Cohomology of the brieskorn-orlik-solomon algebras
Cohomology of the brieskorn-orlik-solomon algebras
复制标题
brieskorn-orlik-solomon 代数的上同调
DOI：
10.1080/00927879508825535
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发表时间：
1995
影响因子：
0.7
通讯作者：
S. Yuzvinsky
中科院分区：
文献类型：
--
作者：
S. Yuzvinsky
设V是域F上维数为'的仿射空间，设a = {H1, H2，…， Hn}是V的超平面的非空排列。对于每个H∈A，固定一个仿射泛函αH，使kerαH = H，令αi = αHi。本文的主要特点是由微分形式ωi = dαi/αi∈A1生成的分级F代数A = A(A) =⊕p=0Ap。如果F = C，那么根据Brieskorn的定理[2]，这个代数在到M = V \ \ ni=1Hi的上同构代数的de Rham映射下是同构的。Orlik和Solomon已经给出了这个代数的显式和纯粹的组合描述，详细描述在[7]的第3节。对于每个λ = (λ1)，…， λn)∈F n左乘dλ乘以ωλ =∑n i=1 λiωi定义了一个协链络合物（a, dλ） 0→A0 dλ→A1 dλ→··dλ→a '→0。本文的目的是研究该配合物的上同调H = H（A, dλ）。H的研究是由[4]和[5]驱动的。本文讨论了F = C时的H * (M，L)，其中L是M上的一个局部系统。在超几何函数和Knizhnik-Zamolodchikov方程的理论中应用了上同调。Kohno[5]证明了如果L是平凡束关于连接d+ωλ的平坦截面的局部系统，那么在λ上的一定泛型条件下，对于p < '， H(M，L) = 0。同样，如果A是实数并且在无穷远处横向于超平面，他发现了一个不依赖于λ的H（M，L）基。然后Esnault， Schechtman，和Viehweg[4]证明了在λ上较弱的泛型条件下
Let V be an affine space of dimension ` over a field F and let A = {H1, H2, . . . , Hn} be a non-empty arrangement of hyperplanes of V . For each H ∈ A fix an affine functional αH such that kerαH = H and put αi = αHi . The main character of the paper is the graded F -algebra A = A(A) = ⊕p=0Ap generated by the differential forms ωi = dαi/αi ∈ A1. If F = C then, according to Brieskorn’s theorem [2], this algebra is isomorphic under the de Rham map to the cohomology algebra of M = V \ ⋃ni=1Hi. Explicit and pure combinatorial description of this algebra has been given by Orlik and Solomon [6] and is presented in detail in Section 3 of [7]. For every λ = (λ1, . . . , λn) ∈ F n the left multiplication dλ by ωλ = ∑n i=1 λiωi defines a cochain complex (A, dλ) 0 → A0 dλ → A1 dλ → · · · dλ → A` → 0. The goal of this paper is to study the cohomology H = H(A, dλ) of this complex. The study of H is motivated by [4] and [5]. These papers are concerned with H∗(M,L) for F = C where L is a local system on M . The cohomology is used in theory of hypergeometric functions and Knizhnik-Zamolodchikov equations. Kohno [5] proved that if L is the local system of flat sections of the trivial bundle with respect to the connection d+ωλ, then under a certain genericity condition on λ, H(M,L) = 0 for p < `. Also if A is real and transverse to the hyperplane at infinity, he found a basis of H(M,L) that does not depend on λ. Then Esnault, Schechtman, and Viehweg [4] proved that under a weaker genericity condition on λ