Complex Abelian Varieties
Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and J
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Series editors A. Chenciner S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin ¨ L. Hormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt
Editor-in-Chief M. Berger
J. Coates
S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Christina Birkenhake Herbert Lange
Complex Abelian Varieties Second, Augmented Edition
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Christina Birkenhake Herbert Lange Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstraße 1 1/2 91054 Erlangen Germany e-mail: [email protected] [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 14-02, 14KXX, 32G20, 14H37, 14H40, 14H42, 14F05, 14G25
ISSN 0072-7830 ISBN 978-3-642-05807-3 ISBN 978-3-662-06307-1 (eBook) DOI 10.1007/978-3-662-06307-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH . Violations are liable for prosecution under the German Copyright Law.
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Preface to the Second Edition
During the last 15 years the theory of abelian varieties has seen progress in several directions. We include some of it in this second edition. In fact, there are five new chapters, on automorphisms, on vector bundles on abelian varieties, some new results on line bundles and the theta divisor, and on cycles on abelian varieties. Finally we give an introduction to the Hodge conjecture for abelian varieties. Whereas the first edition was more or less self-contained, this cannot be said any more of the new chapters. We apply several results, the proofs of which would be beyond the scope of the book. To give an example, the theory of abelian varieties relies heavily on Mukai’s Fourier functor and this is expressed in the language of derived cate
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