Hessenberg varieties and hyperplane arrangements

Abstract

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

45 pages. Version 2, a reference is added, and a few sentenses are corrected

References in corpus (8)

• The freeness of ideal subarrangements of Weyl arrangements
• Multiderivations of Coxeter arrangements
• Note on cohomology rings of spherical varieties and volume polynomial
• The equivariant cohomology rings of Peterson varieties in all Lie types
• Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
• The equivariant cohomology rings of Peterson varieties
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• Arrangements of ideal type

Cited by in corpus (10)

• Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
• The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$
• Geometry of regular Hessenberg varieties
• A Saito criterion for holonomic divisors
• Orbit spaces of torus actions on Hessenberg varieties
• Solomon-Terao algebra of hyperplane arrangements
• Singular string polytopes and functorial resolutions from Newton-Okounkov bodies
• Hessenberg varieties, Slodowy slices, and integrable systems
• The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials