Product formula for twisted MacPherson classes

Abstract

Introduction. The topological Euler characteristic Z is multiplicative, i.e., z(X x Y) z(X)z(Y). For a manifold X, a generalization of z(X) to higher dimensional cohomology classes is the Chern cohomology class c*(X), which satisfies the cross-product formula c*(X x Y)= c*(X) x c*(Y). For a (possibly singular) compact complex algebraic variety X, a generalization of z(X) to higher dimensional homology classes is the Schwartz-MacPherson homology class c,(X), which in the smooth case is just the Poincar dual of the usual Chern cohomology class c*(X) and the 0-th component of which is equal to Z (X)[1, 4, 5]. Very recently, in connec- tion with lifting Schwartz-MacPherson classes to intersection homology [2], the first author [3] proved the product formula for Schwartz-MacPherson classes, i.e., c,(X x Y) c,(X) x c,(Y). The second author [6, 71 defined the "twisted" MacPherson class ct,(X), which includes Schwartz- MacPherson class c,(X) as a special case, i.e., cl,(X) c,(X). The 0-th component of ct,(X) is the "stratified weighted" Euler characteristic Z.t(X), which is a degree-dimX polynomial of t, involves Euler characteristic of sigularities also and equals to z(X) when t 1. In [7] the second author showed the multiplicativity of Z t, i.e., xt(X x Y)= zt(X)zt(Y). In this note, by strengthening and modifying the proof of [3] we show the product formula ct,(X x Y) ct,(X) x ct,(Y), thus the product formulae c,(X x Y) c,(X) x c,(Y) and zt(X x Y) .t(X).t(Y) follow as special cases. More generally we show a product formula for the transformation ct, acting on constructible functions with polynomial coefficients (Theorem 4).