Christian Okonek Michael Schneider Heinz SRindler. ector undies on omplex. rojective S aces

Transcription

1 Christian Okonek Michael Schneider Heinz SRindler ector undies on omplex rojective S aces

2 Progress in Mathe~natics Vol. 1: H. Gross, Quadratic Forms in Infinite-Dimensional Vector Spaces. XXII, 4!9 pages,!979 Vol. 2: F. Pham, Singularites des systemes differentiels de Gauss-Manin. Vl 339 pages, l 979 Vol. 3: C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces. VIII, 389 pages,!980

3 Progress in Mathematics 3 Edited by J. Coates and S. Helgason Christian Okonek Michael Schneider Heinz SQindler Vector Bundles on Complex rojeclive Spaces Springer Science+Business Media, LLC

4 Authors Christian Okonek Michael Schneider Heinz Spindler Mathematische lnstitut der Universitat Bunsenstrasse Gottingen Federal Republic of Germany Library of Congress Cataloging in Publication Data Okonek, Christian, Vector bundles on complex projective spaces. (Progress in mathematics ; 3) Bibliography: p. Includes index. 1. Geometry, Algebraic. 2. Vector bundles. 3. Complex manifolds. 4. Geometry, Projective. I. Schneider, Michael (May 5)- joint author. II. Spindler, Heinz, joint author. Ill. Title. IV. Series: Progress in mathematics (Cambridge) ; 3. OA ' ISBN ISBN (ebook) DOI / CIP-Kurztitelaufnahme der Deutschen Bibliothek Okonek, Christian: Vector bundles on complex projective spaces I Christian Okonek; Michael Schneider; Heinz Spindler.- Boston. Basel, Stuttgart : Birkhauser, (Progress in mathematics: 3) ISBN NE: Schneider, Michael:; Spindler, Heinz: All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. Springer Science+Business Media New York 1980 Originally published by Birkhauser Boston 1980 ISBN

5 Introduction These lecture notes are intended as an introduction to the methods of classification of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as possible we have mostly restricted ourselves to the case X = Fn. According to Serre (GAGA) the classification of holomorphic vector bundles is equivalent to the classification of algebraic vector bundles. Here we have used almost exclusively the language of analytic geometry. The book is intended for students who have a basic knowledge of analytic and (or) algebraic geometry. Some fundamental results from these fields are summarized at the beginning. One of the authors gave a survey in the Seminaire Bourbaki 1978 on the current state of the classification of holomorphic vector bundles overfn. This lecture then served as the basis for a course of lectures in Gottingen in the Winter Semester 78/79. The present work is an extended and up-dated exposition of that course. Because of the introductory nature of this book we have had to leave out some difficult topics such as the restriction theorem of Barth. As compensation we have appended to each section a paragraph in which historical remarks are made, further results indicated and unsolved problems presented. The book is divided into two chapters. Each chapter is subdivided into several sections which in turn are made up of a number of paragraphs. Each section is preceeded by a short description of

6 iv its contents. In assembling the list of literature we have done our best to include all the articles about vector bundles over Pn which are known to us. On the other hctnd we have not thought it necessary to include works about the classification of holomorphic vector bundles over curves. The reader interested in this highly developed theory is recommended to read an article by Tjurin (Russian Math. Surveys 1974) or the lecture notes of a course held at Tata Institute by Newstead. Part of the present interest in holomorphic vector bundles comes from the connection to physics. The mathematician who is interested in this connection is recommended to see the ENS-Seminaire of Douady and Verdier. In the final paragraph of the present lecture notes he will also find remarks about that topic and some literature citations. R. M. Switzer has not only translated the manuscript of these notes into english but has also aided us in answering many mathematical questions. For this assitance we wish to thank him heartily. Furthermore we wish to thank Mrs. M. Schneider for doing such a good job with the unpleasant task of typing these notes and H.Hoppe for assembling the index and doing the difficult job of inserting the mathematical symbols.

7 v Contents Introduction Chapter I Holomorphic vector bundles and the geometry of Wn 1. Basic definitions and theorems 1.1. Serre duality, the Bott formula, Theorem A and Theorem B 1.2. Chern classes and dual classes The splitting of vector bundles 2.1. The theorem of Grothendieck 2.2. Jump lines and the first examples 2.3. The splitting criterion of Horrocks 2.4. Historical remarks Uniform bundles The standard construction Uniform r-bundles over W n, r < n A non-homogeneous uniform (3n-1)-bundle over Some historical remarks, further results and open questions w n o 4. Examples of indecomposable (n-1) -bundles over W n Simple bundles 4.2. The null correlation bundle 4.3. The example of Tango 4.4. Concluding remarks and open questions

8 vi 5. Holomorphic 2-bundles and codimension 2 locally complete intersections 5.1. Construction of 2-bundles associated to a locally complete intersection 5.2. Examples 5.3. Historical remarks 9o 9o 1 o Existence of holomorphic structures on topological bundles Topological classification of bundles bundles over JP bundles over JP bundles over JP Concluding remarks over JP n o 137 Chapter II Stability and moduli spaces Stable bundles Some useful results from sheaf theory Stability: definitions and elementary properties Examples of stable bundles Further results and open questions o The splitting behavior of stable bundles 2.1. Construction of subsheaves 2.2. Applications of the theorem of Grauert and Mlilich 2.3. Historical remarks, further results and open questions o9 234

9 vii 3. Monads 3.1. The theorem of Beilinson 3.2. Examples 3.3. A stable 2-bundle over w Historical remarks Moduli of stable 2-bundles 4.1. Construction of the moduli spaces for stable 2-bundles over w Irreducibility of Mw (o,n) 4.3. Examples 4.4. Historical remarks, further results, open problems o