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Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the geometric approach to modern algebraic number theory. In this book, Kempf has focused on the analytic aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of Kempf's presentation is the systematic use of Mumford's theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles. Also, several examples where abelian varieties arise in various branches of geometry are given as a conclusion of the book.
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George R. Kempf Complex Abelian Varieties and Theta Functions Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona George R. Kempf Department of Mathematics John Hopkins University Baltimore, MD 21218, USA Mathematics Subject Classification (1980): 14K20, 14K25, 32C35, 32125, 32N05 ISBN 3-540-53168-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53168-8 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-m-Publication Data Kempf, George Complcx abelian varieties and theta functions/George R Kcmpf p. cm (Universitext) Includes bibliographical references and index ISBN 0-387-53168-8 - ISBN 3-540-53168-8 1. Abelian varieties. 2 Functions, Theta I Title QA564 K45 1990 5163'53 - dc20 90-22573 CIP Tills work is subject to copynght. All rights are reserved, whether the whole or part of the matenal is concerned, specifically the nghts of translation, repnnting, reuse 01 illustrations, recitation, broadcasting, reproduction on microfilms or in other \\ays, and storage m data banks Duplication of this publication or parts thereof is only permitted under the provisions of the German Copynght Law of September 9, 1965, in its current version, and a copynght fee must always be prud Violations fall under the prosecution act of the German Copyright Law © Springer-Verlag Berlin Heidelberg 1991 Printed m Germany 4113140-543210 - Pnnted on acid-free paper Preface The study of abelian varieties began with the one-dimensional case of elliptic curves. As such cill'ves are defined by a general cubic polynomial equation in two variables, their study is basic to all but the simplest mathematic!:>. The modern approach to elliptic curves occurred in the beginning of the nineteen century with the work of Gauss, Abel and Jacobi. Since the classical period there have been many developments in mathematics. There are basically two distinct lines of generalization of an elliptic curve. They are algebraic curves of higher genus' > 1. The other is higher dimensional compact algebraic groups (abelian varieties). This book deals with these higher dimensional objects which surprisingly enough have more similar properties to elliptic curves than curves of higher genus. There are three methods for studying abelian varieties: arithmetic, algebraic and analytic. The arithmetic study properly using both the algebraic and analytic approaches and reduction modulo a prime. Mumford's book [3] presents an adequate introduction to the algebraic approach with some indication of the analytic theory. In this book I have restricted attention to the analytic approach and