Automorphisms and deformations of regular semisimple Hessenberg varieties

TL;DR

This paper studies the deformation theory and automorphism groups of regular semisimple Hessenberg varieties, revealing their moduli, automorphism structures, and cohomological properties, with explicit results in type A and connections to moduli of curves.

Contribution

It provides a detailed analysis of the deformation spaces, automorphism groups, and moduli stacks of regular semisimple Hessenberg varieties, including explicit classifications in type A.

Findings

Deformation space dimension is r-1 in most types, zero in type A2.

Automorphism group contains a maximal torus, and higher cohomology vanishes.

Explicit automorphism groups and moduli descriptions in type A.

Abstract

We show that regular semisimple Hessenberg varieties can have moduli. To be precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension $1$ in the flag variety $G/B$, where $G$ is a simple algebraic group of rank $r$ over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space~$\mathrm{H}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$ except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and $\dim\mathrm{H}^1(X,TX) = 0$.) Moreover, we show that the Kodaira--Spencer map $\mathfrak{g}\to \mathrm{H}^1(X,TX)$ is onto, that the identity component of the automorphism group of $X$ is a maximal torus of $G$, and that $\mathrm{H}^i(X,TX) = 0$ for $i \geq 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G/B$. In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G/B$ are isomorphic. We also compute the automorphism groups explicitly in type~$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric group $S_{n}$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with $n + 1$ marked points. Using this, we describe the moduli stack of the regular semisimple Hessenberg varieties $X$ explicitly as a quotient stack of $M_{0,n+1}$. We prove several analogous results for Hessenberg varieties in generalized flag varieties $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G/B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$.