Hessenberg varieties and hyperplane arrangements

Abstract

Abstract Given a semisimple complex linear algebraic group G {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement 𝒜 I {\mathcal{A}_{I}} , the regular nilpotent Hessenberg variety Hess ⁡ ( N , I ) {\operatorname{Hess}(N,I)} , and the regular semisimple Hessenberg variety Hess ⁡ ( S , I ) {\operatorname{Hess}(S,I)} . We show that a certain graded ring derived from the logarithmic derivation module of 𝒜 I {\mathcal{A}_{I}} is isomorphic to H * ⁢ ( Hess ⁡ ( N , I ) ) {H^{*}(\operatorname{Hess}(N,I))} and H * ⁢ ( Hess ⁡ ( S , I ) ) W {H^{*}(\operatorname{Hess}(S,I))^{W}} , the invariants in H * ⁢ ( Hess ⁡ ( S , I ) ) {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 {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 ) ) {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 * ⁢ ( Hess ⁡ ( N , I ) ) {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when Hess ⁡ ( N , I ) {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of Hess ⁡ ( N , I ) {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for Hess ⁡ ( N , I ) {\operatorname{Hess}(N,I)} , despite the fact that it is a singular variety in general.