Finally, it’s time to get to the definition of a Riemann surface. A **Riemann surface** is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space which is locally homeomorphic to via charts (i.e., homeomorphisms) for open and such that is holomorphic.

**Examples **

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

The Riemann sphere or is a Riemann sphere with the open sets and the charts

The transition map is and thus holomorphic on .

An important example comes from analytic continuation, which I will briefly sketch below. A **function element** is a pair where is holomorphic and is an open disk. Two function elements are said to be **direct analytic continuations** of each other if and on . 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 , we can consider the totality of all equivalence classes of function elements that can be obtained by continuing along curves in . Then is actually a Riemann surface.

Indeed, we must first put a topology on . If with centered at , then let a neighborhood of be given by all function elements for ; these form a basis for a suitable topology on . Then the coordinate projections form appropriate local coordinates. In fact, there is a globally defined map , whose image in general will be a proper subset of .

**Basic facts **

Since we have local coordinates, we can define a map 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 ; for meromorphicity, this becomes .

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 1Let be a regular map. If is compact and is nonconstant, then is surjective and compact.

To see this, note that is compact, and by the open mapping theorem, open, so the result follows by connectedness of .

**Complexified differentials **

Since a Riemann surface is a smooth 2-dimensional manifold, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle , but it is more natural to take the **complexified cotangent bundle** , which I will in the future just abbreviate ; this will not be confusing since I will only do this when I talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.

I think the natural setting of all this leads to the theory of almost complex manifolds, which sometime soon I will try to learn and/or blog about.

If is a local coordinate on , defined say on , define the (complex) differentials

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

for the complexified tangent space.

I now claim that we can split the tangent space , where the former consists of multiples of and the latter of multiples of ; 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 for open and (just for convenience), we can write

where , which we will often abbreviate as . If is holomorphic and conformal sending , we have

in particular, preserves the decomposition of .

Given smooth, we can consider the projections of the 1-form onto and , respectively; these will be called . Similarly, we define the corresponding operators on 1-forms: to define , first project onto (the reversal is intentional!) and then apply , and vice versa for .

In particular, if we write in local coordinates , then

and

To see this, we have tacitly observed that .

Admittedly this isn’t the best place to stop. Next time, I’ll discuss more about how one can manipulate 1- and 2-forms. Pretty soon we can then get to some big theorems.

December 4, 2009 at 9:47 pm

Great post! Which big theorems will you be talking about?

December 4, 2009 at 10:07 pm

Thanks; I’m currently aiming for Serre duality, Riemann-Roch, Abel’s theorem, uniformization, the solution of the Dirichlet problem (via Perron’s method probably), and the Hodge decomposition of . Of course, how far I’ll get in the series depends largely on how quickly I can learn some of these topics…

December 4, 2009 at 10:50 pm

Great! I would recommend that you also cover the Riemann-Hurwitz theorem; it is an absolutely fundamental tool. The proof is fairly elementary, and relies on the existence of nice triangulations of all Riemann surfaces. Another useful and fun tool is the Plucker formula (http://eom.springer.de/P/p072900.htm; http://en.wikipedia.org/wiki/Adjunction_formula_(algebraic_geometry)#Examples). After knowing Riemann-Roch, one can compute the dimension of the space of holomorphic 1-forms on an elliptic curve (I won’t spoil the answer). Similarly, one can compute the degree of the canonical divisor. Under the assumption that the characteristic of the field of definition is not two or three (two for obvious reasons), one obtains the relation . You will see that . Looking at the other spaces , the above relation tells you that, remarkably, the dimension increases by 1 as increases (this is telling you something about certain linear conditions for vanishing). Now, you simply pick certain generators (meromorphic functions) for these spaces. By observing their degree, you derive dependence relations between them in dimension 6. Ultimately, you can derive the Weierstrass representation of the curve, a cubic. However, Plucker’s formula (also known as the degree-genus formula) tells you this fact immediately given that the topological genus of an elliptic curve is 1. (The genus 1 Riemann surface must be pointed at some P, firstly to provide an identity for the group structure, and secondly to make Aut(X,P) finite, making certain moduli problems well-defined.)

There are many more instances where these theorems are useful; I have not alluded to the ubiquity of the Hurwitz formula, but any sort of branching problem is bound to employ it. Anyway, deriving the equation via Riemann-Roch is more enlightening, so I suppose you should just do both. 🙂

December 5, 2009 at 9:25 am

Thanks for the recommendation! You’re of course right about the Riemann-Hurwitz theorem; Farkas and Kra emphasize it early on using triangulations, though they too give a later proof using Riemann-Roch (which I think is the same as Hartshorne’s, amusingly).

I’ll have to look further into the Plucker formula; I heard about in Griffiths and Harris but didn’t actually get anywhere in that book (which was probably due to my knowing no differential geometry when I looked at it).

Your comment on obtaining the Weierstrass form of an elliptic curve reminds me of something I vaguely learned last year when I took a course on elliptic curves with Silverman’s book; then again, I didn’t actually know most of these things and was using Riemann-Roch entirely as a black box.

December 5, 2009 at 2:01 pm

Which topics in the theory of elliptic curves are you interested in?

December 5, 2009 at 10:25 pm

The highlights of the course were the Mordell-Weil theorem, Siegel’s theorem, and a bit of complex multiplication.

As a result, I had a fair bit of interest in Diophantine geometry in general for a while but I think I should first understand algebraic geometry. Silverman’s book doesn’t presuppose much if you are willing to trust certain things. I tried to read his Diophantine Geometry for a while, but I didn’t get very far owing to my not knowing about abelian varieties.