An application of bivariant theory to Milnor classes

Abstract

The notion of the Milnor number of an isolated singularity of a hypersurface has been generalized to the so-called “Milnor class” in such a way that the degree of the zero-dimensional component of the Milnor class is nothing but the Parusiński generalized Milnor number. The Milnor class of a local complete intersection in a smooth compact complex analytic manifold is defined, up to sign, by the difference of the Chern–Schwartz–MacPherson class and the virtual class or the Fulton–Chern class. In this paper, for certain reasonable morphisms we relate the Milnor classes of the target variety and the source variety via certain classes of the morphism, in particular, using the bivariant theory due to Fulton and MacPherson.