Vector Bundles and Differential Equations Proceedings, Nice, France
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Progress in Mathematics 7 Edited by
J. Coates and
S. Helgason
Vector Bundles and Differential Equations
Proceedings, Nice, France June 12-17, 1979
Edited by Andre Hirschowitz
Birkhauser Boston, Basel, Stuttgart
Editor Professor Andre Hlrschowltz Universite de Nice Institut de Mathematiques et Sciences physiques Parc Val rose 06034 Nice Cedex France
Library of Congress Cataloging in Publication Data Main entry under title: Vector bundles and differential equations. (Progress in mathematics; 7) Bibliography: p. 1. Vector bundles-Congresses. 2. Differential equations-Congresses. I. Hirschowitz, A. II. Series: Progress in mathematics (Cambridge); 7. QA612.63.v42 514'.224 80-19583
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Vector bundles and differential equations: proceedings, Nice, June 12-17, 1979/ ed. by A. Hirschowitz. - Boston, Basel, Stuttgart: Birkhauser, 1980. (Progress in mathematics; 7) NE: Hirschowitz, Andre [Hrsg.] 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. © Birkhciuser Boston, 1980
ISBN 978-0-8176-3022-5 ISBN 978-1-4684-9415-0 (eBook) DOl 10.1007/978-1-4684-9415-0
TAB L E
o
F
CON TEN T S
PREFACE
v
BARTH:
COUNTING SINGULARITIES OF QUADRATIC FORMS ON VECTOR BUNDLES
BOURGUIGNON:
GROUPE DE JAUGE ELARGI ET CONNEXIONS STABLES
2I
LES FIBRES UNIFORMES DE RANG AU PLUS n SUR SONT CEUX QU'ON CROIT
37
ELENCWAJG/ HIRSCHOWITZI SCHNEIDER:
FORSTER/ HIRSCHOWITZ/ SCHNEIDER: TYPE DE SCINDAGE GENERALISE POUR LES FIBRES STABLES HARTSHORNE:
HULEK:
ON THE CLASSIFICATION OF ALGEBRAIC SPACE CURVES
65
83
ON THE CLASSIFICATION OF STABLE RANK-r VECTOR BUNDLES OVER THE PROJECTIVE PLANE
113
LE POTIER:
STABILITE ET AMPLITUDE SUR P2(1t)
145
TRAUTMANN:
ZUR BERECHNUNG VON YANG-MILLS POTENTIALEN DURCH HOLOMORPHE VEKTORBUNDEL
183
PRE F ACE
This volume contains eight lectures resulting from papers delivered at the conference IIJournees mathematiques sur les Fibres vectoriels et Equations differentielles" held in Nice. France from June 12 through June 17.1979. The conference was sponsored by the Societe Mathematique de France.
Partial support was provided by:
Comite Doyen Jean Lepine de la Ville de Nice. Conseil General des Alpes Maritimes.
COUNTING
SINGULARITIES
OF
QUADRATIC
Wolf
FORMS
ON
VECTOR
BUNDLES
BARTH
O. INTRODUCTION. The study of surfaces in P3 with many nodes
(=
ordinary double points) is a
beau-
tiful classical topic, which recently found much attention again [3, 4J. AI I systematic ways to produce such surfaces seem related to symmetric matrices of homogeneous polynomials or, more generally, to quadratic forms on vector bundles: If the form q on the bundle E is generic, then q is of maximal rank on an open set. The rank of q is one less on the discriminant hypersurface {det q = o}, which represents the class 2c 1 (E~). This hypersurface is nonsingular in cod
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