2016/11/01 by Abe, Takuro, Horiguchi, Tatsuya, Masuda, Mikiya +2 · 3 citations
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1611.00269
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))W, 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 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)→ H^*(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^*(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.