2024/05/28 by Patrick Brosnan, Brosnan, Patrick, Laura Escobar +11 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Nonlinear Waves and Solitons #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2405.18313
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 ℂ and B is a Borel subgroup. We show that the space~H1(X,TX) of first order deformations of X has dimension r-1 except in type A2. (In type A2, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and \dimH1(X,TX) = 0.) Moreover, we show that the Kodaira--Spencer map \mathfrakg→ H1(X,TX) is onto, that the identity component of the automorphism group of X is a maximal torus of G, and that Hi(X,TX) = 0 for i ≥ 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~An-1 in the terms of stabilizer subgroups of the action of the symmetric group Sn on the moduli space M0,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 M0,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 Sn on M0,n.