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.
Author(s): Jean-Paul Brasselet, José Seade, Tatsuo Suwa (auth.)
Series: Lecture Notes in Mathematics 1987
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2009
Language: English
Pages: 232
Tags: Several Complex Variables and Analytic Spaces; Dynamical Systems and Ergodic Theory; Manifolds and Cell Complexes (incl. Diff.Topology); Global Analysis and Analysis on Manifolds; Algebraic Geometry
Front Matter....Pages i-xx
The Case of Manifolds....Pages 1-29
The Schwartz Index....Pages 31-41
The GSV Index....Pages 43-69
Indices of Vector Fields on Real Analytic Varieties....Pages 71-83
The Virtual Index....Pages 85-96
The Case of Holomorphic Vector Fields....Pages 97-113
The Homological Index and Algebraic Formulas....Pages 115-128
The Local Euler Obstruction....Pages 129-141
Indices for 1-Forms....Pages 143-166
The Schwartz Classes....Pages 167-184
The Virtual Classes....Pages 185-192
Milnor Number and Milnor Classes....Pages 193-200
Characteristic Classes of Coherent Sheaves on Singular Varieties....Pages 201-213
Back Matter....Pages 215-231
