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On Ginzburg's bivariant Chern classes

Author: Shoji Yokura
Journal: Trans. Amer. Math. Soc. 355 (2003), 2501-2521
MSC (2000): Primary 14C17, 14F99, 55N35
Published electronically: February 6, 2003
MathSciNet review: 1974000
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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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Additional Information

Shoji Yokura
Affiliation: Department of Mathematics and Computer Science, Faculty of Science, University of Kagoshima, 21-35 Korimoto 1-chome, Kagoshima 890-0065, Japan

Keywords: Bivariant theory; Chern-Schwartz-MacPherson class; Constructible function; Convolution
Received by editor(s): January 20, 2002
Published electronically: February 6, 2003
Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 12640081), the Japanese Ministry of Education, Science, Sports and Culture
Article copyright: © Copyright 2003 American Mathematical Society