Complex Tori
A complex torus is a connected compact complex Lie group. Any complex 9 9 torus is of the form X =
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Christina Birkenhake . Herbert Lange
Complex Tari
Springer Science+Business Media, LLC
Christina Birkenhake Fakultăt fUr Mathematik und Physik Universităt Bayreuth D-95440 Bayreuth, Germany
Herbert Lange Mathematisches Institut Universităt Erlangen-Niimberg D-91054 Erlangen, Germany
Library of Congress Cataloging-in-Publication Data Birkenhake, Christina. Complex tori I Christina Birkenhake, Herbert Lange. p. cm. - (Progress in mathematics ; v. 177) Includes bibliographical references and index. ISBN 978-1-4612-7195-6 ISBN 978-1-4612-1566-0 (eBook) DOI 10.1007/978-1-4612-1566-0 1. Complex manifolds. 2. Torus (Geometry) 1. Lange, H. (Herbert), 1943II. Title. III. Series: Progress in mathematics (Boston, Mass.) ; VoI. 177. QA613.B455 1999 514'.3~c21 99-32326 CIP
AMS Subject Classifications: 32J\ 8, 32G20, 14K30, 32M05, 14M 17 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Birkhauser Boston in 1999 Softcover reprint of the hardcover lst edition 1999 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-7195-6
SPIN 19901498
Formatted from authors' files by TEXniques, Inc., Cambridge, MA.
987 654 3 2 1
Contents
Introduction 1
Complex Tori 1 Homomorphisms of Complex Tori 2 Line Bundles . . . . . . . The Neron-Severi Group . . 3 4 The Dual Complex Torus. . 5 Extensions of Complex Tori 6 Complementary Subtori and Shafarevich Extensions . 7 Simple and Indecomposable Complex Tori . . . . 8 The Endomorphism Algebra of a Complex Torus. 9 The Theorem of Oort and Zarhin . . . . . . . 10 The Space of all Complex Tori of Dimension 9
2 N ondegenerate Complex Tori 1 Polarizations of Index k .. 2 Moduli Spaces of Nondegenerate Complex Tori. 3 The Rosati Involution . . . . . . . . . . . . . . 4 The Dual Polarization .. . . . . . . . . . . . . Poincare's Reducibility Theorem for Nongenerate 5 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . .. The Algebraic Dimension. . . . . . . . . . . . . . . . . .. 6 Picard Number and Algebraic Dimension of Complex Tori 7 of Dimension 2 . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
3 Embeddings into Projective Space 1 Kahler Theory of Line Bundles on Complex Tori . 2 Harmonic Forms with Values in a Nondegener
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