Notes on the cohomology of partial Hessenberg varieties

TL;DR

This paper investigates the cohomology of partial Hessenberg varieties by relating them to the well-studied cohomology of Hessenberg varieties in full flag varieties, revealing new connections to hyperplane arrangements and combinatorial conjectures.

Contribution

It provides a novel approach to understanding the cohomology of partial Hessenberg varieties through their relation to full flag Hessenberg varieties.

Findings

Established a relationship between the cohomology of partial and full flag Hessenberg varieties.

Uncovered connections to hyperplane arrangements.

Linked Hessenberg varieties to the Stanley-Stembridge conjecture.

Abstract

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.