Moduli of Abelian Varieties
This is a book aimed at researchers and advanced graduate students in algebraic geometry, interested in learning about a promising direction of research in algebraic geometry. It begins with a generalization of parts of Mumford's theory of the equations d
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1644
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1644
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Allan Adler S. Ramanan
Moduli of Abelian Varieties
Springer
Authors Allan Adler Cherokee Station P.O. Box 20276 New York, NY 10021, USA e-mail: [email protected] Sundararaman Ramanan Tata Institute of Fundamental Research Homi Bhabha Road 400 005 Mumbai, India e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Adler, Allan: Moduli of Abelian varieties / Allan Adler; S. Ramanan. Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London ; Milan; Paris; Santa Clara; Singapore; Tokyo; Springer, 1996 (Lecture notes in mathematics ; 1644) ISBN 3-540-62023-0 NE: Ramanan, Sundararaman:; GT
Mathematics Subject Classification (1991): 11041, IIF27, IIF32, II F46, llG05, IlGI8, 14-02, 14020, 14E15, 14130, 14135, 14J60, 14KIO, 14K25, 14M12, 14M15, 14N05, 14NIO, 16W99, 20C33, 20C34 ISSN 0075-8434 ISBN 3-540-62023-0 Springer-Verlag Berlin Heidelberg New York 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the first author SPIN: 10520167 46/3142-543210 - Printed on acid-free paper
Table of Contents
§O
Introduction
1
§1 §2 §3 §4 §5 §6 §7 §8
Standard Heisenberg Groups Heisenberg Groups Representations of Heisenberg Groups Some Homomorphisms Between Abelian Groups Some Homomorphisms Between Heisenberg Groups The Homomorphism T(x,l) The Linear Operator O(x,l) O(x,l) as an Intertwining Operator Characterization of the Homomorphism T(x,l)
8 8
§9 §1O §11 §12 §13 §14
Heisenberg groups of Line Bundles on Abelian Varieties18 Heisenberg Groups of Sheaves on Abelian Varieties 18 The Mumford Functor 19 Strongly Symmetric Line Bundles 20 23 Some Commutative Diagrams in the Category A Application of the Mumford Functor to the Preceding Diagrams24 -I Characterization of 3j, 3j , hh ,id) and h(s2,id) 29
Ch.I
Ch.II
Ch.III
Ch.IV
Ch.V
9
10 11 12 13 14 16
Theta Structures and the Addition Formula §15 Symmetric Theta Structures §16 The
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