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 AI , the regular nilpotent Hessenberg variety Hess(N, I), and the regular semisimple Hessenberg variety Hess(S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H∗(Hess(N, I)) and H∗(Hess(S, I)) , the invariants in H∗(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 ofW 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) → H∗(Hess(N, I)) announced by Dale Peterson and an affirmative answer to a conjecture of SommersTymoczko are immediate consequences. We also give an explicit ring presentation of H∗(Hess(N, I)) in types B, C, and G. Such a presentation was already known in type A or when Hess(N, I) is the Peterson variety. Moreover, we find the volume polynomial of Hess(N, I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess(N, I), despite the fact that it is a singular variety in general.