Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups
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 transformation from the BT of constructible functions to a certain operational bivariant Chow theory. For a constructible function \(\alpha\) on a variety \(X\), bivariant with respect to a morphism \(f:X\rightarrow Y\), they get a collection \(\gamma (\alpha)\) (which is an element of a bivariant Chow group) \((\gamma (\alpha))(g):A(Y')\rightarrow A(X\times _YY')\), one for each morphism \(g:Y'\rightarrow Y\), which are compatible with proper push-forwards and intersection products. Compatibility with flat pull-back is not required.
scientific article; zbMATH DE number 1747326