  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.

Algebraic Geometry;Algebraic Topology