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Kobayashi-Hitchin correspondence and Donaldson-Tian-Yau conjecture on generalized complex geometry

Research Project

Project/Area Number 18H01120
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionOsaka University

Principal Investigator

Goto Ryushi  大阪大学, 大学院理学研究科, 教授 (30252571)

Co-Investigator(Kenkyū-buntansha) 満渕 俊樹  大阪大学, その他部局等, 名誉教授 (80116102)
大川 新之介  大阪大学, 大学院理学研究科, 准教授 (60646909)
石田 政司  大阪大学, 大学院理学研究科, 教授 (50349023)
榎 一郎  大阪大学, 大学院理学研究科, 准教授 (20146806)
Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥17,160,000 (Direct Cost: ¥13,200,000、Indirect Cost: ¥3,960,000)
Fiscal Year 2022: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2021: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Keywords一般化された複素構造 / 一般化されたケーラー構造 / 一般化された接触構造 / ポアソン構造 / 小林ーヒッチン対応 / スカラー曲率 / モーメント写像 / 多重安定性 / 一般化された佐々木構造 / 一般化されたケーラー幾何 / 一般化されたリッチ流 / ポアソン多様体 / 一般化されたケーラー幾何学 / 一般化されたケーラー多様体 / アインシュタイン計量 / 自己同型群 / 簡約群 / 松島・リヒネロビッツの定理 / 小林・ヒッチン対応 / アインシュタイン・エルミート計量 / 正則ベクトル束 / ポアソン モジュール / 安定ベクトル束 / アインシュタイン・ケーラー計量 / モーメントマップ
Outline of Final Research Achievements

Generalized complex geometry and generalized Kahler geometry are deeply related to various fields such as non-Kahler geometry (bi-Hermitian geometry), non-commutative algebraic geometry, geometric partial differential equations, real 4-dimensional differential topology, etc. Recent developments in this research area have received widespread attention. The main research results are the following three.
(1) We establish the Kobayashi-Hitchin correspondence for generalized holomorphic vector bundles on generalized Kahler manifolds. (2) We established the Matsushima-Lichnerovitz theorem in generalized Kahler geometry. (3) We constructed a generalized Sasaki structure over odd-dimensional semi-simplery groups.

Academic Significance and Societal Importance of the Research Achievements

一般化された複素構造および一般化されたケーラー構造は通常の複素構造,シンプレクティック構造を特別な場合として含む多様体の幾何構造である.ポアソン幾何,ノンケーラー幾何(双エル ミート幾何),非可換代数幾何, 幾何学的偏微分方程式,実4次元の微分トポロジーなど,様々な分野と深く関連しており,この研究分野の最近の大きな進展が注目されている.本研究の中心的なテーマは一般化されたケーラー多様体について,研究代表者が考案したモー メント写像を用い定義したスカラー曲率に用いて安定性との同値性を調べ研究するものである.その意味で,研究目的は明確であり,独創性は際立っている.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (27 results)

All 2023 2022 2021 2020 2018

All Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 7 results) Presentation (13 results) (of which Int'l Joint Research: 10 results,  Invited: 12 results) Funded Workshop (7 results)

  • [Journal Article] The Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type2023

    • Author(s)
      R. Goto
    • Journal Title

      International Mathematics Research Notices

      Volume: . Issue: 2 Pages: 1496-1567

    • DOI

      10.1093/imrn/rnad038

    • Related Report
      2022 Annual Research Report 2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kaehler manifolds of symplectic type2022

    • Author(s)
      Ryushi Goto
    • Journal Title

      Mathematische Annalen

      Volume: 384 Issue: 1-2 Pages: 1-51

    • DOI

      10.1007/s00208-021-02299-z

    • Related Report
      2022 Annual Research Report 2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type2022

    • Author(s)
      R. Goto
    • Journal Title

      Rivista di Matematica della Universia di Parma

      Volume: 13 Pages: 611-649

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type 5.発行年 2022年2022

    • Author(s)
      R. Goto
    • Journal Title

      Rivista di Matematica della Universia di Parma

      Volume: 13 Pages: 611-649

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type2021

    • Author(s)
      R.goto
    • Journal Title

      Mathematische Annalen

      Volume: .

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Scalar curvature as moment map in generalized Kahler geometry.2020

    • Author(s)
      Ryushi Goto
    • Journal Title

      J. Symplectic Geom.

      Volume: 18 Issue: 1 Pages: 147-190

    • DOI

      10.4310/jsg.2020.v18.n1.a4

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Scalar curvature as moment map in generalized K\"ahler geometry2020

    • Author(s)
      R. Goto
    • Journal Title

      The Journal of Symplectic Geometry

      Volume: 18

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] Generalized Kahler structures and generalized Sasakian structures2023

    • Author(s)
      R. Goto
    • Organizer
      Supergravity, Generalized Geometry and Ricci Flow, Simons Center, Stony Brook University
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generalized Kahler structures and generalized Sasakian structures2023

    • Author(s)
      R. Goto
    • Organizer
      Supergravity, Generalized Geometry and Ricci Flow, Simons Center, Stony Brook Universit
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generalized contact structures and generalized Sasakian structures2022

    • Author(s)
      R. Goto
    • Organizer
      Correspondences of various geometries September 30 - October 2 , 2022 Nara Women’s University
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generalized contact structures and generalized Sasakian structures2022

    • Author(s)
      R. Goto
    • Organizer
      Correspondences of various geometries September 30 - October 2 , 2022 Nara Women’s University(招待講演)(国際学会)
    • Related Report
      2021 Annual Research Report
  • [Presentation] Scalar curvature as moment map in generalized Kahler geometry2021

    • Author(s)
      R. Goto
    • Organizer
      Cohomology of Complex Manifolds and Special Structures, II Levico
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type2021

    • Author(s)
      R. Goto
    • Organizer
      PRCSG Pacific Rim Complex and Symplectic Geometry Conference( 7/12)
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type2021

    • Author(s)
      R. Goto
    • Organizer
      Workshop on Special Geometries on Riemannian Manifolds, Montreal (10/13)
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type2021

    • Author(s)
      R. Goto
    • Organizer
      Generalized connections and curvature.,One day workshop (11/15
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type2021

    • Author(s)
      R. Goto
    • Organizer
      Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 一般化されたケーラー幾何学2020

    • Author(s)
      Ryushi Goto
    • Organizer
      2020 日本数学会 秋季学会 企画特別講演(招待講演)
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Scalar curvature is moment map in generalized Kahler geometry.2018

    • Author(s)
      R. Goto
    • Organizer
      Cohomology of Complex Manifolds and Special Structures Levico Terme (Trento),
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Einstein-Hermitian vector bundles over generalized Kahler manifolds2018

    • Author(s)
      R. Goto
    • Organizer
      The 24th Symposium on Complex Geometry , Kanazawa
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Einstein-Hermitian vector bundles over generalized Kahler manifolds2018

    • Author(s)
      R. Goto
    • Organizer
      第4回日中幾何学研究集会
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] 第27回複素幾何研究集会2021

    • Related Report
      2020 Annual Research Report
  • [Funded Workshop] 第6回日中幾何学研究集会2021

    • Related Report
      2020 Annual Research Report
  • [Funded Workshop] The 6th Workshop "Complex Geometry and Lie Groups" 15-20 February 20212021

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] 第26回複素幾何シンポジウム2020

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Complex Geometry and Lie groups at Firentze, Italy2018

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] 第4回日中幾何学研究集会 合肥 中国2018

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] 第24回複素幾何シンポジウム 金沢2018

    • Related Report
      2018 Annual Research Report

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Published: 2018-04-23   Modified: 2024-01-30  

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