The Schwartz Index

Abstract

The index of a tangent vector field in a singular point is well-defined on manifolds, as described in the previous chapter. When working with singular analytic varieties, it is necessary to give a sense to the notion of “tangent” vector field and, once this is done, it is natural to ask what should be the notion of “the index” at a singularity of the suitable vector field. Indices of vector fields on singular varieties were first considered by M.-H. Schwartz in [139,141] (see also [33,142]) in her study of the Poincarè–Hopf Theorem and Chern classes for singular varieties. For her purpose there was no point in considering vector fields in general, but only a special class of vector fields that she called “radial,” which are obtained by the important process of radial extension. In this chapter we explain the definition of the corresponding index as it was defined by M.-H. Schwartz for vector fields constructed by radial extension. Complete description and constructions will be found in [28]. We define a natural extension of this index for arbitrary (stratified) vector fields on singular varieties. This index is sometimes called “radial index” in the literature, but we prefer to call it here the Schwartz index. The Schwartz index for arbitrary stratified vector fields was first defined by H. King and D. Trotman in [96], and later independently in [6, 49, 149]. In [30, 31] this index was interpreted in differential-geometric terms and this was used to study its relations with various characteristic classes for singular varieties. This is discussed in [28] and in Chap. 10 below.