A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes

TL;DR

This paper presents a general formula for the defect in the Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes in regular embeddings, extending previous results and providing new insights into Milnor-classes.

Contribution

It introduces a generalized formula for the defect in the Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes in regular embeddings, using constructible functions and specialization techniques.

Findings

Derived a formula for the Milnor-class of local complete intersections.

Reproduced known results for hypersurfaces as a special case.

Extended Riemann-Roch formulas to broader classes of embeddings.

Abstract

We give a general formula for the defect appearing in the Verdier-type Riemann-Roch formula for Chern-Schwartz-MacPherson classes in the case of a regular embedding. Our proof of this formula uses the constructible function version of Verdier's specialization functor, together with a specialization property of Chern-Schwartz-MacPherson classes and the corresponding Riemann-Roch theorem for smooth morphisms. As a very special case we get a formula for the Milnor-class of a local complete intersection in a smooth manifold, which in the case of a hypersurface gives back a result of Parusinski-Pragacz.