Affine pavings of Hessenberg varieties for semisimple groups

TL;DR

This paper proves that certain Hessenberg varieties associated with regular elements in semisimple groups can be paved by affines, extending previous results and showing they have no odd-dimensional cohomology.

Contribution

It establishes affine pavings for Hessenberg varieties related to regular elements and reduces the problem for arbitrary elements to the nilpotent case.

Findings

Hessenberg varieties for regular nilpotent elements are paved by affines.

Hessenberg varieties for regular elements have no odd-dimensional cohomology.

Partial reduction from arbitrary elements to nilpotent elements in paving Hessenberg varieties.

Abstract

In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent and arbitrary elements of \mathfrak{gl}_n(\C) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases the Hessenberg variety has no odd dimensional cohomology.