Development of toric topology

• 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
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• 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 / 旗多様体 / トーラス軌道 / シューベルト多様体 / 置換群 / トーリック幾何 / コホモロジー剛性問題 / Hessenberg variety / 双曲多様体 / トーリックトポロジー / 組合せ論

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.

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。

Research Products

• [Int'l Joint Research] Ajou University/KAIST/IBS(韓国)
• [Int'l Joint Research] マクマスター大学(カナダ)
• [Int'l Joint Research] Moscow State University/Steklov Institute, Moscow(ロシア連邦)
• [Int'l Joint Research] 南京大学(中国)
• [Journal Article] THE COHOMOLOGY RINGS OF REGULAR SEMISIMPLE HESSENBERG VARIETIES FOR h = (h(1),n,...,n) (2019)
  • Author(s): H. Abe, T. Horiguchi, M. Masuda
  • Journal Title: J. Comb. Volume: 17 Issue: 01 Pages: 1-24
  • DOI: 10.4310/joc.2019.v10.n1.a2
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2016856
  • Peer Reviewed
• [Journal Article] THE COHOMOLOGY RINGS OF REGULAR NILPOTENT HESSENBERG VARIETIES IN LIE TYPE A (2018)
  • Author(s): Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda
  • Journal Title: International Mathematics Research Notices Volume: 16 Issue: 05 Pages: 1-52
  • DOI: 10.1093/imrn/rnx275
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2016845
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] CLASSIFICATION OF REAL BOTT MANIFOLDS AND ACYCLIC DIGRAPHS (2017)
  • Author(s): Suyoung Choi, Mikiya Masuda, Sang-il Oum
  • Journal Title: Transaction of American Mathematical Society Volume: 10 Issue: 06 Pages: 1-25
  • DOI: 10.1090/tran/6896
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2016736
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Torsion in the cohomology of torus orbifolds (2017)
  • Author(s): Hideya Kuwata, Mikiya Masuda, Haozhi Zeng
  • Journal Title: Chinese Annals of Mathematics Volume: 未定
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Cohomological rigidity of manifolds defined by right-angled 3-dimensional polytopes (2017)
  • Author(s): V. Buchstaber, N. Erokhovets, M. Masuda, T. Panov, S. Park
  • Journal Title: Uspekhi Mat. Nauk Volume: 72 Issue: 2(434) Pages: 3-66
  • DOI: 10.4213/rm9759
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Volume polynomials and duality algebras of multi-fans (2016)
  • Author(s): Anton Ayzenberg, Mikiya Masuda
  • Journal Title: Arnold Mathematical Journal Volume: 15 Issue: 11 Pages: 1-45
  • DOI: 10.1007/s40598-016-0048-4
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2016838
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Generic torus orbit closures in Schubert varieties (2018)
  • Author(s): Mikiya Masuda
  • Organizer: Algebraic Topology, Combinatorics, and Mathematical Physics
  • Int'l Joint Research / Invited
• [Presentation] Cohomological rigidity problems in toric topology (2018)
  • Author(s): Mikiya Masuda
  • Organizer: Glances@Manifolds III
  • Int'l Joint Research / Invited
• [Presentation] The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\ldpts,n)$ (2018)
  • Author(s): Mikiya Masuda
  • Organizer: Hessenberg Varieties in Combinatorics, Geometry and Representation Theory, Banff
  • Int'l Joint Research / Invited
• [Presentation] Generic torus orbit closures in Schubert varieties (2018)
  • Author(s): Mikiya Masuda
  • Organizer: Hyperplane arrangement
  • Int'l Joint Research / Invited
• [Presentation] Cohomology of Hessenberg varieties (2018)
  • Author(s): Mikiya Masuda
  • Organizer: IBS workshop
  • Int'l Joint Research / Invited
• [Presentation] Volume polynomials of regular nilpotent Hessenberg varieties (2017)
  • Author(s): Mikiya Masuda
  • Organizer: Princeton-Rider Workshop On the Homotopy Theory of Polyhedral Products
  • Int'l Joint Research / Invited
• [Presentation] Cohomology of regular Hessenberg varieties (2016)
  • Author(s): Mikiya Masuda
  • Organizer: Glances@Manifolds 2
  • Place of Presentation: クラコフ（ポーランド）
  • Year and Date: 2016-08-09
  • Int'l Joint Research / Invited