Complex Surfaces and Connected Sums of Complex Projective Planes
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603 Boris Moishezon
Complex Surfaces and Connected Sums of Complex Projective Planes
Springer-Verlag Berlin Heidelberg NewYork 19.77
Author Boris Moishezon Mathematics Department TeI-Aviv University TeI-Aviv/Israel
Library of Congress Cataloging in Publication Data
Moishczon, Boris, 1937Complex sumfaces and connected sums of complex projective planes. (Lecture notes in mathematics ; 603) Bibliography: p. Includes index. I. Surfaces, Algebraic. 2. Projective planes. 3. Manifolds (Mathematics) I. Title. If. Series: Lecture notes in mathematics (Berlin) ; 603. QA3.L28 no. 603 ~QA571~ 510'.8s r516'.352~ 77-22/36
AMS Subject Classifications (1970): 14J 99, 32J15, 57A15, 57 D55 ISBN 3-540-08355-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-08355-3 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE
OF C O N T E N T S
Page
Introduction
1
Part I.
6
Topology of simply-connected algebraic surfaces of given degree n
§i.
A topological comparison theorem for fibers of holomorphic functions on complex threefolds
~2.
A topological comparison theorem for elements of linear systems on complex threefolds
35
§3.
Comparison of topology of simply-connected projective surfaces of degree n and non-singular hypersurfaces of degree n in ~pg.
60
§4.
Simply-connected algebraic surfaces of general type
66
§5.
Topological normalization of simply-connected algebraic surfaces
69
Appendix I.
Generic projections of algebraic surfaces into ~p3
6
72
§i.
A theorem of F. Severi
82.
Duality Theorem and Corollaries of it
82
§3.
Proof of Theorem 3 (§3, Part I)
99
§i.
Deformations of elliptic surfaces with~'non-stable '' singular fibers
116
§2.
Lefshetz
162
Part If.
Elliptic Surfaces
fibrations of 2-toruses
72
116
Kodaira fibrations of 2-toruses
196
Topology of simply-connected elliptic surfaces
222
Appendix If.
R. Livne.
A Theorem about the modular group
223
Bibliography
231
Subject Index
23~
Introduction In [i] Wall proved the following theorem: If VI,V 2 are simply-connected h-cobordant such that
compact 4-manifolds,
which are
to each other, then there exists an integer k ~ O V 1 / k(S2×S 2) is diffeomorphic
(/ is the connected
to V 2 ~ k(S2×S 2)
sum operation).
It follows almost immediately is a simply-connected
from this result that if V
compact 4-manifold,
integer k ~ O such that V ~ (k+l)P ~ k Q
then there exists an
is diffeomorphic
to
LP ~ mQ for some ~,
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