Vector fields on singular varieties (Q1037613)

This book is mainly devoted to the theory of indices of vector fields on singular spaces and their relations to Chern classes. The study in this area started in the early 1960's, mainly thanks to Marie-Hélène Schwartz and Robert MacPherson, and has been continued by several others, including the authors of the monograph under review. As a guiding principle of this writing can be seen the interplay between, on the one hand, the topological approach, relying on obstruction theory in the spirit of \textit{N. Steenrod}'s book [The topology of fibre bundles (1951; Zbl 0054.07103)] and, on the other hand, the geometric approach, based on the Chern-Weil theory. The book consists of 13 chapters. In Chapter 1, the authors review mainly for smooth manifolds some versions of selected results on indices of vector fields and characteristic classes. [The reviewer remarks that by a ``nonzero vector field''\, they mean an everywhere nonzero vector field.] They also outline for manifolds the two approaches mentioned above: localization via obstruction theory, related to relative characteristic classes, and localization via the Chern-Weil theory, related to the theory of residues. In Chapter 2, the authors begin to deal with vector fields on singular analytic varieties; they extend to arbitrary stratified vector fields on singular varieties the Schwartz index, originally defined just for a special class of vector fields called radial. Chapter 3 is devoted to the GSV (``G''\, stands for Gómez-Mont, ``S''\, for Seade, and ``V''\, for Verjovski) and related indices. While Chapters 2 and 3 focus on indices of vector fields on complex analytic varieties, Chapter 4 studies properties of indices analogous to the GSV and Schwartz indices for vector fields on real analytic singular varieties. Chapter 5 concentrates on the virtual index which was introduced by D. Lehmann, M. Soares, and T. Suwa for holomorphic vector fields on complex analytic varieties, and then it was extended to continuous vector fields. Chapter 6 is devoted to holomorphic vector fields and one-dimensional singular foliations. In Chapter 7, the authors discuss the homological index introduced by X. Gómez-Mont for holomorphic vector fields, and they also present the Eisenbud-Levin-Khimshiashvili signature formula for the index of real analytic vector fields as well as its generalization to singular varieties. The main topic of Chapter 8 is the local Euler obstruction, introduced by MacPherson for constructing characteristic classes of singular complex algebraic varieties. Finally, Chapters 10-13 deal with several generalizations of Chern classes in the context of singular varieties from the viewpoint of localization theory, by means of indices of vector fields; titles of these chapters are The Schwartz Classes (Ch. 10), The Virtual Classes (Ch. 11), Milnor Number and Milnor Classes (Ch. 12), and Characteristic Classes of Coherent Sheaves on Singular Varieties (Ch. 13).