On Ginzburg’s bivariant Chern classes

Yokura, Shoji

doi:10.1090/S0002-9947-03-03252-5

On Ginzburg’s bivariant Chern classes
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Trans. Amer. Math. Soc. 355 (2003), 2501-2521 Request permission

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 theory $s\mathbb {AH}$ 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 theory $\mathbb {GF}$ of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from $\mathbb {GF}$ to $s\mathbb {AH}$ satisfying the “normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.

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