Hessenberg varieties and hyperplane arrangements

Given a semisimple complex linear algebraic group {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement {\mathcal{A}_{I}}, the regular nilpotent Hessenberg variety {\operatorname{Hess}(N,I)}, and the regular semisimple Hessenberg variety {\operatorname{Hess}(S,I)}. We show that a certain graded ring derived from the logarithmic derivation module of {\mathcal{A}_{I}} is isomorphic to {H^{*}(\operatorname{Hess}(N,I))} and {H^{*}(\operatorname{Hess}(S,I))^{W}}, the invariants in {H^{*}(\operatorname{Hess}(S,I))} under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety {G/B}.This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map {H^{*}(G/B)\to H^{*}(\operatorname{Hess}(N,I))} announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for {\operatorname{Hess}(N,I)}, despite the fact that it is a singular variety in general.

Manifolds of Isospectral Matrices and Hessenberg Varieties

We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.