Geometry of regular Hessenberg varieties

TL;DR

This paper investigates the geometric and cohomological properties of regular Hessenberg varieties in complex semisimple Lie algebras, establishing vanishing of higher cohomology and reducedness of fibers, with applications.

Contribution

It proves vanishing of higher cohomology for irreducible regular Hessenberg varieties and shows fibers in a flat family are reduced, advancing understanding of their geometric structure.

Findings

Higher cohomology groups of the structure sheaf vanish for irreducible regular Hessenberg varieties.

The scheme-theoretic fibers over closed points in the flat family are reduced.

Applications demonstrate the relevance of these geometric properties.

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. For a regular element $x$ in $\mathfrak{g}$ and a Hessenberg space $H\subseteq \mathfrak{g}$, we consider a regular Hessenberg variety $X(x,H)$ in the flag variety associated with $\mathfrak{g}$. We take a Hessenberg space so that $X(x,H)$ is irreducible, and show that the higher cohomology groups of the structure sheaf of $X(x,H)$ vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.