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.
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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. In the first chapter we consider Riemann surfaces as covering spaces and develop a few basics from topology which are needed for this.
Then we construct the Riemann surfaces which arise via analytic continuation of function germs. In particular this includes the Riemann surfaces of algebraic functions. As well we look more closely at analytic functions which display a special multi-valued behavior. Examples of this are the primitives of holomorphic i-forms and the solutions of linear differential equations.
The second chapter is devoted to compact Riemann surfaces. The main classical results, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion problem, are presented. Sheaf cohomology is an important technical tool. But only the first cohomology groups are used and these are comparatively easy to handle.
The main theorems are all derived, following Serre, from the finite dimensionality of the first cohomology group with coefficients in the sheaf of holomorphic functions. This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces. The reviewer is inclined to think that it may well become a favorite. Only valid for books with an ebook version.
Read this book on SpringerLink. Recommended for you. PAGE 1.
Lectures on Riemann Surfaces. Otto Forster. 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.
Lectures on Riemann Surfaces
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. Meromorphic functions, first properties of morhisms of Riemann surfaces. Exercises from Lecture 2 ps-file , pdf-file. Lecture 3, Tuesday, September 30, Elementary properties of morphisms of Riemann surfaces.