On a naturality of Chern-Mather classes

Abstract

Introduction. The Chern-Schwartz-MacPher- son class (or more precisely natural transformation) is the unique natural transformation from the co- variant constructible function functor to the covari- ant integral homology functor satisfying the normal- ization that the value of the characteristic function of a nonsingular compact complex analytic variety is equal to the Poincare dual of the total Chern cohomology class of the tangent bundle. The ex- istence of such a transformation was conjectured by Deligne and Grothendieck and was proved by MacPherson (10). The novelity of MacPherson's proof is introducing the notion of local Euler obstruc- tion (which was independently introduced by Kashi- wara (7) also) and assigning the Chern-Mather class to this local Euler obstruction, not to the character- istic function. Although the Chern-Mather class is a very geometrically simple homology class, the assign- ment of the Chern-Mather class to the characteristic function does not give such a natural transformation. It is often said that few functorial proper- ties are know for the Chern-Mather class (e.g., see (6, Note, page 377, right after Example 19.1.7)), al- though the assignment of the Chern-Mather class to the local Euler obstruction is perfectly natural, which is the main part of MacPherson's proof. In this paper, using this fine naturality of this assingment, we interpret the Chern-Mather class in the same way as the Chern-Schwartz-MacPherson class. Further- more, by introducing the notion of a q-deformed local Euler obstruction which unifies the charac- teristic function and the local Euler obstruction, we give a q-deformed Chern-Schwartz-MacPherson class natural transformation, which specializes to the Chern-Mather class natural transformation for q = 0 and the Chern-Schwartz-MacPherson class natural transformation for q = 1.