The Schwartz Classes

Abstract

As mentioned before, the first generalization of Chern classes to singular varieties is due to M.-H. Schwartz, using obstruction theory and radial frames. These classes are the primary obstructions to constructing a special type of stratified frames on V that she called radial frames. To avoid possible misunderstandings, here we prefer to call them frames constructed by radial extension, as in the case of vector fields. We refer to [28, 33] for details of the construction and we content ourselves with summarizing here their main properties. It was shown in [33] that these classes correspond, by Alexander isomorphism, to the MacPherson classes, that we discuss briefly in the last section of this chapter.

In this chapter we provide a viewpoint for studying Schwartz–MacPherson classes which is particularly close to the theory of indices of vector fields that we develop in this book, both from the topological and the differential geometric sides. In the first three sections, we discuss the Schwartz index of frames and a method for defining the Schwartz classes of singular varieties using arbitrary stratified frames, not necessarily constructed by radial extension. As a corollary we obtain that the Schwartz classes are the primary obstruction to constructing a stratified frame (any frame, not necessarily radial) on the skeleton of the appropriate dimension and for an appropriate cellular decomposition: if such a frame exists, then the corresponding Schwartz class vanishes (the converse is false in general).

In Sect. 4, we use the methods of [31], joint work with D. Lehmann, for constructing localized Schwartz classes in both the topological and differential geometric contexts, via Chern–Weil theory and using stratified frames. The last section discusses briefly MacPherson and Mather classes (see [28, 117]).