Bivariant Chern classes and Grothendieck transformations

Abstract

The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a sense gives a positive answer to a corresponding question posed by Fulton and MacPherson. This construction generalizes and unifies previous attemps, in particular the main result of the paper math.AG/0209299.