the Residue Theorem
$$
2\pi i\sum_{p\in X} \res_p(f\omega)=\int_{\p P}f\omega.
$$
If $p\in\alpha_i$ is identified with $p'\in\alpha_i^{-1}$
then $f(p')-f(p)=b_i$ so $\int_{\alpha_i\cup\alpha_i'}f\omega=b_iw_i$.
\end{proof}


\para Now choose a basis $\phi_1,\ldots,\phi_g$ for $\Omega(X)$ and use vector notation
$$
   \Phi=(\phi_1,\ldots,\phi_g), \qquad F(p)=\int_o^p\Phi\in\C^g,
$$
and for $i=1,2,\ldots,2g$ define $A_i,B_i\in\C^g$ by
$$
   A_i=\int_{\alpha_i}\Phi, \qquad B_i=\int_{\beta_i}\Phi.
$$
For $x=(x_1,\ldots,x_{2g}\in\C^{2g}$ define $A(x),B( x)\in\C^g$ by
$$
   A(x)=\sum_{i=1}^{2g} x_iA_i, \qquad
   B(x)=\sum_{i=1}^{2g} x_iB_i,
$$
and for $y\in\C^g$ define $A^\#(y), B^\#(y)\in\C^{2g}$ by
$$
A^\#(y)=(y\cdot A_1,\ldots,y\cdot A_{2g}),\qquad
B^\#(y)=(y\cdot B_1,\ldots,y\cdot B_{2g})
$$
where $(y_1,\ldots,y_g)\cdot(z_1,\ldots,z_g)=\sum_iy_iz_i$. Thus
$$
   A^\#(y)(x)=y\cdot A(x), \qquad
   B^\#(y)(x)=y\cdot B(x)
$$
for $x\in\C^{2g}$, $y\in\C^g$.
\arap

\begin{lemma} The sequence
$$
  0\to\C^g\mapright{A^\#}\C^{2g}\mapright{B}\C^g\to 0
$$
is exact (and similarly if $A$ and $B$ are reversed).
\end{lemma}

\begin{proof} To see that $B:\C^{2g}\to\C$ is surjective we show that $B^\#$ is injective.
Choose $x=(x_1,\ldots,x_g)\in\C^g$ with $B^\#(x)=0$; we will  show that $x=0$.
Let $\omega=\sum_k x_k\phi_k$.
Then $\omega$ is holomorphic and $\int_{\beta_i}\omega=0$ for all $i$.
Hence $p\mapsto\int_o^p\omega$ is single valued and thus constant,
so $\omega=0$ so (as the $\phi_i$ are independent) $x=0$.

To see that $B\circ A^\#=0$ assume $x=A^\#(y)$. Let $\omega=\sum_Iy_i\phi_i$.
The $\omega$ is holomorphic so, by the Lemma, $\sum_i w_iB_i=0$
where $w_i=\int_{\alpha_i}\omega=A^\#(y)_i=x_i$, i.e. $B(w)=B(x)=0$.

Exactness at $\C^{2g}$ follows since the proof that $B$ is surjective
showed that $B^\#$ injective and the same argument shows that $A^\#$ is injective.
\end{proof}


\para Now we prove Abel's Theorem, i.e. that $u$ is injective.
Suppose that $D\in\Div_0(X)$ and that $u(D)=0$; we must find $f\in\cM^*(X)$
with $D=(f)$. Let $D=\sum_{p\in X} n_pp$. As $\sum_p n_p=0$ there
is a meromorphic differential $\omega$ with $\res_p(\omega)=n_p$.
Suppose first that $\int_{\alpha_j}\omega\in 2\pi i\Z$.
Then $f(q)=\int_o^q\omega$ is well defined, meromorphic, and satisfies $(f)=D$.
It remains to reduce to the case where $w_j:=\int_{\alpha_j}\omega\in 2\pi i\Z$.
By the Lemma
$$
\frac{-1}{2\pi i}\sum_jw_jB_j=\sum_p\res_p(F\omega)=\sum_pn_p F(p)=u(D)\in\Lambda
$$
and thus it has the form $\sum_jm_jB_j$ where $m_j\in\Z$. Hence
$\sum_j(w_j+2\pi i m_j)B_j=0$ so $w_j+2\pi i m_j=A^\#(y)_j$ for some
$y\in\C^g$, Now $\psi=\omega-\sum_jy_j\phi_j$ has the same poles and residues
as $\omega$ and $\int_{\alpha_j}\psi\in2\pi i\Z$.
\arap


\para Now we prove the Jacobi Inversion Theorem, i.e. that $u$ is surjective.
It is enough to show that the image of $u$ contains an open set
since $u$ is a homomorphism of groups. For this choose $q_1,\ldots,q_g\in X$
such that $\omega(q_1)=\cdots\omega(q_j)=0$, $\omega\in\Omega(X)\implies\omega=0$.
Then the derivative of $u$ at $D=\sum_j q_j$ is injective and hence by the Inverse Function
Theorem there are neighborhoods $V_i$ of $q)i$ such that $u$ maps
$V_1\times V_g$ onto an open set.
\arap

\begin{lemma} Fix $o\in X$ and define $\iota:X^g\to \Pic_0(X)$ by
$$
\iota(p_1,\ldots,p_g)=\sum_{j=1}^g(p_j-o).
$$
Then  $\iota$ is surjective.
\end{lemma}

\begin{proof}
Choose $D\in\Div_0(X)$. Riemann Roch gives
$$
\ell(-\iota(p_1,\ldots,p_g)+D)\ge \ell(go+D)\ge \deg(go+D)+1-g=1
$$
so there is an $f$ with
$(f)\ge\iota(p_1,\ldots,p_g) -D$. Since both sides have degree zero
we must have equality, i.e. $\iota(p_1,\ldots,p_g)=D$ in $\Pic_0(X)$.
\end{proof}

\para Finally we show that $A_1,\ldots A_{2g}$ are independent over $\R$,
i.e. that $\Lambda$ is a lattice. If this fails, $\Lambda$ lies in a hyperplane
in $\R^{2g-1}$ and $\C^q/\Lambda$ is not compact. Both
$\iota$ and $u$ are surjective and hence also the composition
$u\circ\iota$. But $X^g$ is compact so it  follows that
that the image $\C^g/\Lambda$ of $u\circ\iota$ is compact as required.
\arap

\begin{thebibliography}{99}

\bibitem{FORSTER} O. Forster:
{\em Lectures on Riemann Surfaces},
Springer Graduate Texts in Math {\bf 81} 1981.

\bibitem{GRIFFITHS}
P. A. Griffiths: {\em Introduction to Algebraic Curves},
AMS Translations of Math. Monographs {\bf 76} 1989.

\bibitem{GH}
P. A. Griffiths \& J. Harris: {\em Principles of Algebraic
Geometry}, Wiley Interscience, 1978.

\bibitem{GUNNING} R. C. Gunning:
{\em Lectures on Riemann Surfaces},
Princeton Mathematical Notes, 1966.

\bibitem{NARISIMHAN}
R. Narisimhan:
{\em Compact Riemann Surfaces}
Lectures in Math ETH Z\"urich, Birkh\"auser, 1992.

\bibitem{REYSSAT} E. Reyssat:
{\em Quelques Aspects des Surface de Riemann},
Birkh\"auser, 1989.

\end{thebibliography}

\end{document}
