Mather classes of Schubert varieties via small resolutions

TL;DR

This paper develops a new method to compute the Chern-Mather classes of Schubert varieties in orthogonal Grassmannians using small resolutions and integrals involving Pfaffians, offering an alternative to Nash blowups.

Contribution

It introduces a novel approach to express Chern-Mather classes via small resolutions and integrals, extending the method to orthogonal and Lagrangian Grassmannians.

Findings

Explicit formulas for Chern-Mather classes in orthogonal Grassmannians.

Application of equivariant localization for integral computation.

Proposals for Kazhdan-Lusztig classes in related Grassmannians.

Abstract

We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian and odd orthogonal Grassmannian.