Chern-Schwartz-MacPherson classes in the point of view of Obstruction Theory and Lipschitz framework

TL;DR

This paper revisits the obstruction-theoretic approach to defining Chern-Schwartz-MacPherson classes, demonstrating a simplified construction within the Lipschitz framework that enhances understanding of singular varieties.

Contribution

It introduces a streamlined method for defining Chern classes of complex analytic varieties using obstruction theory in the Lipschitz setting, reviving and simplifying Schwartz's original ideas.

Findings

Simplified construction of Chern classes in Lipschitz framework

Reinterpretation of Schwartz's obstruction theory approach

Enhanced understanding of singular varieties' characteristic classes

Abstract

Since Chern and Grothendieck, Chern's characteristic class theory has made significant progress. In particular with regard to the classes of singular varieties. Conjectured by Grothendieck and Deligne and demonstrated by MacPherson, Chern classes of singular varieties have been defined in several ways, such as using polar varieties, Lagrangian theory... However, the initial definition using obstruction theory, due to Marie-H\'el\`ene Schwartz, has been forgotten. Despite the simple ideas that enabled the obstruction definition, their implementation using Whitney stratifications requires delicate and technical constructions. In the present article, we show that in the Lipschitz framework, the ideas of Marie-H\'el\`ene Schwartz lead to a simplified definition and construction of Chern classes of complex analytic varieties.