Positivity in Algebraic Geometry I Classical Setting: Line Bundles a
This two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a
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A Series of Modern Surveys in Mathematics
Editorial Board M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton H.W. Lenstra, Jr., Leiden J.Tits, Paris D. B. Zagier, Bonn/Paris G.M. Ziegler, Berlin Managing Editor R. Remmert, Münster
Volume 48
Robert Lazarsfeld
Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series
1 23
Prof. Robert Lazarsfeld Department of Mathematics University of Michigan Ann Arbor, MI 48109, USA e-mail: [email protected]
Library of Congress Control Number: 2004109578 Mathematics Subject Classification (2000): Primary: 14-02 Secondary: 14C20, 14F05, 14F17, 32L10, 32J99, 14J17
ISBN 978-3-540-2258-7 ISBN 978-3-642-18808-4 (eBook) DOI 10.1007/978-3-642-18808-4 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 microfilms or in any other ways, 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer Berlin Heidelberg New York 2004 Softcover reprint of the hardcover 1st edition 2004 Typesetting: Computer to film by the authors’ data Printed on acid-free paper 41/3142XT - 5 4 3 2 1 0
To Lee Yen, Sarah, and John
Preface
The object of this book is to give a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Our focus lies on a number of questions that grew up with the field during the period 1950–1975. The sheaf-theoretic methods that revolutionized algebraic geometry in the fifties — notably the seminal work of Kodaira, Serre, and Grothendieck — brought into relief the special importance of ample divisors. By the mid sixties a very satisfying theory of positivity for line bundles was largely complete, and first steps were taken to extend the picture to bundles of higher rank. In a related direction, work of Zariski and others led to a greatly deepened understanding of the behavior of linear series on algebraic varieties. At the border with topology the classical theorems of Lefschetz were understood from new points of view, and extended in surprising ways. Hartshorne’s book [276] and the survey articles in the Arcata proceedings [281] give a good picture of the state of affairs as of the mid seventies. The years since then have seen continued interest and activity in these matters. Work initiated during the earlier period has matured and found new applications. More importantly, the flowering of higher dimensional geometry has led to fresh perspectives and — especially in connection with vanishing theorems
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