Singular Chern Classes of Schubert Varieties via Small Resolution

TL;DR

This paper presents a method to compute Chern-Schwartz-MacPherson classes of Schubert varieties in Grassmannians using small resolutions, enabling new calculations of Chern-Mather classes, local Euler obstructions, and proving cases of a positivity conjecture.

Contribution

It introduces a novel approach using small resolutions for calculating characteristic classes of Schubert varieties, improving upon previous methods.

Findings

Computed CSM classes for new Schubert varieties

Derived formulas for Chern-Mather classes and Euler obstructions

Proved new cases of the positivity conjecture

Abstract

We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.