Understanding Integration I: Riemann Surfaces

I’m back! And now, posting from Kavli IPMU in Japan.  Now, I’m going to start a series on theta functions, Jacobians, Pryms, and abelian varieties more generally, hopefully with some applications, with my goal being at least one post a week, and eventually establishing a regular posting schedule again.  But today, we’ll start with basics, something that should be completely understandable to graduate students and advanced undergraduates.

We start with contour integration in the complex plane.  As described to undergrads, the objects involved are a curve and a function that is holomorphic in some neighborhood of , and the operation of integration gives us back a complex number depending on both.

Let’s be a bit more careful.  Let be complex numbers and a meromorphic function on the plane.  Then the integral from to of is not, itself, well-defined: there are many possible numbers we can get.  In some sense, there’s a “fundamental value” (note: this is rhetorical, there is no preferred value or path, though in many situations, there’s an “obvious” choice), but then we can also get that number plus any integer linear combination of the residues of at the poles.

In fact, this is getting down to the core of what integration is.  First we need to think about the domain: let be the open subset of $\mathbb{C}$ where is actually a holomorphic function.  Then we need to understand loops in , but only up to homology, which counts how the loops go around each puncture, and only that information, which is what we need to actually compute the contribution of the residues.  So, at the moment, integration appears to be a pairing .

Now, we’re algebraic geometers here, despite talking about integration.  So we want to work with things that, quite honestly, are not naturally holomorphic on domains in the complex plane.  Or at least, the domain of holomorphy isn’t going to be in the plane.  For instance, though there’s no problem with , we have a bit more of an issue with .  For , we can choose to take a branch cut from 0 to infinity in order to make it well defined.  Or, we can realize that as it wants to assign two values to each nonzero number, we can take a double cover of the punctured plane to get a legitimate and nice domain.

And so this starts us studying Riemann surfaces.  We’ll be slightly informal and just say that a Riemann surface is a Hausdorff space (second countable, I believe) such that locally looks like open sets in the complex plane.  Just a manifold such that the transition maps are holomorphic to and from domains in .

In this context, everything still works: we can take any path , though we’ll restrict to homology classes of loops, because the indeterminacy turns out to be what’s interesting, and parametrize it, then use that parametrization to integrate a holomorphic function along it.  There’s only one problem.

Compact Riemann surfaces have no non-constant holomorphic functions.  We can actually prove this from basic complex analysis: Liouville’s theorem says that any bounded entire function must be constant, and we can look at a chart on our Riemann surface.  Any holomorphic function on the surface is bounded, by compactness, so it’s bounded on the chart, which transports it to a bounded entire function.  Thus, it is constant on the chart, and so on the whole Riemann surface.  The correct proof is even simpler.  The real and imaginary parts of are both real functions, and as the domain is compact, they attain a maximum.  Also, as is holomorphic, the real and imaginary parts are harmonic on each chart.  As the maximum of a harmonic function must occur on the boundary, and every point is in the interior of some chart, the real and imaginary parts must be constant, and so must be.

In the next post, we’ll talk about a solution to this problem, one that’s significantly better than just looking at meromorphic functions.