Panoptic Study of Hyperplane arrangements

2016 Fiscal Year Final Research Report

• Project/Area Number: 24244001
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hokkaido University
• Principal Investigator: Terao Hiroaki 北海道大学, 理学(系)研究科(研究院), 名誉教授 (90119058)
• Co-Investigator(Kenkyū-buntansha): 吉永 正彦 北海道大学, 理学(系)研究科(研究院), 准教授 (90467647)
• Co-Investigator(Renkei-kenkyūsha): ABE Takuro 九州大学, マス・フォア・インダストリ研究所, 准教授 (50435971) ; KAMIYA Hidehiko 大阪経済大学, 経済学部, 教授 (50300687) ; KOHNO Toshitake 東京大学, 数理科学研究科, 教授 (80144111) ; TAKEMURA Akimichi 滋賀大学, データサイエンス教育研究センター, 教授 (10171670) ; TERAOSOMA Tomohide 東京大学, 大学院数理科学研究科, 教授 (50192654) ; MATSUMOTO Kohji 名古屋大学, 多元数理科学研究科, 教授 (60192754)
• Research Collaborator: Cuntz Michael ハノーファー大学, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, 教授 ; Hoge Torsten ルール大学ボーフム, 数学部門, 講師 ; NAKASHIMA Norihiro 東京電機大学, 情報環境学部, 助教 (90732115) ; Roehrle Gerhard ルール大学ボーフム, 数学部門, 教授 ; Schenck Hal イリノイ大学, 数学部門, 教授
• Project Period (FY): 2012-04-01 – 2017-03-31
• Keywords: 代数的組合せ論 / 超平面配置

Outline of Final Research Achievements

The following three essential themes were selected to grasp the vastness of the research of hyperplane arrangements. The obtained results are as follows:
(A) An explicit formula for a basis of the derivation module of the Shi arrangements coming from the Weyl arrangement of type A was obtained. In the formula, the Bernoulli polynomials naturally appear. We proved that any ideal subarrangement is free and that the height distribution and the exponents are dual to each other. (B) Hypergeometric integrals are regarded as a period map on the moduli space of arrangements of hyperplanes and can be described in terms of local system cohomology. In this context, a new method was obtained to compute the cohomology groups of the Aomoto complex. (C) The multiple addition theorem was proved. Consequently we learned that the class of hyperplane arrangements in which the so-called Terao conjecture holds true is quite large.

Free Research Field

超平面配置