Motivic Bivariant Characteristic Classes

Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation in the algebraic context for k of characteristic zero and in the compact complex analytic context. It unifies the well-known three characteristic class transformations of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a bivariant relative Grothendieck group K_0(V/-) and in the algebraic context (in any characteristic) two Grothendieck transformations mC_y resp. T_y defined on K_0(V/-). Evaluating at y=0, we get a motivic lift T_0 of Fulton-MacPherson's bivariant Riemann-Roch transformation. The associated covariant transformations agree for k of characteristic zero with our motivic Chern- and Hirzebruch class transformations defined on K_0(V/X). Finally, evaluating at y=-1, we get for k of characteristic zero, a motivic lift T_{-1} of Ernstr\"om-Yokura's bivariant Chern class transformation.