Hodge rigidity of Chern classes

TL;DR

This paper investigates the structure of Chern--Schwartz--MacPherson classes of Schubert cells, showing that under certain conditions, each component corresponds to an irreducible subvariety, extending previous results.

Contribution

It extends Huh's results by relaxing regularity assumptions, proving irreducibility of components for a broader class of Schubert cells and varieties.

Findings

Each homogeneous component is represented by an irreducible subvariety.

Results apply to all cominuscule Schubert cells of classical type.

Analogous results are obtained for certain symplectic Grassmannians and flag varieties.

Abstract

In this paper, we study the homogeneous components of the Chern--Schwartz--MacPherson (CSM) classes of Schubert cells. We prove that, under suitable conditions, each such component is represented by an irreducible subvariety. In particular, our result extends Huh's result \cite{Huh} by relaxing the regularity assumption on log resolutions. As a consequence, the conclusion holds for all cominuscule Schubert cells of classical type and for a large family of exceptional cases. We also obtain analogous results for certain Schubert varieties in symplectic Grassmannians and flag varieties.