Topological Methods in Algebraic Geometry
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MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE HERAUSGEGEBEN VON
J. L. DOO B . E. HEINZ· F. HIRZEBRUCH E.HOPF· H.HOPF· W.MAAK· S. MAC LANE W.MAGNUS· F.K.SCHMIDT· K.STEIN GESCHAFTSFUHRENDE HERAUSGEBER
B. ECKMANN UND B. L. VAN DER WAERDEN ZURICH
BAND 131
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1966
TOPOLOGICAL METHODS IN ALGEBRAIC GEOMETRY
F. HIRZEBRUCH UNIVERSITY OF BONN
THIRD ENLARGED EDITION
NEW APPENDIX AND TRANSLATION FROM THE SECOND GERMAN EDITION BY R. L. E. SCHWARZENBERGER UNIVERSITY OF WARWICK
WITH AN ADDITIONAL SECTION BY A. BOREL INSTITUTE FOR ADVANCED STUDY, PRINCETON
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1966
Geschäftsführende Herausgeber:
Prof. Dr. B. Eckmann Eidgenössische Technische Hochschule Zürich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universität Zürich
All rights reserved, especially that of translation into foreign languages. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard or any other means) without written permission from the Publishers Copyright 1956 and 1962 by Springer-Verlag OHG in Berlin • Göttingen • Heidelberg © by Springer-Verlag Berlin Heidelberg 1966 Ursprünglich erschienen bei Springer-Verlag Berlin Heidelberg New York 1966 Softcover reprint of the hardcover 3rd edition 1966
Library of Congress Catalog Card Number 66-14573
ISBN 978-3-662-30630-7 ISBN 978-3-662-30697-0 (eBook) DOI 10.1007/978-3-662-30697-0
Titel No. 5114
To my teachers Heinrich Behnke and Heinz Hop]
Preface to the first edition In recent years new topological methods, especially the theory of sheaves founded by J. LERAY, have been applied successfully to algebraic geometry and to the theory of functions of several complex variables. H. CARTAN and J.-P. SERRE have shown how fundamental theorems on holomorphically complete manifolds (STEIN manifolds) can be formulated in terms of sheaf theory. These theorems imply many facts of function theory because the domains of holomorphy are holomorphically complete. They can also be applied to algebraic geometry because the complement of a hyperplane section of an algebraic manifold is holomorphically complete. J.-P. SERRE has obtained important results on algebraic manifolds by these and other methods. Recently many of his results have been proved for algebraic varieties defined over a field of arbitrary characteristic. K. KODAIRA and D. C. SPENCER have also applied sheaf theory to algebraic geometry with great success. Their methods differ from those of SERRE in that they use techniques from differential geometry (harmonic integrals etc.) but do not make any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind on algebraic manifolds with the help of sheaf theory. I was able to work together with K. KODAIRA and D. C. SPENCER during a stay at the Institute for Advanced Study at Princeton from 1952 to 1954
