The Virtual Classes

Abstract

The constructions described in the previous chapter, mostly based on [31, 33, 139], provide geometric insights of the Schwartz–MacPherson classes via obstruction theory and localization. These approaches are useful for understanding what the classes measure from the viewpoint of indices of vector fields and frames. The Fulton–Johnson classes [59,60] provide another way of generalizing the Chern class of complex manifolds to the case of singular varieties. In the context we consider, they coincide with the virtual classes (see Sect. 11.1).

In this chapter we define and study the virtual classes from a viewpoint similar to the one we used in the previous chapter for the SchwartzMacPherson classes. This is based on our articles [31, 34], joint work with D. Lehmann.

If the variety V is globally defined by a function on M, the virtual classes can be localized topologically and one can interpret them as “weighted” Schwartz classes. That is explained in Sect. 11.3 where we prove the Proportionality theorem of [34] for this index. This theorem is analogous to, and inspired by, the similar theorem of [33] for the Schwartz index, proved in the previous chapter.

In Sect. 11.4, the localization of virtual classes is performed using the differential geometric method of [31], i.e., Chern–Weil theory, and using stratified frames. In that context, we construct localized “Fulton–Johnson classes” at the singular set of the given frames. While Sects. 11.1–11.3 are of a local nature, Sect. 11.4 is global.