Hessenberg varieties and hyperplane arrangements
Hessenberg varieties and hyperplane arrangements
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DOI：
10.1515/crelle-2018-0039
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发表时间：
2016-11
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影响因子：
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通讯作者：
T. Abe;T. Horiguchi;M. Masuda;S. Murai;Takashi Sato
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文献类型：
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作者：
T. Abe;T. Horiguchi;M. Masuda;S. Murai;Takashi Sato
摘要给定一个半单复线性代数群G {{G}}和G的正根上的一个下理想I，得到了理想排列I {\mathcal{a}_{I}}、正则幂零Hessenberg变Hess (N，I) {\operatorname{Hess}(N，I)}和正则半单Hessenberg变Hess (S，I) {\operatorname{Hess}(S，I)}。我们表明,一定分次环来源于对数𝒜派生模块我{\ mathcal{一}_{我}}同构H *(赫斯(N,我)){H ^ {*} (\ operatorname{赫斯}(我)N,)}和H *(赫斯(S I)) W H ^ {{*} (\ operatorname{赫斯}(S I)) ^ {W}},的不变量H *(赫斯(S I)) {H ^ {*} (\ operatorname{赫斯}(S I))}外尔集团的下一个动作W (g .一般说谎类型,显示了这种同构推广了Borel的著名定理，证明了W的协不变代数同构于标志簇G/B {G/B}的上同构环。海森伯格变分和超平面排列之间的这种令人惊讶的联系使我们能够得出许多有趣的结果。例如，Dale Peterson提出的限制映射H * * (G/B)→H * * (Hess (N，I)) {H^{*}(G/B)\到H^{*}(\operatorname{Hess}(N，I))}的满射性以及对Sommers和Tymoczko猜想的肯定回答都是直接结果。我们也给出了H * * (Hess (N，I)) {H^{*}(\operatorname{Hess}(N，I))}在类型B， C和g中的显式环表示。当Hess (N，I) {\operatorname{Hess}(N，I)}是Peterson变元时，这种表示在类型a中已经已知。此外，我们找到了Hess (N，I) {\operatorname{Hess}(N，I)}的体积多项式，并看到了Hess (N，I) {\operatorname{Hess}(N，I)}的硬Lefschetz性质和Hodge-Riemann关系，尽管它通常是一个奇异变量。
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 {\mathcal{A}_{I}} , the regular nilpotent Hessenberg variety Hess ( N , I ) {\operatorname{Hess}(N,I)} , and the regular semisimple Hessenberg variety Hess ( S , I ) {\operatorname{Hess}(S,I)} . We show that a certain graded ring derived from the logarithmic derivation module of 𝒜 I {\mathcal{A}_{I}} is isomorphic to H * ( Hess ( N , I ) ) {H^{*}(\operatorname{Hess}(N,I))} and H * ( Hess ( S , I ) ) W {H^{*}(\operatorname{Hess}(S,I))^{W}} , the invariants in H * ( Hess ( S , I ) ) {H^{*}(\operatorname{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 ) ) {H^{*}(G/B)\to H^{*}(\operatorname{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 ) ) {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when Hess ( N , I ) {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of Hess ( N , I ) {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for Hess ( N , I ) {\operatorname{Hess}(N,I)} , despite the fact that it is a singular variety in general.