Lectures on Algebraic Geometry I Sheaves, Cohomology of Sheaves, and
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebrai
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Günter Harder
Lectures on Algebraic Geometry I Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces 2nd revised Edition
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Prof. Dr. Günter Harder Max-Planck-Institute for Mathematics Vivatsgasse 7 53111 Bonn Germany [email protected]
Mathematics Subject Classification 14-01, 14A01, 14F01, 14H01, 14K01
1st Edition 2008 2nd revised Edition 2011 All rights reserved © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011 Editorial Office: Ulrike Schmickler-Hirzebruch | Barbara Gerlach Vieweg+Teubner Verlag is a brand of Springer Fachmedien. Springer Fachmedien is part of Springer Science+Business Media. www.viewegteubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printed on acid-free paper Printed in Germany ISBN 978-3-8348-1844-7
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Preface I want to begin with a defense or apology for the title of this book. It is the first part of a two volume book. The two volumes together are meant to serve as an introduction into modern algebraic geometry. But about two thirds of this first volume concern homological algebra, cohomology of groups, cohomology of sheaves and algebraic topology. These chapters 1 to 4 are more an introduction into algebraic topology and homological algebra than an introduction into algebraic geometry. Only in the last Chapter 5 we will see some algebraic geometry. In this last chapter we apply the results of the previous sections to the theory of compact Riemann surfaces. Even this section does not look like an introduction into modern algebraic geometry, large parts of the material covered looks more like 19’th century mathematics. But historically the theory of Riemann surfaces is one of the roots of algebraic geometry. We will prove the Riemann-Roch theorem and we will discuss the structure of the divisor class group. These to themes are ubiquitous in algebraic geometry. Finally I want to say that the theory of Riemann surfaces is also in these days a very active area, it plays a fundamental role in recent developments. The moduli space of Riemann surfaces attracts the attention of topologists, number theorists and of mathematical physicists. To me this seems to be enough justification to begin an introduction to algebraic geometry by discussing Riemann surfaces at the beginning. Only in the second volume we will lay the foundation
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