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\begin{document}

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\section*{Riemann surface}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_facts}{Basic facts}\dotfill \pageref*{basic_facts} \linebreak
\noindent\hyperlink{complexified_differentials}{Complexified differentials}\dotfill \pageref*{complexified_differentials} \linebreak
\noindent\hyperlink{PicardGroupOfHolomorphicLineBundles}{Picard group of holomorphic line bundles}\dotfill \pageref*{PicardGroupOfHolomorphicLineBundles} \linebreak
\noindent\hyperlink{central_theorems}{Central theorems}\dotfill \pageref*{central_theorems} \linebreak
\noindent\hyperlink{homotopy_type}{Homotopy type}\dotfill \pageref*{homotopy_type} \linebreak
\noindent\hyperlink{branched_covers}{Branched covers}\dotfill \pageref*{branched_covers} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A Riemann surface is a $1$-dimensional [[algebraic geometry|algebro-geometric]] object with good properties. The name `surface' comes from the classical case, which is $1$-dimensional over the [[complex numbers]] and therefore $2$-dimensional over the [[real numbers]].

There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Classically, a \textbf{Riemann surface} is a [[connected space|connected]] complex-$1$-dimensional [[complex manifold]], in the strictest sense of `manifold'. In other words, it's a [[Hausdorff space|Hausdorff]] [[second countable space]] $M$ which is locally [[homeomorphism|homeomorphic]] to the [[complex plane]] $\mathbb{C}$ via charts (i.e., [[homeomorphisms]]) $\phi_i:U_i \to V_i$ for $U_i \subset M, V_i \subset \mathbb{C}$ open and such that $\phi_j \circ \phi_i^{-1}: V_i \cap V_j \to V_i \cap V_j$ is [[holomorphic function|holomorphic]].

There are generalizations e.g. over [[local field]]s in [[rigid analytic geometry]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Evidently an [[open subspace]] of a Riemann surface is a Riemann surface. In particular, an open subset of $\mathbb{C}$ is a Riemann surface in a natural manner.

The [[Riemann sphere]] $P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}$ or $S^2$ is a Riemann surface with the open sets $U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}$ and the charts

\begin{displaymath}
\phi_1 =z, \;\phi_2 = \frac{1}{z}.
\end{displaymath}
The transition map is $\frac{1}{z}$ and thus holomorphic on $U_1 \cap U_2 = \mathbb{C}^*$.

An important example comes from [[analytic continuation]], which we will briefly sketch below. A \textbf{function element} is a pair $(f,V)$ where $f: V \to \mathbb{C}$ is holomorphic and $V \subset \mathbb{C}$ is an open disk. Two function elements $(f,V), (g,W)$ are said to be \textbf{direct analytic continuations} of each other if $V \cap W \neq \emptyset$ and $f \equiv g$ on $V \cap W$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element $\gamma = (f,V)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $\mathbb{C}$. Then $X$ is actually a Riemann surface.

Indeed, we must first put a topology on $X$. If $(g,W) \in X$ with $W=D_r(w_0)$ centered at $w_0$, then let a neighborhood of $g$ be given by all function elements $(g_w, W')$ for $w \in W, W' \subset W$; these form a basis for a suitable topology on $X$. Then the coordinate projections $(g,W) \to w_0$ form appropriate local coordinates. In fact, there is a globally defined map $X \to \mathbb{C}$, whose image in general will be a proper subset of $\mathbb{C}$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{basic_facts}{}\subsubsection*{{Basic facts}}\label{basic_facts}

Since we have local coordinates, we can define a map $f: X \to Y$ of Riemann surfaces to be \textbf{holomorphic} or \textbf{regular} if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map $f: X \to \mathbb{C}$; for [[meromorphic function|meromorphicity]], this becomes $f: X \to S^2$.

Many of the usual theorems of elementary [[complex analysis]] (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a [[Laurent series|Laurent expansion]], etc.

\begin{theorem}
\label{}\hypertarget{}{}
Let $f: X \to Y$ be a regular map. If $X$ is compact and $f$ is nonconstant, then $f$ is surjective and $Y$ compact.

\end{theorem}
To see this, note that $f(X)$ is [[compact space|compact]], and an open subset by the [[open mapping theorem]], so the result follows by connectedness of $Y$.

\hypertarget{complexified_differentials}{}\subsubsection*{{Complexified differentials}}\label{complexified_differentials}

Since a Riemann surface $X$ is a $2$-dimensional [[smooth manifold]] in the usual (real) sense, it is possible to do the usual [[differential form|exterior calculus]]. We could consider a 1-form to be a section of the (usual) [[cotangent bundle]] $T^*(X)$, but it is more natural to take the \textbf{complexified cotangent bundle} $\mathbb{C} \otimes_{\mathbb{R}} T^*(X)$, which we will in the future just abbreviate $T^*(X)$; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.

If $z = x + i y$ is a local coordinate on $X$, defined say on $U \subset X$, define the (complex) differentials

\begin{displaymath}
d z = d x + i d y , \;d\bar{z} = d x - i d y.
\end{displaymath}
These form a basis for the complexified cotangent space at each point of $U$. There is also a dual basis

\begin{displaymath}
\frac{\partial}{\partial z }  := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \;
\frac{\partial}{\partial \bar{z} }  := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)
\end{displaymath}
for the complexified tangent space.

We now claim that we can split the tangent space $T(X) = T^{1,0}(X) + T^{0,1}(X)$, where the former consists of multiples of $\frac{\partial}{\partial z}$ and the latter of multiples of $\frac{\partial}{\partial \bar{z}}$; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given $g: U \to \mathbb{C}$ for $U \subset \mathbb{C}$ open and $0 \in U$ (just for convenience), we can write

\begin{displaymath}
g(z) = g(0) + Az + A' \bar{z} + o(|z|)
\end{displaymath}
where $A =   \frac{\partial g }{\partial  z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)$, which we will often abbreviate as $g_z(0), g_{\bar{z}}(0)$. If $\psi: U' \to U$ is holomorphic and conformal sending $z_0 \in U' \to 0 \in U$, we have

\begin{displaymath}
g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|);
\end{displaymath}
in particular, $\phi$ preserves the decomposition of $T_0(\mathbb{C})$.

Given $f: X \to \mathbb{C}$ smooth, we can consider the projections of the 1-form $df$ onto $T^{1,0}(X)$ and $T^{0,1}(X)$, respectively; these will be called $\partial f, \overline{\partial} f$. Similarly, we define the corresponding operators on 1-forms: to define $\partial \omega$, first project onto $T^{0,1}(M)$ (the reversal is intentional!) and then apply $d$, and vice versa for $\overline{\partial} \omega$.

In particular, if we write in local coordinates $\omega = u d z + v d\bar{z}$, then

\begin{displaymath}
\partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},
\end{displaymath}
and

\begin{displaymath}
\overline{\partial} \omega  = d( u d z) = u_{\bar{z}} d\bar{z} \wedge d z.
\end{displaymath}
To see this, we have tacitly observed that $d v = v_z d z + v_{\bar{z}} d\bar{z}$.

\hypertarget{PicardGroupOfHolomorphicLineBundles}{}\subsubsection*{{Picard group of holomorphic line bundles}}\label{PicardGroupOfHolomorphicLineBundles}

The [[Picard group]] of a Riemann surface is the group of [[holomorphic line bundles]] in it. Introductions include (\hyperlink{Bobenko}{Bobenko, section 8}).

See also at \emph{[[Narasimhan–Seshadri theorem]]} and at \emph{\href{moduli+space+of+connections#FlatConnectionsOverATorus}{moduli space of connections -- Flat connections over a torus}}.

\hypertarget{central_theorems}{}\subsubsection*{{Central theorems}}\label{central_theorems}

In the theory of Riemann surfaces, there are several important theorems. Here are two:

\begin{itemize}%
\item The [[Riemann-Roch theorem]], which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;


\item The [[uniformization theorem]], which partially classifies Riemann surfaces.



\end{itemize}
\hypertarget{homotopy_type}{}\subsubsection*{{Homotopy type}}\label{homotopy_type}

A [[compact topological space|compact]] Riemann surface of [[genus]] $g \geq 2$ is a [[homotopy 1-type]]. The [[fundamental groupoid]] is a [[Fuchsian group]].

(\href{http://mathoverflow.net/a/93340/381}{MO discussion})

\hypertarget{branched_covers}{}\subsubsection*{{Branched covers}}\label{branched_covers}

By the [[Riemann existence theorem]], every connected compact [[Riemann surface]] admits the [[structure]] of a [[branched cover of the Riemann sphere]]. (\href{http://mathoverflow.net/a/53286/381}{MO discussion})

\hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy}

[[!include function field analogy -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[annulus]], [[2-sphere]], [[trinion]]


\item [[Teichmüller space]]


\item [[moduli space of Riemann surfaces]]


\item [[super Riemann surface]]


\item [[stable vector bundle]]


\item [[elliptic curve]]


\item [[worldsheet]]


\item [[beta-gamma system]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Historical references include

\begin{itemize}%
\item [[Hermann Weyl]], \emph{Die Idee der Riemannschen Fl\"a{}che}, 1913 (\_The concept of a Riemann surface\_) (on the book, by Peter Schreiber, 2013: \href{http://mathineurope.eu/en/home/47-information/math-calendar/971-1913-publication-of-the-concept-of-a-riemann-surface-by-hermann-weyl}{web})

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Alexander Bobenko]], \emph{Compact Riemann Surfaces} lecture notes (\href{http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf}{pdf})


\item [[Eberhard Freitag]], \emph{Riemann surfaces -- Sheaf theory, Riemann Surfaces, Automorphic forms}, 2013 (\href{http://www.rzuser.uni-heidelberg.de/~t91/skripten/riemfl.pdf}{pdf})



\end{itemize}
[[!redirects Riemann surfaces]]



\end{document}