Compact Riemann Surfaces

Abstract

In this chapter, we consider some facts concerning holomorphic line bundles (and their holomorphic sections) on compact Riemann surfaces. We first consider conditions that guarantee the existence of holomorphic sections with prescribed values. Unlike the open Riemann surface case (in which one has Theorem 3.11.5), a holomorphic line bundle need not have the positivity required for such a section to exist. For example, the space of holomorphic functions on a compact Riemann surface X has dimension 1, and a negative holomorphic line bundle on X has no nontrivial global holomorphic sections. After the above considerations, we consider the fact that a holomorphic line bundle is positive if and only if its degree is positive, and we then consider finiteness of Dolbeault cohomology. The Riemann–Roch formula is then proved (using finiteness). Finally, we consider the Serre duality theorem and the Hodge decomposition theorem, and some of their consequences. Different approaches to the above facts appear in, for example, Forster (Lectures on Riemann Surfaces, Springer, 1981) and Narasimhan (Compact Riemann Surfaces, Birkhauser, 1992).