On a naturality of Chern-Mather classes
Let \(X\) denote a possibly singular algebraic variety and \({\mathcal F}(x)\) the group of constructible functions on \(X\). \textit{R. D. MacPherson} [Ann. Math. (2) 100, 423-432 (1974; Zbl 0311.14001)] showed the existence of a unique natural transformation \(C_*:{\mathcal F}\to H_*\) such that for \(X\) nonsingular \(C_*(\mathbf{1}_X)=c(Tx)\cap [X]\) where \(C(TX)\) is the total Chern class and \([X]\) the fundamental homology class of \(X\). \(C_*\) is known as the Chern-Schwartz-MacPherson class of \(X\). The novelty of MacPherson's proof is introducing the notion of local Euler obstruction and assigning the Chern-Mather class to this local Euler obstruction, not to the characteristic function. It is often said that few ``functorial'' properties are known for the Chern-Mather class, although 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. Furthermore, by introducing the notion of a ``\(q\)-deformed'' local Euler obstruction which unifies the characteristic 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 \(q=1\).
scientific article; zbMATH DE number 1443388
scientific article