On the Chern classes of singular complete intersections

Abstract

We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz–MacPherson and Fulton–Johnson classes, cSM(X) and cFJ(X). Their difference (up to sign) is the total Milnor class M(X), a gener-alization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann–Roch type formulae for the total classes cSM(X) and cFJ(X), and use these to prove a surprisingly simple formula for the total Milnor class when X is defined by a finite number of local complete intersection X1,.....,Xr in a complex manifold, satisfying certain transversality conditions. As applications, we obtain a Parusiński–Pragacz type formula and an Aluffi type formula for the Milnor class, and a description of the Milnor classes of X in terms of the global Lê classes of the Xi.