Equivariant characteristic classes of singular hypersurfaces

Abstract

In this paper, we introduce definitions for the integrated equivariant Milnor number μIG and the equivariant Milnor class ℳG(Z), for singular hypersurfaces. We prove that the μIG are constant on the strata in a Whitney stratification of Z, along with the correlation ℳG(Z) = ℳG,0(Z) = 1 |G|Σi=1kμ IG(x i) for hypersurfaces hosting isolated singularities x1,...,xk, where ℳG,0(Z) denotes the 0th equivariant Milnor class of Z. We also introduce the equivariant Fulton-Johnson class of singular hypersurfaces. We give an equivariant version of Verdier's specialization morphism in homology, and also for constructible functions. This is used for finding a relation between equivariant Fulton-Johnson and Schwartz-MacPherson classes.