### 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}, 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 A_{I} 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 {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 and 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 and 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.

Original language | English |
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Journal | Journal fur die Reine und Angewandte Mathematik |

DOIs | |

Publication status | Accepted/In press - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal fur die Reine und Angewandte Mathematik*. https://doi.org/10.1515/crelle-2018-0039

**Hessenberg varieties and hyperplane arrangements.** / Abe, Takuro; Horiguchi, Tatsuya; Masuda, Mikiya; Murai, Satoshi; Sato, Takashi.

Research output: Contribution to journal › Article

*Journal fur die Reine und Angewandte Mathematik*. https://doi.org/10.1515/crelle-2018-0039

}

TY - JOUR

T1 - Hessenberg varieties and hyperplane arrangements

AU - Abe, Takuro

AU - Horiguchi, Tatsuya

AU - Masuda, Mikiya

AU - Murai, Satoshi

AU - Sato, Takashi

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 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 {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 and 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 and 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 {{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 {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 and 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 and 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.

UR - http://www.scopus.com/inward/record.url?scp=85060703145&partnerID=8YFLogxK

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U2 - 10.1515/crelle-2018-0039

DO - 10.1515/crelle-2018-0039

M3 - Article

AN - SCOPUS:85060703145

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

ER -