nLab
Riemann surface

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 $1$-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 $ℂ$ 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 field?s 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? $P^1(ℂ):=ℂ\cup \{\mathrm{\infty }\}$ or $S^2$ is a Riemann sphere with the open sets $U_1=ℂ,U_2=ℂ-\{0\}\cup \{\mathrm{\infty }\}$ and the charts

(1)$${\varphi }_1=z,{\varphi }_2=\frac{1}{z}.$$\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 $(f,V)$ where $f:V\to ℂ$ is holomorphic and $V\subset ℂ$ is an open disk. Two function elements $(f,V),(g,W)$ 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 $\gamma =(f,V)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $ℂ$. 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 $(g_w,W\prime )$ for $w\in W,W\prime \subset W$; 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 $X\to ℂ$, 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 $f(X)$ 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 $T^{*}(X)$, but it is more natural to take the complexified cotangent bundle $ℂ{\otimes }_{ℝ}{T}^{*}(X)$, 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 $z=x+iy$ is a local coordinate on $X$, defined say on $U\subset X$, define the (complex) differentials

(2)$$dz=dx+idy,d\overline{z}=dx-idy.$$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)$$\frac{\partial }{\partial z}:=\frac{1}{2}(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}),\frac{\partial }{\partial \overline{z}}:=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})$$\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 $T(X)=T^{\mathrm{1,0}}(X)+T^{\mathrm{0,1}}(X)$, 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)$$g(z)=g(0)+\mathrm{Az}+A\prime \overline{z}+o(\mid z\mid )$$g(z) = g(0) + Az + A' \bar{z} + o(|z|)

where $A=\frac{\partial g}{\partial z}(0),A\prime =\frac{\partial g}{\partial \overline{z}}(0)$, which we will often abbreviate as $g_z(0),g_{\overline{z}}(0)$. If $\psi :U\prime \to U$ is holomorphic and conformal sending $z_0\in U\prime \to 0\in U$, we have

(5)$$g(\varphi (\zeta ))=g(\varphi (0))+A\varphi \prime (z_0)(\zeta -z_0)+A\prime \overline{\varphi \prime (z_0)(\zeta -z_0)}+o(\mid z\mid );$$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 $T_0(ℂ)$.

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

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

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

and

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

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

Further directions

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.