A note of characteristic class for singular varieties

TL;DR

This paper explores characteristic classes of singular varieties, expressing the Schwartz-MacPherson class via $$-class and Chern classes, and applies this to compute Euler characteristics of complements.

Contribution

It introduces a new expression for the Schwartz-MacPherson class using $$-class and differential forms, advancing understanding of singular varieties' characteristic classes.

Findings

Expressed Schwartz-MacPherson class in terms of $$-class and Chern classes

Derived a formula for Euler characteristic of complements of singular varieties

Provided tools for studying characteristic classes in singular algebraic geometry

Abstract

In this work we study characteristic classes of possibly singular varieties embedded as a closed subvariety of a nonsingular variety. In special, we express the Schwartz-MacPherson class in terms of the $\mu$-class and Chern class of the sheaves of logarithmic and multi-logarithmic differential forms. As an application we show an expression for Euler characteristic of a complement of a singular variety.