Characteristic Classes of Hypersurfaces and Characteristic Cycles

TL;DR

This paper introduces a new formula for the Chern-Schwartz-MacPherson class of hypersurfaces in nonsingular compact complex varieties, generalizing previous results on Euler characteristics through two distinct approaches.

Contribution

It provides a novel formula for characteristic classes of hypersurfaces, utilizing characteristic cycles and Verdier's specialization, with simplified proofs and new related formulas.

Findings

New formula for Chern-Schwartz-MacPherson class of hypersurfaces

Two approaches: characteristic cycle theory and Verdier's specialization

Simplified proof of Aluffi's formula and additional related formulas

Abstract

We give a new formula for the Chern-Schwartz-MacPherson class of a hypersurface in a nonsigular compact complex analytic variety. In particular this formula generalizes our previous result on the Euler characteristic of such a hypersurface. Two different approaches are presented. The first is based on the theory of characteristic cycle and the works of Sabbah, Briancon-Maisonobe-Merle, and Le-Mebkhout. In particular, this approach leads to a simple proof of a formula of Aluffi for the above mentioned class. The second approach uses Verdier's specialization property of the Chern-Schwartz-MacPherson classes. Some related new formulas are also given.