Motivic Bivariant Characteristic Classes

الأصل · EN

Let K₀(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 K₀(V/-) and in the algebraic context (in any characteristic) two Grothendieck transformations mCy resp. Ty defined on K₀(V/-). Evaluating at y=0, we get a motivic lift T₀ 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 K₀(V/X). Finally, evaluating at y=-1, we get for k of characteristic zero, a motivic lift T₋₁ of Ernström-Yokura's bivariant Chern class transformation.