Forster: Riemann Surfaces

Time and Room: Tue, Fri 11-13, E 27
Starts on Tuesday, October 21, 2003, at 11:15h

Problem sessions: Tuesday 14-16, E 47

What this course is about: Every serious study of analytic functions of one complex variable will need Riemann surfaces. For example, "multi-valued" functions like square root or logarithm can be treated in a satisfactory way using Riemann surfaces covering the complex plane. Abstractly speaking, a Riemann surface is simply a complex 1-dimensional manifold (which looks locally like an open set in the complex plane). Some topics treated in this course: Definitions and basic properties. Construction of Riemann surfaces associated to algebraic functions (the square root is the most elementary example). Quotients of Riemann surfaces by discontinuous automorphism groups (this allows an elegant treatment of modular functions and forms). Divisors, line bundles, Theorem of Riemann-Roch

Prerequisites: A first course on the theory of analytic functions of one complex variable (e.g. Funktionentheorie I). Basic notions of algebra and topology.

For: Studierende der Mathematik (und Theoretischen Physik) im Hauptstudium. Students of the International Master Program in Mathematics

Contents

1. Definition of Riemann surfaces

cf. [1], Sect. 1

2. Elementary properties of holomorphic mappings

cf. [1], Sect. 2

3. Branched and unbranched coverings

cf. [1], Sect. 4

4. Fundamental group, universal covering, covering transformations

cf. [1], Sects 3 and 5

5. Algebraic functions

cf. [1], Sect. 8

6. Sheaves, divisors, line bundles

cf. [1], Sects 6, 29

7. Cohomology groups

cf. [1], Sects 12-14

8. The Riemann-Roch theorem

cf. [1], Sects 15, 16, 17

Literature

1. O. Forster: Lectures on Riemann Surfaces. Springer Verlag
2. Farkas/Kra: Riemann Surfaces. Springer Verlag
3. Gunning: Lectures on Riemann Surfaces. Mathematical Notes. Princeton University Press

Otto Forster (), 2003-07-07/2003-12-07