Indices of Vector Fields on Singular Varieties: An Overview

Abstract

Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincaré-Hopf when the ambient variety is a smooth manifold. One has the Schwartz index, the local Euler obstruction, the GSV-index, the virtual index and the homological index (and probably other indices too). In this article we give a brief overview of all these indices and the relations amongst them. We also indicate their relations with the various generalisations to complex analytic singular varieties of the Chern classes of complex manifolds.