Hessenberg varieties and hyperplane arrangements

被引:22
作者
Abe, Takuro [1 ]
Horiguchi, Tatsuya [2 ]
Masuda, Mikiya [3 ]
Murai, Satoshi [4 ]
Sato, Takashi [5 ]
机构
[1] Kyushu Univ, Inst Math Ind, Fukuoka 8190395, Japan
[2] Osaka Univ, Dept Pure & Appl Math, Grad Sch Informat Sci & Technol, 1-5 Yamadaoka, Suita, Osaka 5650871, Japan
[3] Osaka City Univ, Dept Math, Sumiyoshi Ku, Osaka 5588585, Japan
[4] Waseda Univ, Sch Educ, Dept Math, Shinjuku Ku, Nishi Waseda 1-6-1, Tokyo 1698050, Japan
[5] Osaka City Univ, Sumiyoshi Ku, Adv Math Inst, 3-3-138 Sugimoto, Osaka 5588585, Japan
来源
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2020年 / 764卷
关键词
EQUIVARIANT COHOMOLOGY RINGS; GEOMETRY; COMBINATORICS; EXPONENTS; TOPOLOGY;
D O I
10.1515/crelle-2018-0039
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a semisimple complex linear algebraic group 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. 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 Iiodge-Riemann relations hold for Hess(N, I), despite the fact that it is a singular variety in general.
引用
收藏
页码:241 / 286
页数:46
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