I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, where $X$ is a compact Riemann surface and $\mathcal{O}$ is the sheaf of holomorphic functions. Forster proves this by reducing the problem into considering restriction map of Cech cohomology groups given by a shrinking sequence of relatively compact open coverings(this is done by choosing a proper coordinate patch and using Leray's theorem and Dolbeault's lemma), and introduces $L^2$-norms on such groups. Well, first of all, what's the point in introducing the $L^2$-norm, by which I mean, is that suggesting any interesting topological properties such as compactness or anything else? What's the essential problem lying in this theorem? Secondly, is there any other way of proving this finiteness theorem? I would appreciate your help!
1 Answer
Wells, Differential Analysis on Complex Manifolds, gives the complete proof using the standard approach for complex manifolds, through Hermitian metrics. Forster's book is great, but odd. Forster avoids the use of Hermitian metrics entirely, trying to rely as much as possible on sheaf cohomology. In the algebraic category, that suffices to prove finiteness of cohomology. Another approach, along the lines of Forster's, is that you could prove that compact Riemann surfaces have lots of meromorphic functions, using the potential theory arguments that Forster gives. He builds up a theory of subharmonic functions and then uses Perron's method to prove the existence of harmonic functions satisfying some boundary conditions. If I remember correctly, Forster indicates how to use these results to prove the existence of meromorphic functions separating points. You can then show that every compact Riemann surface admits a holomorphic embedding in complex projective space as the solutions of some algebraic equations (the Kodaira embedding theorem), and then you can use algebraic geometry directly, along the lines of Shafarevitch's Basic Algebraic Geometry. Griffiths' Introduction to Algebraic Curves also provides a proof (I think) of the finiteness of the first cohomology once you have the algebraic embedding. The point of introducing the L2 norm is that you want to be able to use functional analysis to control the projection operation from the cochains to the cohomology. Forster shows that, roughly, you make a finite dimensional error when you restrict a cochain to a smaller open set (lemma 14.3) and this is the source of the finite dimensionality of the cohomology.
