Chern classes and transversality for singular spaces

TL;DR

This paper compares various notions of transversality for singular complex spaces and proves a refined intersection formula for their Chern-Schwartz-MacPherson classes, extending known results to broader contexts.

Contribution

It introduces a new Verdier-Riemann-Roch theorem for non-characteristic pullbacks and establishes a refined intersection formula for singular spaces.

Findings

Refined intersection formula for Chern-Schwartz-MacPherson classes.

Comparison of transversality notions for singular spaces.

Proof of a new Verdier-Riemann-Roch theorem.

Abstract

In this paper we compare different notions of transversality for possible singular complex algebraic or analytic subsets of an ambient complex manifold and prove a refined intersection formula for their Chern-Schwartz-MacPherson classes. In case of a transversal intersection of complex Whitney stratified sets, this result is well known. For splayed subsets it was conjectured (and proven in some cases) by Aluffi and Faber. Both notions are stronger than a micro-local "non-characteristic intersection" condition for the characteristic cycles of (associated) constructible functions, which nevertheless is enough to imply the asked refined intersection formula for the Chern-Schwartz-MacPherson classes. The proof is based the multiplicativity of Chern-Schwartz-MacPherson classes with respect to cross products, as well as a new Verdier-Riemann-Roch theorem for "non-characteristic pullbacks".