Differential Topology of Complex Surfaces Elliptic Surfaces with pg=
This book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up
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1545
John W. Morgan Kieran G. O'Grady
Differential Topology of Complex Surfaces Elliptic Surfaces with pg= 1: Smooth Classification With the collaboration of Millie Niss
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors John W. Morgan Department of Mathematics Columbia University New York, NY 10027, USA Kieran G. O'Grady Institute for Advanced Study Olden Lane Princeton NJ 08540, USA
Mathematics Subject Classification (1991): 14-XX, 57-XX, 53-XX
ISBN 3-540-56674-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56674-0 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Morgan, John W., 1946- .Differential topology of complex surfaces: elliptic surfaces with Pg=l: smooth classification 1John W. Morgan, Kieran G. O'Grady; with the collaboration of Millie Niss. p. em, - (Lecture notes in mathematics; 1545) On t.p. "g" is subscript. Includes bibliographical references and index. ISBN 0-387-56674-0 1. Elliptic surfaces. 2. Differential topology. I. O'Grady, Kieran G., 1958- . II. Niss, Millie. III. Title. IV. Series: Lecture notes in mathematics (Springer-Verlag); 1545. QA3.L28 no. 1545 [QA573] 510 s - dc20 [516.3'52] 93-16063 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, 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 1993 Printed in Germany Typesetting: Camera-ready by authorleditor 46/3140-543210 - Printed on acid-free paper
Contents 1 Introduction 1.1 Statement of the main results 1.2 Background . . . . . . . . 1.3 Outline of the paper . . . 1.4 Conventions and notation
1 2 3 6 8
2
Unstable polynomials of algebraic surfaces 12 2.1 Introduction........................... 12 2.1.1 Generalities on the v-map . . . . . . . . . . . . . . . 14 15 2.2 A stratification of parameter spaces for vector bundles on 5 2.2.1 An inductive procedure that defines the type of a bundle near E 16 2.2.2 Definition of the stratification by type near E 18 19 2.2.3 The pushforward to 5 .. . . . . . . . 2.3 The stratification of Mc+k(S, if) 19 2.3.1 The case of polarizations near to 11"* H . . . . 20 21 2.3.2 The morphism from xt,st to M c+k- ltl(5 , H) 2.4 The At construction 21 22 2.4.1 The construction of At(V) . . . . 2.4.2 Proof of Proposition 2.4.1 . . . . 24 2.5 Analysis of the strata of M c+k (5, if (r)) 25 2.5.1 The strata xt,st . 25 26 2.5.2 The strata Xt,ss . . . . . 27 2.6 Proofs of the theorems. . . . . . 2.6.1 Proof of Theorem 2.1.1 . 27 28 2.6.2 Relative moduli spaces . 30 2.6.3 The relativ
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