Motivic bivariant characteristic classes

The relative Grothendieck group $K_0(\text{m}V\mathrm{/}X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the “scissor relation”. The motivic Hirzebruch class ${T_y}*: K_0(\m V /X) \to H^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_([X \xrightarrow {\op {id}X} X])$oftheisomorphismclassoftheidentity$X \xrightarrow {id_X} X$ is the Poincar'e dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It “unifies” the well-known three characteristic classes of singular varieties: MacPherson’s Chern class, Baum-Fulton-MacPherson’s Todd class (or Riemann-Roch) and Goresky-MacPherson’s L-class or Cappell-Shaneson’s L-class. In this paper we construct a bivariant relative Grothendieck group ${\text{bK}}_0(\text{m}V\mathrm{/}X\to Y)$ so that it equals the original relative Grothendieck group $K_0(\text{m}V\mathrm{/}X)$ when $Y$ is a point. We also construct a unique Grothendieck transformation $T_y: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ[y]$ satisfying a certain normalization condition for a smooth morphism so that it equals the motivic Hirzebruch class ${T_y}: K_0(\m V /X) \to H_^{BM}(X) \otimes \bQ[y]$when$Y$isapoint\mathrm{.}When$y =0$,$T_0: \bK_0(\m V/X \to Y) \to \bH(X \to Y) \otimes \bQ$ is a “motivic” lift of Fulton-MacPherson’s bivariant Riemann-Roch $\ga_{td}^{\op {FM}}:\bK_{alg}(X \to Y) \to \bH(X \to Y) \otimes \bQ$.