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, $c^{SM}(X)$ and $c^{FJ}(X)$. Their difference (up to sign) is the total Milnor class ${\mathcal M}(X)$, a generalization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes $c^{SM}(X)$ and $c^{FJ}(X)$, 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 $X_1,\cdot \ldots \cdot,X_r$ in a complex manifold, satisfying certain transversality conditions. As applications we obtain a Parusi\'{n}ski-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 L\^e classes of the $X_i$.