A formula for Segre classes of singular projective varieties

Abstract

The Segre class of a singular projective variety X is that of the normal cone of the diagonal in the product X x X. This class was introduced by K. W. Johnson and W. Fulton to study immersions and embeddings. In our previous work we related the Segre classes and the Chern-Mather classes for hypersurfaces with codimension one singularities and Xn c P2 with isolated singularities. In this paper we generalize these results to the case of Xn c FN with singularities of codimension N — n (N < In). The notion of Segre classes (of cones) has become of increasing importance as a key ingredient for constructing or analyzing various invariants, e.g., in intersection theory and group representation theory, etc. The Segre class treated in this note is the Segre class of a singular projective variety, which was introduced by K. W. Johnson (and W. Fulton) to study immersions and embeddings of singular projective varieties [4]. This is the Segre class of the normal cone C^{X x X) of the diagonal Δ in the product X x X. We call this class Johnson's Segre class, denoted by S*(X). Another well-studied characteristic class of a singular variety is MacPherson's Chern class, the existence of which was conjectured by Deligne and Grothendieck. R. MacPherson [7] constructed this Chern class, using Chern-Mather classes and introducing the notion of local Euler obstruction. A. Dubson [2] gave a more concrete description for MacPherson's Chern class C*(X) : Let S?χ be a (in fact, any) Whitney stratification of X with the smooth part of X as the top-dimensional stratum and let C^(X) denote the Chern-Mather class of X. Then ms •