Vector Fields on Singular Varieties

ISBN-13: 9783642052040 / Angielski / Miękka / 2009 / 232 str.

Jean-Paul Brasselet; Josa(c) Seade; Tatsuo Suwa

Opis

Vector?eldsonmanifoldsplaymajorrolesinmathematicsandothersciences. In particular, the Poincar e Hopf index theorem and its geometric count- part, the Gauss Bonnettheorem, giveriseto the theoryof Chernclasses, 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 notionofindexofavector?eld, andofChernclasses, ifthespaceacquiress- gularities.Thequestionwasexploredbyseveralauthorswithvariousanswers, 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 Jean-Paul Brasselet Cuernavaca Jos e Seade Tokyo Tatsuo Suwa September 2009 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, ICTP at Trieste, Italia, IMPA at Rio de Janeiro, and USP at S ao Carlos in Brasil, to name a few, and we would like to thank them for their generous hospitality and support. Thanks are also due to people who helped us in many ways, in particular our co-authors of results quoted in the book: Marcelo Aguilar, Wolfgang Ebeling, Xavier G omez-Mont, Sabir Gusein-Zade, L eDung Tran g, Daniel Lehmann, David Massey, A.J. Parameswaran, Marcio Soares, Mihai Tibar, Alberto Verjovsky, andmanyother colleagueswho helped usin variousways."

Recenzje

From the reviews:

"This book is dedicated to the study of indices of vector fields and flows around an isolated singularity, or stationary point, in the cases where the underlying space is either a manifold or a singular variety. ... The book gives a thorough presentation of the results, old and new, related to indices of vector fields on singular varieties and is a valuable reference for both the specialist and the non-specialist." (M. G. Soares, Mathematical Reviews, Issue 2011 d)

Spis treści

The Case of Manifolds.- The Schwartz Index.- The GSV Index.- Indices of Vector Fields on Real Analytic Varieties.- The Virtual Index.- The Case of Holomorphic Vector Fields.- The Homological Index and Algebraic Formulas.- The Local Euler Obstruction.- Indices for 1-Forms.- The Schwartz Classes.- The Virtual Classes.- Milnor Number and Milnor Classes.- Characteristic Classes of Coherent Sheaves on Singular Varieties.

Blurb

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 ‘good’ notion of the index of a vector field, and of Chern classes, if the underlying space becomes singular. The question has been explored by several authors resulting in 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.