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Complex geometric structures on homogeneous and locally homogeneous manifolds

Research Project

Project/Area Number 16K05123
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionOsaka University (2020-2021)
Niigata University (2016-2019)

Principal Investigator

HASEGAWA KEIZO  大阪大学, 理学研究科, 招へい教授 (00208480)

Co-Investigator(Kenkyū-buntansha) 神島 芳宣  城西大学, 理学部, 客員教授 (10125304)
塚田 和美  お茶の水女子大学, 名誉教授 (30163760)
守屋 克洋  筑波大学, 数理物質系, 助教 (50322011)
Project Period (FY) 2016-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords局所共形ケーラー構造 / Vaisman構造 / 佐々木構造 / ユニモジュラー・リー群 / 冪零リー群 / Heisenbergリー群 / CR構造 / 局所共形ケーラー多様体 / 佐々木多様体 / Vaismann多様体 / ユニモジュラー群 / ユニモジュラーリー群 / 等質ケーラー多様体 / 等質佐々木多様体 / 複素幾何学 / 等質多様体 / ケーラー構造 / 複素構造
Outline of Final Research Achievements

We have shown that a compact homogeneous locally conformally Kaehler manifold is a holomorphic fiber bundle over a complex flag manifold with fiber a 1-dimensional complex torus; and its structure is of Vaisman type, that is, the associated Lee form is parallel. Later, we partially extended the result, obtaining the structure theorem for Homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups. In particular, we have shown that a unimodular Vaisdman Lie group is, up to modification, R x sl(2), R x su(2) or R x n, where n is the Heisenberg Lie algebra. Later, we could have determined all possible modifications for these Lie algebras, thus obtaining a complex classification of unimodular Vaisman Lie groups. Similarly, we have seen that a unimodular Sasaki Lie algebra is sl(2), su(2), or a modification of n.

Academic Significance and Societal Importance of the Research Achievements

複素幾何学の分野において,ケーラー構造は最も基本的な幾何構造である。近年,非ケーラー複素多様体の発見に始まって,現在では非ケーラー幾何学としての研究分野も確立しつつあり,その重要性も認識されている。非ケーラー多様体の中でもケーラー構造に近いものとして,局所共形ケーラー構造を持つものがあり,複素曲面に限れば非ケーラー曲面の多くはこの局所共形ケーラー構造をもつことが知られている。本研究において,高次元の局所共形ケーラー多様体として,等質多様体を研究対象にし,特に,コンパクト群や冪零群を含むユニモジュラー・リー群の等質多様体について,その不変な局所共形ケーラー構造,Vaisman構造を決定した。

Report

(7 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (28 results)

All 2022 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (10 results) Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 6 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (8 results) (of which Int'l Joint Research: 4 results,  Invited: 6 results) Funded Workshop (3 results)

  • [Int'l Joint Research] University of Hamburg(ドイツ)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] ハンブルグ大学(ドイツ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ハンブルグ大学(ドイツ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ロシア・アカデミー(ロシア連邦)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ハンブルグ大学(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] ロシア・アカデミー(ロシア連邦)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] ハンブルグ大学(ドイツ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] ロシア・アカデミー(ロシア連邦)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] V. Cortes/University od Hamburg(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] D. V. Alekseevsky/Russian Academy of Sciences(ロシア連邦)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Polar varieties and bipoloar surfaces of minimal surfaces in the n-sphere2022

    • Author(s)
      Katsuhiro Moriya
    • Journal Title

      Annals of Global Analysis and Geometry

      Volume: 61 Pages: 21-36

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Gauss maps of transversally complex submanifolds of a quaternion projective spaces2021

    • Author(s)
      Kazumi Tsukada
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 73 Pages: 1-28

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lagrangian submanifolds of <i>S</i><sup>6</sup> and the associative Grassmann manifold2020

    • Author(s)
      Enoyoshi Kanako、Tsukada Kazumi
    • Journal Title

      Kodai Mathematical Journal

      Volume: 43 Issue: 1 Pages: 170-192

    • DOI

      10.2996/kmj/1584345693

    • NAID

      130007812085

    • ISSN
      0386-5991, 1881-5472
    • Year and Date
      2020-03-15
    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups2019

    • Author(s)
      D. Alekseevsky, K. Hasegawa and Y. Kamishima
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 8 Pages: 1-14

    • DOI

      10.1017/nmj.2019.34

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On quaternioic 3 CR-structure and pseudo-Riemannian metric2018

    • Author(s)
      Yoshinobu Kamishima
    • Journal Title

      Applied Mathematics

      Volume: 9 Pages: 114-129

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Transversally complex submanifolds of a quaternion projective space2017

    • Author(s)
      Kazumi Tsukada
    • Journal Title

      Springer Proceedings in Mathematics and Statistics

      Volume: 203 Pages: 223-233

    • Related Report
      2017 Research-status Report
  • [Journal Article] Compact Locally conformally Kaehler manifolds,2016

    • Author(s)
      K. Hasegawa and Y. Kamishima,
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 53 Pages: 683-703

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] Sasaki and CR Lie groups - construction and classification problems2021

    • Author(s)
      Keizo Hasegawa
    • Organizer
      Special Geometry, Mirror Symmetry and Integrable Systems (Waseda Univeristy)
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Unimodular Sssaki and CR Lie groups2020

    • Author(s)
      長谷川敬三
    • Organizer
      幾何学セミナー,大阪大学理学研究科
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Complete classification of unimodular Sasaki and vaisman Lie groups2019

    • Author(s)
      長谷川 敬三
    • Organizer
      第25回国際複素幾何学シンポジウム(金沢)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Locally homogeneous aspherical Sasaki manifolds2019

    • Author(s)
      神島芳宣
    • Organizer
      Taipei Conference on Geometric Invariance and Partial Differential Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] On locally homogeneous aspherical Kaehler manifolds2018

    • Author(s)
      神島芳宣
    • Organizer
      水戸幾何セミナー, 茨城大学理学部
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups2017

    • Author(s)
      Keizo Hasegawa
    • Organizer
      Differential Geometry Seminar, University of Hamburg
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Smooth rigidity of compact aspherical locally homogeneous manifolds and Application to Geometric structures2017

    • Author(s)
      Yoshinobu Kamishima
    • Organizer
      JNU-KAIST Geometric Topology Fair: 韓国
    • Related Report
      2017 Research-status Report
  • [Presentation] Locally conformally Kaehler homogeneous spaces of unimodular Lie groups,2016

    • Author(s)
      K. Hasegawa
    • Organizer
      The 8-th TIMS-OCAMI-WASEDA Joint International Workshop on Differential Geometry and Geometric Analysis
    • Place of Presentation
      早稲田大学,東京
    • Year and Date
      2016-12-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Complex Geometry and Lie Groups2021

    • Related Report
      2020 Research-status Report
  • [Funded Workshop] Complex Geometry and Lie Groups, Ferenze, Italy2018

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Quaternionic Differential Geometry and its related Topics2016

    • Place of Presentation
      御茶ノ水女子大学,東京
    • Year and Date
      2016-09-07
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2023-01-30  

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