Vector fields on Singular Varieties
Vector fields on manifolds play a major role in mathematics and other sciences. In particular, the Poincaré-Hopf index theorem gives rise to the theory of Chern classes, key manifold-invariants in geometry and topology. It is natural to ask what is the ‘g
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1987
Jean-Paul Brasselet · José Seade · Tatsuo Suwa
Vector Fields on Singular Varieties
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Jean-Paul Brasselet
Tatsuo Suwa
IML-CNRS, Case 907, Luminy 13288 Marseille Cedex 9, France [email protected]
Department of Mathematics Hokkaido University Sapporo 060-0810, Japan [email protected]
José Seade Instituto de Matemáticas Unidad Cuernavaca Universidad Nacional Autónoma de México C.P. 62210, Cuernavaca, Morelos, México [email protected]
ISBN: 978-3-642-05204-0 e-ISBN: 978-3-642-05205-7 DOI: 10.1007/978-3-642-05205-7 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009940925 Mathematics Subject Classification (2000): 32S65, 37F75, 57R20, 57R25, 58K45, 14B05, 14C17, 14J17, 32S05, 32S55, 58K65 c Springer-Verlag Berlin Heidelberg 2009 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
Vector fields on manifolds play major roles in mathematics and other sciences. In particular, the Poincar´e–Hopf index theorem and its geometric counterpart, the Gauss–Bonnet theorem, give rise to the theory of Chern classes, key invariants of manifolds in geometry and topology. One has often to face problems where the underlying space is no more a manifold but a singular variety. Thus it is natural to ask what is the “good” notion of index of a vector field, and of Chern classes, if the space acquires singularities. The question was explored by several authors with various answers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern–Weil theory. The interplay between these two methods is one of the main features of the monograph. Marseille Cuernavaca Tokyo September 2009
Jean-Paul Brasselet Jos´e Seade Tatsuo Suwa
v
Acknowledgements
Parts of this monograph were written while the authors were staying at various institutions, such as Hokkaido University and Niigata University in Japan, CIRM, Universit´e de la Mediterran´ee and IML at Marseille, France, the Instituto de Matem´ aticas of UNAM at Cuernavaca, Mexico, I
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