Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups
type: Item
instance of: scholarly article
title: Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups
zbMATH Open document ID: 0994.14004
author: Lars Ernström
author: Shoji Yokura
published in: Selecta Mathematica. New Series
publication date: 2002-05-29T00:00:00Z
review text: Deligne and Grothendieck have conjectured the existence of a unique natural transformation \(c_*\) from the constructible function functor to the homology functor such that the value \(c_*(1_X)\) of the characteristic function of a smooth complex algebraic variety \(X\) is the Poincaré dual of the total Chern cohomology class \(c(T_X)\cap [X]\). The conjecture has been affirmatively solved by \textit{R. MacPherson} [Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001)], see also \textit{G. Kennedy} [Commun. Algebra 18, No. 9, 2821-2839 (1990; Zbl 0709.14016)], \textit{M. Kwieciński} [``Sur le transformé de Nash et la construction du graphe de MacPherson'' (Thèse, Université de Provence 1994)] and \textit{C. Sabbah} [Astérisque 130, 161-192 (1985; Zbl 0598.32011)]. The distinguished value of the characteristic function of a variety under this transformation (called the Chern-Schwartz-MacPherson one) is isomorphic to the Schwartz class via the Alexander duality. A bivariant theory (BT) assigns to each morphism of complex algebraic varieties an abelian group. A BT is equipped with three commuting operations: product, proper push-forward and pull-back where product is associative, push-forward and pull-back are functorial, and there is a projection formula. A Grothendieck transformation between two BTs is a collection of homomorphisms preserving the above three operations. The notion of BT has been introduced by \textit{W. Fulton} and \textit{R. MacPherson} [Mem. Am. Math. Soc. 243, 165 p. (1981; Zbl 0467.55005)]. They have conjectured the existence of a Grothendieck transformation from the BT of constructible functions to the bivariant homology theory in the category of complex algebraic varieties which extends the original Chern-Schwartz-MacPherson transformation. The conjecture was solved by \textit{J.-P. Brasselet} [Astérisque 101-102, 7-22 (1981; Zbl 0529.55009)] for a certain reasonable category. The authors show that there exists a unique Grothendieck transfo…
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 1747326
zbMATH Keywords: algebraic variety
zbMATH Keywords: bivariant theory
zbMATH Keywords: intersection product
zbMATH Keywords: Chern-Schwartz-MacPherson class
zbMATH Keywords: specialization
zbMATH Keywords: Grothendieck transformation
zbMATH Keywords: characteristic function
zbMATH Keywords: Chow group
zbMATH Keywords: push-forward
zbMATH Keywords: pull-back
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: s00029-002-8100-8
OpenAlex ID: W1967063988
DOI: 10.1007/S00029-002-8100-8
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description: scientific article; zbMATH DE number 1747326