nLab
Riemann surface

Context

Complex geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Manifolds and cobordisms

Contents

Idea

A Riemann surface is a 11-dimensional algebro-geometric object with good properties. The name ‘surface’ comes from the classical case, which is 11-dimensional over the complex numbers and therefore 22-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-11-dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space MM which is locally homeomorphic to the complex plane \mathbb{C} via charts (i.e., homeomorphisms) ϕ i:U iV i\phi_i:U_i \to V_i for U iM,V iU_i \subset M, V_i \subset \mathbb{C} open and such that ϕ jϕ i 1:V iV jV iV j\phi_j \circ \phi_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 \mathbb{C} is a Riemann surface in a natural manner.

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

(1)ϕ 1=z,ϕ 2=1z. \phi_1 =z, \;\phi_2 = \frac{1}{z}.

The transition map is 1z\frac{1}{z} and thus holomorphic on U 1U 2= *U_1 \cap U_2 = \mathbb{C}^*.

An important example comes from analytic continuation?, which we will briefly sketch below. A function element is a pair (f,V)(f,V) where f:Vf: V \to \mathbb{C} is holomorphic and VV \subset \mathbb{C} is an open disk. Two function elements (f,V),(g,W)(f,V), (g,W) are said to be direct analytic continuations of each other if VWV \cap W \neq \emptyset and fgf \equiv g on VWV \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 γ=(f,V)\gamma = (f,V), we can consider the totality XX of all equivalence classes of function elements that can be obtained by continuing γ\gamma along curves in \mathbb{C}. Then XX is actually a Riemann surface.

Indeed, we must first put a topology on XX. If (g,W)X(g,W) \in X with W=D r(w 0)W=D_r(w_0) centered at w 0w_0, then let a neighborhood of gg be given by all function elements (g w,W)(g_w, W') for wW,WWw \in W, W' \subset W; these form a basis for a suitable topology on XX. Then the coordinate projections (g,W)w 0(g,W) \to w_0 form appropriate local coordinates. In fact, there is a globally defined map XX \to \mathbb{C}, whose image in general will be a proper subset of \mathbb{C}.

Basic facts

Since we have local coordinates, we can define a map f:XYf: 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:Xf: X \to \mathbb{C}; for meromorphicity, this becomes f:XS 2f: 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:XYf: X \to Y be a regular map. If XX is compact and ff is nonconstant, then ff is surjective and YY compact.

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

Complexified differentials

Since a Riemann surface XX is a 22-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)T^*(X), but it is more natural to take the complexified cotangent bundle T *(X)\mathbb{C} \otimes_{\mathbb{R}} T^*(X), which we will in the future just abbreviate T *(X)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+iyz = x + i y is a local coordinate on XX, defined say on UXU \subset X, define the (complex) differentials

(2)dz=dx+idy,dz¯=dxidy.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 UU. There is also a dual basis

(3)z:=12(xiy),z¯:=12(x+iy) \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 1,0(X)+T 0,1(X)T(X) = T^{1,0}(X) + T^{0,1}(X), where the former consists of multiples of z\frac{\partial}{\partial z} and the latter of multiples of z¯\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:Ug: U \to \mathbb{C} for UU \subset \mathbb{C} open and 0U0 \in U (just for convenience), we can write

(4)g(z)=g(0)+Az+Az¯+o(z) g(z) = g(0) + Az + A' \bar{z} + o(|z|)

where A=gz(0),A=gz¯(0)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 z¯(0)g_z(0), g_{\bar{z}}(0). If ψ:UU\psi: U' \to U is holomorphic and conformal sending z 0U0Uz_0 \in U' \to 0 \in U, we have

(5)g(ϕ(ζ))=g(ϕ(0))+Aϕ(z 0)(ζz 0)+Aϕ(z 0)(ζz 0)¯+o(z); 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, ϕ\phi preserves the decomposition of T 0()T_0(\mathbb{C}).

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

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

(6)ω=d(vdz¯)=v zdzdz¯, \partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},

and

(7)¯ω=d(udz)=u z¯dz¯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 zdz+v z¯dz¯d v = v_z d z + v_{\bar{z}} d\bar{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 April 5, 2014 03:36:45 by Urs Schreiber (185.37.147.12)