Remarks on bivariant constructible functions
Jean‐Paul Brasselet, Shoji Yokura · Advanced studies in pure mathematics · 2018
IntroductionThe so-called Chern-Schwartz-MacPherson class (or transformation) is the unique natural transformation from the covariant functor of constructible functions to the integral homology covariant functor, satisfying a certain normalization condition (see [14], and also [3], [10].[20].)The bivariant theory has been introduced by W.Fulton and R.MacPherson [9], and they conjectured (or posed as a question) the existence of a Grothendieck transformation from the bivariant theory of constructible functions to the bivariant homology theory in the category of complex algebraic varieties, which specializes to the original Chern-Schwartz-MacPherson transformation.The conjecture has been solved by Brasselet for a certain reasonable category [2] (see also [19] and [24]).In this paper we report some consequences of this Brasselet's theorem, concerning bivariant constructible functions (i.e., constructible functions satisfying the local Euler condition) and some related results and we also pose some problems. §1. Constructible functions and Chern-Schwartz-MacPherson classesA constructible set of an analytic variety X is obtained from analytic subvarieties of X by a finite number of unions, intersections and complements.A constructible function on a compact complex analytic