Holomorphic Vector Bundles over Compact Complex Surfaces
The purpose of this book is to present the available (sometimes only partial) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic structures in a given topological vector bundle over a compact co
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Subseries: Mathematisches Institut der Universitat und Max-Planck-Institut fiir Mathematik Bonn - vol. 22 Adviser: F. Hirzebruch
1624
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Vasile Brinzanescu
Holomorphic Vector Bundles over Compact Complex Surfaces
Springer
Author Vasile Brinzanescu Institute of Mathematics of Romanian Academy P.O. Box 1-764 70700 Bucharest, Romania
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Brizanescu, Vasile: Holomorphic vector bundles over compact complex surfaces / Vasile Brizanescu. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1624) ISBN 3-540-61018-9 NE:GT
Mathematics Subject Classification (1991): 32L10, 32J15, 32G13, 14D20, 14J05, 32L07, 53C07, 55R1O ISBN 3-540-61018-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part ofthe 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 Typesetting: Camera-ready TEX output by the author SPIN: 10479984 46/3142-543210 - Printed on acid-free paper
To Ligia and Oana
Introduction
The theory of holomorphic vector bundles over compact complex analytic manifolds is not completely developed. Still, in the case of compact complex surfaces, very important results were obtained in the last twenty years. The main purpose of this book is to present the available (sometimes only partial !) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic vector bundles over compact complex surfaces, with special emphasis to the case of nonalgebraic surfaces. Thus, the book is an introduction to the subject of holomorphic vector bundles over the nonalgebraic surfaces and the intersection with the classical books by C. Okonek, M. Schneider, H. Spindler Vector bundles on complex projective spaces, S. Kobayashi Differential geometry of complex vector bundles, R. Friedman, J.W. Morgan Smooth four-manifolds and complex surfaces is reasonably small. The existence problem asks whether in a given (COO) topological vector bundle E over a compact complex surface X it exists a holomorphic structure. For an algebraic surface X, Schwarzenberger [Swl ] has shown that every topological vector bundle whose first Chern class lies in the Neron-Severi group of X, admits an algebraic structure. In the case of a nonalgebraic surface, the an
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