AAG: Deformations and automorphisms of Hessenberg varieties

Abstract: Hessenberg varieties are certain subvarieties of flag varieties defined in terms of two pieces of data: a combinatorial parameter, known either as a Hessenberg function or a Hessenberg space, and an element of a Lie algebra, which is just a matrix in the most basic an fundamental case.  The Hessenberg varieties associated to regular semisimple matrices (or, more generally, regular semisimple elements of a Lie algebra) are remarkable in that they are smooth and their cohomology groups have a direct group-theoretic description, which only depends on the combinatorial defining data. In fact, the varieties themselves are cellular. However, moving the matrix around gives a monodromy action on the cohomology, which has deep combinatorial significance.  (This connection to combinatorics was noticed by Shareshian and Wachs, and has been exploited extensively since then.)

For a long time, it wasn't known whether or not moving the matrix actually deforms the Hessenberg variety as an algebraic variety. I'll report on joint work with Escobar, Hong, Lee, Lee, Mellit and Sommers showing that, in general, it does. In fact, for one particular piece of combinatorial data, we are able to completely determine deformations and the automorphisms of all regular semisimple Hessenberg varieties.  Phrased another way, we are able to completely determine the moduli stack.

Host: Matt Kerr