Motivic Bivariant Characteristic Classes
Abstract
Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasiprojective) 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 wellknown three characteristic class transformations of singular varieties: MacPherson's Chern class, BaumFultonMacPherson's Todd class and the Lclass of GoreskyMacPherson and CappellShaneson. 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 FultonMacPherson's bivariant RiemannRoch 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 ErnströmYokura's bivariant Chern class transformation.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.2166
 Bibcode:
 2011arXiv1110.2166S
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology
 EPrint:
 This is an improved version with more general results of the paper arXiv:1005.1124 of Shoji Yokura with the same title. Version 2 is the final version to appear in Advances in Mathematics. More comments on the relation to the complex cobordism of Quillen and the algebraic cobordism of LevineMorel added