Addendum to bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

Abstract

This note is an addendum to our previous paper [EY] and some unexplained notation and definitions in this addendum should be referred to [EY]. In [FM] W. Fulton and R. MacPherson introduced a new formalism called bivariant theories, which are simultaneous generalizations of covariant theories and contravariant theories. A bivariant theory assigns a group not to an object but to a morphism and it has three operations; product, pushforward and pullback. Grothendieck transformations are transformations, preserving these three operations, from one bivariant theory to another, which is a generalization of ordinary natural transformations. Our interests are the so-called Chern–Schwartz–MacPherson class [M] and its bivariant versions. The Chern–Schwartz–MacPherson class is the unique natural transformation from the covaraint functor F of constructible functions to the covariant homology functor H∗, satisfying a normalization condition. In [FM, §10] W. Fulton and R. MacPherson asked whether there exists a (unique) Grothendieck transformation from the bivariant version F of constructible function functor F to the bivariant version H of the homology theory H∗, satisfying a certain normaliztion condition (see [FM, §3.1 and §6.1] for F and H). Such a Grothendieck transformation is called a bivariant Chern–Schwartz–MacPherson class (or simply, a bivariant Chern class). In [B] J.-P. Brasselet showed the existence of a bivariant