Geometry of regular Hessenberg varieties

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.

arXiv comment26 pages

References in corpus (6)

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• Note on cohomology rings of spherical varieties and volume polynomial
• The equivariant cohomology rings of Peterson varieties in all Lie types
• Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
• Hessenberg varieties and hyperplane arrangements
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$

Cited by in corpus (4)

• A survey of recent developments on Hessenberg varieties
• Orbit spaces of torus actions on Hessenberg varieties
• An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties
• Hessenberg varieties, Slodowy slices, and integrable systems