Complex Abelian Varieties
Abelian varieties are special examples of projective varieties. As such theycan be described by a set of homogeneous polynomial equations. The theory ofabelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jac
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Editors
M. Artin S. S. Chern J. Coates J. M. Frohlich H. Hironaka F. Hirzebruch L. Hormander C. C. Moore J. K. Moser M. Nagata W Schmidt D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt S. Watanabe Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Herbert Lange
Christina Birkenhake
Complex Abelian Varieties
Springer-Verlag Berlin Heidelberg GmbH
Herbert Lange Christina Birkenhake Mathematisches Institut Universität Erlangen-Nümberg Bismarckstr. 1 1/2 W-8520 Erlangen, FRG
Mathematics Subject Classification (1980): 14-02, 14KXX, 30FlO, 32G20 ISBN 978-3-662-02790-5 DOI 10.1007/978-3-662-02788-2
ISBN 978-3-662-02788-2 (eBook)
Library of Congress Cataloging-in-Publication Data Lange, H. (Herbert), 1943- Complex Abelian varieties / Herbert Lange, Christina Birkenhake. p. cm. -- (Grundlehren der mathematischen Wissenschaften; 302) Includes bibliographical references and index. I. Abelian Varieties. 2. Riemann surfaces. I. Birkenhake, Christina. 11. Title. III. Series. QA564.L32 1992 516.3'53--dc20 92-23806
This work is subject to copyright. All rights are reserved, whether the wh oie or part ofthe material is concerned, specifically the rights oftranslation, 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 ofthe German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992 Softcover reprint of the hardcover 1st edition 1992 Typesetting: Camera ready by authors 41/3140-543210 - Printed on acid-free paper
Contents Introduction Notation Chapter 1 Complex Tori § 1 Complex Tori § 2 Homomorphisms § 3 Cohomology of Complex Tori § 4 The Hodge Decomposition Exercises Chapter 2 Line Bundles on Complex Tori § 1 Line Bundles on Complex Tori § 2 The Appell-Humbert Theorem § 3 Canonical Factors § 4 The Dual Complex Torus § 5 The Poincare Bundle Exercises Chapter 3 Cohomology of Line Bundles § 1 Characteristics § 2 Theta Functions § 3 The Positive Semidefinite Case § 4 The Vanishing Theorem § 5 Cohomology of Line Bundles § 6 The Riemann-Roch Theorem Exercises Chapter 4 Abelian Varieties § 1 Polarized Abelian Varieties § 2 The Riemann Relations § 3 The Decomposition Theorem § 4 The Gauss Map § 5 Projective Embeddings § 6 Symmetric Line Bundles § 7 Symmetric Divisors § 8 Kummer Varieties § 9 Morphisms into Abelian Varieties § 10 The Pontryagin Product § 11 Homological Versus Numerical Equivalence Exercises
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Contents
Chapter 5 Endomorphisms of Abelian Varieties § 1 The Rosati Involution § 2 Polarizations § 3 Norm-Endomorphisms and Symmetric Idempotents § 4 Endomorphisms
