On Ginzburg’s bivariant Chern classes

Abstract

The convolution product is an important tool in geometric representation theory. Ginzburg constructed the “bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a “constructible function version" of one of Ginzburg’s results; motivated by its proof, we introduce another bivariant algebraic homology theorysAHs\mathbb {AH}on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theoryGF\mathbb {GF}of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation fromGF\mathbb {GF}tosAHs\mathbb {AH}satisfying the “normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.