Development of toric topology

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05152
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka City University
• Principal Investigator: Masuda Mikiya 大阪市立大学, 大学院理学研究科, 教授 (00143371)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: トーリック多様体 / ヘッセンバーグ多様体 / 凸多面体 / トーラス群作用 / 同変コホモロジー

Outline of Final Research Achievements

I studied geometry and topology of torus actions and related combinatorics. In particular I have obtained the following results.
(1) Convex polytopes which can be realized as right angle polytopes in the 3-dimensional hyperbolic space are called Pogorelov polytopes and small covers over them become hyperbolic 3-manifolds. We have shown that they can be distinguished by their Z/2-cohomology rings. (2) We have shown that the cohomology rings of regular nilpotent Hessenberg varieties are isomorphic to rings obtained from logarithmic derivations on hyperplane arrangements and extended known results on the cohomology rings for type A to any Lie type. (3) We have classified toric manifolds over an n-cube with one vertex cut.

Free Research Field

トポロジー、組合せ論、代数幾何

Academic Significance and Societal Importance of the Research Achievements

（１）３次元双曲多様体は、Mostow剛性により基本群で区別できるが、Loebell type の双曲多様体に限れば、基本群よりずっと弱い不変量であるZ/2係数コホモロジー環で区別できることを示したのは、特筆すべき結果と思う。（２）Regular nilpotent Hessenberg varietyのコホモロジー環と超平面配置を結びつけたのは驚く結果と思う。（３）分類結果の副産物として、射影的代数多様体の構造と非射影的代数多様体の構造の両方をもつ微分可能多様体が、トーリック多様体の範疇に存在することが分かったことは新しい知見である。