Joachim Wehler

Department of Mathematics

LMU MÃ¼nchen

Winter Term 2019/20

Riemann Surfaces

Winter Term 2019/20 (Lecture 4+2)

Intended audience and prerequisites

Performance record## 1. Description

The lecture "Riemann surfaces" is a first graduate course in analysis. The lecture continues the undergraduate lecture on complex analysis of one variable.

A Riemann surface is a manifold with a complex structure. The manifold is locally biholomorphic to an open set in the complex plane C. Hence a Riemann surface has complex dimension = 1, i.e. real dimension = 2.

Each Riemann surface is either compact or a Stein manifold. Complex analysis on Stein manifolds generalizes complex analysis on domains in the plane. On Stein manifolds there exist many holomorphic functions. Differently, a compact Riemann surface has no holomorphic functions besides the constants. Hence one topic on compact Riemann surfaces are meromorphic functions. They can be considered holomorphic maps to the Riemann sphere. The Riemann sphere and the complex torus are the most simple compact Riemann surfaces.The lecture introduces and investigates a series of new concepts and a whole new language. These are fundamental also in other domains of pure mathematics:

- The concept of a
**manifold**: A Hausdorff space locally homeomorphic to an affine space. - Charts and transition functions to introduce a
**complex structure**on a manifold. - The language of
**sheaves**: A tool for gluing local data to global objects. -
**Cohomology**of sheaves: To compute possibly obstructions of the gluing process. -
**Line bundles**, their sheaves of sections, and their cohomology groups.

The first sheaves under consideration are respectively the sheaves of continuous functions, of holomorphic functions and of meromorphic functions. Further important examples are the sheaves of differential forms and the sheaf of holomorphic vector fields.

All sections of a sheaf form a vector space. The main results on compact Riemann surfaces are the finiteness theorem for sections in holomorphic line bundles, the dimension formula (Theorem of Riemann-Roch) and the Serre duality theorem.

## 2. Intended audience and prerequisites

Graduate students of mathematics and theoretical physics (TMP), advanced bachelor students.

The lectures in class are planned: Monday and Wednesday, 10-12 o'clock a.m.

Prerequisites: Complex analysis of one variable, differential forms, covering spaces (algebraic topology).

## 3. Performance record

Master mathematics (WP 37, WP 36, WP 34), Master TMP.

## 4. References

- Forster, Otto: Riemann Surfaces. Springer.
- Gunning, Robert: Lectures on Riemann Surfaces.
- Lamotke, Klaus: Riemannsche FlÃ¤chen. Springer.

## 4. Problem sessions

In addition to the lecture in class, each week a problem session will be held. The basis is a series of problems for homework. The students are encouraged to present their solution in class.## 5. Examination (ModulprÃ¼fung)

The examination will be held as oral examination. It takes between 30 and 60 minutes. No auxiliary means are admitted.If you want to participate in the examination, please register by sending me an email until 10.1.2020 with the following data: Name, first name, student number, email address, course of studies, number of semester, module to credit. Note your registration is binding.

Update 11.1.2020: The list for sign up is now closed.

Update 18.1.2020: For the **arrangement of the participants** see Schedule.

## 6. Lecture notes

Script