# nLab Riemann surface

complex geometry

### Examples

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

A Riemann surface is a $1$-dimensional 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.

## Definition

Classically, a Riemann surface is a connected complex-$1$-dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space $M$ which is locally homeomorphic to the complex plane $ℂ$ via charts (i.e., homeomorphisms) ${\varphi }_{i}:{U}_{i}\to {V}_{i}$ for ${U}_{i}\subset M,{V}_{i}\subset ℂ$ open and such that ${\varphi }_{j}\circ {\varphi }_{i}^{-1}:{V}_{i}\cap {V}_{j}\to {V}_{i}\cap {V}_{j}$ is holomorphic.

There are generalizations e.g. over local fields in rigid analytic geometry.

## Examples

Evidently an open subspace of a Riemann surface is a Riemann surface. In particular, an open subset of $ℂ$ is a Riemann surface in a natural manner.

The Riemann sphere? ${P}^{1}\left(ℂ\right):=ℂ\cup \left\{\infty \right\}$ or ${S}^{2}$ is a Riemann sphere with the open sets ${U}_{1}=ℂ,{U}_{2}=ℂ-\left\{0\right\}\cup \left\{\infty \right\}$ and the charts

(1)${\varphi }_{1}=z,\phantom{\rule{thickmathspace}{0ex}}{\varphi }_{2}=\frac{1}{z}.$\phi_1 =z, \;\phi_2 = \frac{1}{z}.

The transition map is $\frac{1}{z}$ and thus holomorphic on ${U}_{1}\cap {U}_{2}={ℂ}^{*}$.

An important example comes from analytic continuation?, which we will briefly sketch below. A function element is a pair $\left(f,V\right)$ where $f:V\to ℂ$ is holomorphic and $V\subset ℂ$ is an open disk. Two function elements $\left(f,V\right),\left(g,W\right)$ are said to be direct analytic continuations of each other if $V\cap W\ne \varnothing$ 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 $\gamma =\left(f,V\right)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $ℂ$. Then $X$ is actually a Riemann surface.

Indeed, we must first put a topology on $X$. If $\left(g,W\right)\in X$ with $W={D}_{r}\left({w}_{0}\right)$ centered at ${w}_{0}$, then let a neighborhood of $g$ be given by all function elements $\left({g}_{w},W\prime \right)$ for $w\in W,W\prime \subset W$; these form a basis for a suitable topology on $X$. Then the coordinate projections $\left(g,W\right)\to {w}_{0}$ form appropriate local coordinates. In fact, there is a globally defined map $X\to ℂ$, whose image in general will be a proper subset of $ℂ$.

## Basic facts

Since we have local coordinates, we can define a map $f:X\to Y$ of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map $f:X\to ℂ$; for 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 expansion, etc.

###### Theorem

Let $f:X\to Y$ be a regular map. If $X$ is compact and $f$ is nonconstant, then $f$ is surjective and $Y$ compact.

To see this, note that $f\left(X\right)$ is compact, and an open subset by the open mapping theorem?, so the result follows by connectedness of $Y$.

## 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 exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle ${T}^{*}\left(X\right)$, but it is more natural to take the complexified cotangent bundle $ℂ{\otimes }_{ℝ}{T}^{*}\left(X\right)$, which we will in the future just abbreviate ${T}^{*}\left(X\right)$; 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 $z=x+iy$ is a local coordinate on $X$, defined say on $U\subset X$, define the (complex) differentials

(2)$dz=dx+idy,\phantom{\rule{thickmathspace}{0ex}}d\overline{z}=dx-idy.$d z = d x + i d y , \;d\bar{z} = d x - i d y.

These form a basis for the complexified cotangent space at each point of $U$. There is also a dual basis

(3)$\frac{\partial }{\partial z}:=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right),\phantom{\rule{thickmathspace}{0ex}}\frac{\partial }{\partial \overline{z}}:=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right)$\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)

for the complexified tangent space.

We now claim that we can split the tangent space $T\left(X\right)={T}^{1,0}\left(X\right)+{T}^{0,1}\left(X\right)$, where the former consists of multiples of $\frac{\partial }{\partial z}$ and the latter of multiples of $\frac{\partial }{\partial \overline{z}}$; 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 $g:U\to ℂ$ for $U\subset ℂ$ open and $0\in U$ (just for convenience), we can write

(4)$g\left(z\right)=g\left(0\right)+\mathrm{Az}+A\prime \overline{z}+o\left(\mid z\mid \right)$g(z) = g(0) + Az + A' \bar{z} + o(|z|)

where $A=\frac{\partial g}{\partial z}\left(0\right),A\prime =\frac{\partial g}{\partial \overline{z}}\left(0\right)$, which we will often abbreviate as ${g}_{z}\left(0\right),{g}_{\overline{z}}\left(0\right)$. If $\psi :U\prime \to U$ is holomorphic and conformal sending ${z}_{0}\in U\prime \to 0\in U$, we have

(5)$g\left(\varphi \left(\zeta \right)\right)=g\left(\varphi \left(0\right)\right)+A\varphi \prime \left({z}_{0}\right)\left(\zeta -{z}_{0}\right)+A\prime \overline{\varphi \prime \left({z}_{0}\right)\left(\zeta -{z}_{0}\right)}+o\left(\mid z\mid \right);$g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|);

in particular, $\varphi$ preserves the decomposition of ${T}_{0}\left(ℂ\right)$.

Given $f:X\to ℂ$ smooth, we can consider the projections of the 1-form $\mathrm{df}$ onto ${T}^{1,0}\left(X\right)$ and ${T}^{0,1}\left(X\right)$, 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 ${T}^{0,1}\left(M\right)$ (the reversal is intentional!) and then apply $d$, and vice versa for $\overline{\partial }\omega$.

In particular, if we write in local coordinates $\omega =udz+vd\overline{z}$, then

(6)$\partial \omega =d\left(vd\overline{z}\right)={v}_{z}dz\wedge d\overline{z},$\partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},

and

(7)$\overline{\partial }\omega =d\left(udz\right)={u}_{\overline{z}}d\overline{z}\wedge dz.$\overline{\partial} \omega = d( u d z) = u_{\bar{z}} d\bar{z} \wedge d z.

To see this, we have tacitly observed that $dv={v}_{z}dz+{v}_{\overline{z}}d\overline{z}$.

## Theorems

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

• The Riemann-Roch theorem, which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;
• The uniformization theorem?, which partially classifies Riemann surfaces.

## References

Revised on February 18, 2013 23:04:58 by Urs Schreiber (80.81.16.253)