On the uniqueness of bivariant Chern class and bivariant Riemann–Roch 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. However, its uniqueness has been unresolved since then. In this paper we show that restricted to morphisms whose target varieties are possibly singular but (rational) homology manifolds (such as orbifolds), the bivariant Chern classes (with rational coefficients) are uniquely determined. And also we discuss some related things and problems. In the final sections we construct a unique bivariant Chern class γ : F → H 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 the uniqueness questions concerning Grothendieck transformations posed by Fulton and MacPherson.

On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

“…This uniqueness theorem was further extended to a bit more general theorem in [6,28]; the target varieties of morphisms can be singular, but "homologically" still nonsingular in a sense. In this section, we give a quick review (without proof) of some results of our previous work [6,28].…”

“…[6]) Let Y be an algebraic variety over a base field k of characteristic zero, which is an Alexander variety in the sense of [19,36,37]. Then its fundamental class [Y ] ∈ A * (Y ) ⊗ Q is a strong orientation with respect to the operational bivariant Chow theory with rational coefficients.…”

“…[6]) Let R be a Noetherian ring, and let Y be a complex variety, which is an R-homology manifold (cf. [3]).…”

Characteristic classes of proalgebraic varieties and motivic measures

“…As shown by Brasselet, Schürmann and Yokura [12] (cf Brasselet, Schürmann and Yokura [13], Ernström and Yokura [29; 30], Schürmann [61] and Yokura [77]), a natural transformation between two covariant functors commuting with exterior products is always extended to a Grothendieck transformation between their associated bivariant theories. Therefore we get the following:…”

Higher bivariant Chow groups and motivic filtrations

“…The problem of cycle class maps has been considered more deeply and in greater generality by several authors, in particular, by Brasselet-Schürmann-Yokura [4,5], Ginzburg [8,9] and Yokura [12,13]. However, we shall not concern ourselves with those intricacies, since our construction in Section 3 is geared towards the following two main objectives of this paper:…”