Lecture 1, Tuesday, September 16, Definition of Riemann surfaces, first examples. Holomorhic functions. Exercises from Lecture 1 ps-file , pdf-file. Lecture 2, Tuesday, September 23, Basic properies of holomorphic functions.

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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 O. Forster, Lectures on Riemann Surfaces.

Graduate Texts in Math. Iena, Riemann Surfaces. Notes from lectures at the University of Luxembourg Miranda, Algebraic Curves and Riemann Surfaces. Graduate Studies in Math. Narasimhan, Compact Riemann Surfaces. Lectures in Math.

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## Riemann surfaces

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.

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## Lectures on Riemann Surfaces

It seems that you're in Germany. We have a dedicated site for Germany. This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Munster. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent. The book is divided into three chapters.

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