TY - JOUR

T1 - Hessenberg varieties and hyperplane arrangements

AU - Abe, Takuro

AU - Horiguchi, Tatsuya

AU - Masuda, Mikiya

AU - Murai, Satoshi

AU - Sato, Takashi

N1 - Publisher Copyright:
© De Gruyter 2020.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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.

AB - 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.

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U2 - 10.1515/crelle-2018-0039

DO - 10.1515/crelle-2018-0039

M3 - Article

AN - SCOPUS:85060703145

SN - 0075-4102

VL - 2020

SP - 241

EP - 286

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 764

ER -