On the Chern classes of singular complete intersections

TL;DR

This paper develops new formulas for Chern classes of singular complete intersections, linking Milnor classes to Lê classes, and extends classical Riemann-Roch results to singular varieties.

Contribution

It introduces simple formulas for Milnor classes of singular complete intersections using Verdier-Riemann-Roch type theorems, connecting local invariants to global classes.

Findings

Derived Verdier-Riemann-Roch formulas for singular Chern classes.

Established a simple formula for Milnor classes under transversality conditions.

Linked Milnor classes to global Lê classes of the defining varieties.

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 $X_i$.