Motivic bivariant characteristic classes

Let K0(V/X) be the relative Grothendieck group of varieties over X∈Obj(V), with V=Vk(qp) (resp. V=Vcan) the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation Ty⁎:K0(V/X)→H⁎(X)⊗Q[y] in the algebraic context for k of characteristic zero, with H⁎(X)=CH⁎(X) (resp. in the complex algebraic or analytic context, with H⁎(X)=H2⁎BM(X)). It “unifies” the well-known three characteristic class transformations of singular varieties: MacPhersonʼs Chern class, Baum-Fulton-MacPhersonʼs Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a bivariant relative Grothendieck group K0(V/X→Y) for V=Vk(qp) (resp., Vcan) so that K0(V/X→pt)=K0(V/X) in the algebraic context with k of characteristic zero (resp., complex analytic context).We also construct in the algebraic context (in any characteristic) two Grothendieck transformations mCy=Λymot:K0(Vqp/X→Y)→Kalg(X→Y)⊗Z[y] and Ty:K0(Vqp/X→Y)→H(X→Y)⊗Q[y] with Kalg(f) the bivariant algebraic K-theory of f-perfect complexes and H the bivariant operational Chow groups (or the even degree bivariant homology in the case k=C). Evaluating at y=0, we get a “motivic” lift T0 of Fulton-MacPhersonʼs bivariant Riemann-Roch transformation τ:Kalg→H⊗Q. The covariant transformations mCy:K0(Vqp/X→pt)→G0(X)⊗Z[y] and Ty⁎:K0(Vqp/X→pt)→H⁎(X)⊗Q[y] agree for k of characteristic zero with our motivic Chern and Hirzebruch class transformations defined on K0(Vqp/X). Finally, evaluating at y=−1, for k of characteristic zero we get a “motivic” lift T−1 of Ernström-Yokuraʼs bivariant Chern class transformation γ:F˜→CH.