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2023 Fiscal Year Final Research Report

Research on the Solomon-Terao complexes by using D-module theory

Research Project

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Project/Area Number 18KK0389
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionRikkyo University (2023)
Kyushu University (2018-2022)

Principal Investigator

ABE TAKURO  立教大学, 理学部, 教授 (50435971)

Project Period (FY) 2019 – 2023
Keywords超平面配置 / 対数的ベクトル場 / 自由配置 / Solomon-寺尾理論 / Liouville複体 / 完全交差性 / Cohen-Macaulay性 / Ziegler予想
Outline of Final Research Achievements

In this research, we studied the Solomon-Terao polynomial theory, which attracts many interests recently, from the viewpoint of the D-modules, in particular, that of so called the Liouville complex theory due to Uli Walther. On this approach, a joint work with Castro and Narvaez in Sevilla, we observed that the freenss of arrangements coincides with the Cohen-Macaulayness of the Liouville algebra, and a specializaion of the Liouville complex coincides with the Solomon-Terao complex. This shows that in a very high possibility, the Liouville complex theory could be regarded as a two-variable version of the Solomon-Terao complex theory. This enlarges the research of this area drastically. Also, in a joint work with Graham Denahm in Western university, we proved the Ziegler's conjecture on the logarithnmic differential forms. This was conjectured about 30 years before, and we investigated a theory to show it.

Free Research Field

超平面配置、代数学

Academic Significance and Societal Importance of the Research Achievements

本研究では、直線の有限集合の一般化である超平面配置の代数を幾何・表現論の視点から解析・一般化することを目指した。まずSolomon-寺尾理論について説明する。超平面配置の代数は超平面に接するベクトル場、流れのようなものの集合である対数的ベクトル場の研究である。この対数的ベクトル場と組み合わせ論及び幾何と繋ぐものがSolomon-寺尾理論であった。これは代数的な定義を持っているが、これに対して近年Walther氏により導入されたD加群的視点を持つLiouville複体理論を融合することで、Solomon-寺尾理論に新たな視点を導入することが、本研究では達成された。

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Published: 2025-01-30  

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