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Lectures on Riemann Surfaces

  • Otto Forster

Part of the Graduate Texts in Mathematics book series (GTM, volume 81)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Otto Foster
    Pages 1-95
  3. Otto Foster
    Pages 96-174
  4. Otto Foster
    Pages 175-235
  5. Back Matter
    Pages 243-256

About this book

Introduction

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. And the proof of this is based on the fact that one can locally solve inhomogeneous Cauchy­ Riemann equations and on Schwarz' Lemma.

Keywords

Riemann surfaces Riemann-roch theorem Riemannsche Fläche Surfaces differential equation minimum

Authors and affiliations

  • Otto Forster
    • 1
  1. 1.Mathematishes InstitutUniverität MünchenFederal Republic Of Germany

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-5961-9
  • Copyright Information Springer-Verlag New York 1981
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-5963-3
  • Online ISBN 978-1-4612-5961-9
  • Series Print ISSN 0072-5285
  • Series Online ISSN 0072-5285
  • Buy this book on publisher's site