Mather classes and conormal spaces of Schubert varieties in cominuscule spaces

TL;DR

This paper provides a type-independent formula for the Mather class of Schubert varieties in complex cominuscule flag manifolds, with applications to local Euler obstructions and conormal spaces, and explores positivity and unimodality conjectures.

Contribution

It introduces a novel, type-independent formula for the equivariant Mather class of Schubert varieties in cominuscule spaces, advancing understanding of their geometric invariants.

Findings

Derived explicit formulas for local Euler obstructions.

Computed torus equivariant localizations of conormal spaces.

Formulated and tested conjectures on positivity and unimodality of related polynomials.

Abstract

Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results of Boe and Fu about the characteristic cycles of the intersection homology sheaves of Schubert varieties. We also conjecture that certain `Mather polynomials' are unimodal in general Lie type, and log concave in type A.