Motivic bivariant characteristic classes

Abstract

Let K 0 (V/X) be the relative Grothendieck group of varieties over X ∈ Obj(V), with V = V (qp) k (resp. V = V an c ) 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 * : K 0 (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) = H BM 2 * (X)). It "unifies" the wellknown 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 groupin 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 = Λ mot y :with K alg (f ) the bivariant algebraic Ktheory of f -perfect complexes and H the bivariant operational Chow groups (or the even degree bivariant homology in case k = C). Evaluating at y = 0, we get a "motivic" lift T 0 of Fulton-MacPherson's bivariant Riemann-Roch transformation τ : K alg → H ⊗ Q. The covariant transformations mCy : K 0 (V qp /X → pt) → G 0 (X) ⊗ Z[y] and Ty * : K 0 (V qp /X → pt) → H * (X) ⊗ Q[y] agree for k of characteristic zero with our motivic Chern-and Hirzebruch class transformations defined on K 0 (V qp /X). Finally, evaluating at y = −1, for k of characteristic zero we get a "motivic" lift T −1 of Ernström-Yokura's bivariant Chern class transformation γ :F → CH.

Motivic Bivariant Characteristic Classes and Related Topics

“…where u f := td(T f ) ∈ H * (X) ⊗ Q and T f is the (virtual) tangent bundle of f . See [20, (*) on p.124] for H the bivariant homology in case k = C. For H = CH the bivariant Chow group and k of any characteristic, the above Riemann-Roch formula follows from [19,Theorem 18.2] as explained in [38]. The Riemann-Roch formula implies the following two results: SGA 6-Riemann-Roch Theorem: The following diagram commutes for a proper smooth morphism f :…”

“…As the "confined" and "specialized" maps we take the class Prop of proper and Sm of smooth morphisms, respectively. Theorem 3.1 ( [44], [38]). We define…”

“…The pre-motivic bivariant Grothendieck group M(V/−) has the following universal property: [38]). Let B be a bivariant theory on V such that a smooth morphism f has a nice canonical orientation θ(f ) ∈ B(f ), and let c : V ect(−) → B * (−) be a contravariantly functorial characteristic class of algebraic (or analytic) vector bundles with values in the associated cohomology theory, which is multiplicative in the sense that c (V ) = c (V )c (V ) for any short exact sequence of vector bundles 0 →…”

“…In [38] we obtain in the quasi-projective context (over a base field k of any characteristic) two bivariant analogues…”

“…In this paper we make a survey on the above results [38] as well as on a corresponding universal "oriented" bivariant theory [44], which is a first step on the way to a bivariant-theoretic analogue of Levine-Morel's or Levine-Pandharipande's algebraic cobordism [28,29]. Finally we switch to a differential topological context of smooth manifolds and make a remark on a new geometric bivariant bordism theory based on the notion of a "fiberwise bordism" ( [3], [45]).…”