Riemann Surfaces and Algebraic Curves

Lectures at the HU Berlin, Summer 2018

Problems and Notes

Here are the problem sheets, and some notes on the finiteness theorem:

Problems 01 02 03 04 05 06 07 08 09 10 11 / Notes

Topics

Anybody who attended a class in complex analysis knows that it is much nicer than real analysis, and the same holds for the corresponding notions of manifolds. Riemann surfaces, the one-dimensional complex manifolds, lie at the crossroad between various areas. From a topological point of view, a compact Riemann surface is just an orientable two-dimensional compact topological manifold, and as such it is classified by a single invariant, the number of holes in it:

This number of holes is called the genus of the surface. For example, a sphere has genus zero while a doughnut has genus one. The picture becomes much richer from a complex analytic perspective: It turns out that the genus has an analytic interpretation in terms of holomorphic differential forms, and starting from genus one, the compact Riemann surfaces of a given genus vary in families. Furthermore, we will see that any compact Riemann surface carries a natural structure of an algebraic variety and can thus be seen as an object of topology, analysis and algebraic geometry at the same time. In the lecture we will start with the topological setup and then pass via analysis to algebraic geometry. All prerequisites will be developed on the way, but the topic may also serve as a complement to basic courses in topology or algebraic geometry.

Prerequisites

The lecture will only assume familiarity with basic notions of algebra and complex analysis.

Literature

  1. O. Forster, Lectures on Riemann Surfaces. Graduate Texts in Math. 81 (corrected 4th printing), Springer (1999). [Primus]
  2. O. Iena, Riemann Surfaces. Notes from lectures at the University of Luxembourg (2016). [pdf]
  3. R. Miranda, Algebraic Curves and Riemann Surfaces. Graduate Studies in Math. 5, AMS (1995). [Primus]
  4. W. Schlag, A Course in Complex Analysis and Riemann Surfaces. Graduate Studies in Math. 145, AMS (2014). [Primus]
  5. R. Narasimhan, Compact Riemann Surfaces. Lectures in Math. ETH Zürich, Birkhäuser (1992). [Primus]