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 $\gamma$ and a function $f$ that is holomorphic in some neighborhood of $\gamma$, and the operation of integration gives us back a complex number depending on both.

Let’s be a bit more careful.  Let $p,q$ be complex numbers and $f$ a meromorphic function on the plane.  Then the integral from $p$ to $q$ of $f$ 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 $f$ 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 $U$ be the open subset of $\mathbb{C}$ where $f$ is actually a holomorphic function.  Then we need to understand loops in $U$, 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 $Holo(U)\times H_1(U,\mathbb{Z})\to \mathbb{C}$.

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 $z^2$, we have a bit more of an issue with $\sqrt{z}$.  For $\sqrt{z}$, 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 $\mathbb{C}$.

In this context, everything still works: we can take any path $\gamma$, 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 $f:X\to\mathbb{C}$ are both real functions, and as the domain is compact, they attain a maximum.  Also, as $f$ 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 $f$ 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.

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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5 Responses to Understanding Integration I: Riemann Surfaces

1. v says:

I received a link to this page via e-mail. I don’t recall but I might have subscribed to it in the past. It was just as well, though, because I’m having a problem with differentiating and integrating comb functions. Of course, I’m not at all sure that it would be of interest to the trend started herewith but I thought I might give it a try. Should it be outside of the topic at hand I’ll retire, otherwise I’ll be much interested to hear from a mathematician about that problem. All the best.

• Well, this series probably won’t be too useful for actually trying to differentiate and integrate things. I can’t even say that I know what comb functions are. But here, integration theory is being used as a means to get into some topics in algebraic geometry.

2. I think your proof that holomorphic functions on a compact Riemann surface are constant is flawed. The problem is that it is not obvious that there are charts mapping part of the surface to the entire complex plane. Recall that all simply connected regions in the complex plane are conformally equivalent … except the entire plane itself!

It would be better to note that the real part (or absolute value) of a holomorphic function will have a maximum by compactness, and then use a maximum principle.

• That’s a good point, and a problem I often have, reaching for Liouville when what I need is the maximum principle. It will be corrected shortly.

3. Reblogged this on L'horreur islamique.