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Variational approach to Kobayashi-Hitchin correspondence and Higgs bundles by means of Quot-schemes

Research Project

Project/Area Number 19K14524
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionOsaka Metropolitan University (2022-2023)
Tokyo Institute of Technology (2019-2021)

Principal Investigator

Hashimoto Yoshinori  大阪公立大学, 大学院理学研究科, 准教授 (60836485)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords小林-Hitchin対応 / 正則ベクトル束の安定性 / Kaehler多様体上の標準計量 / Hermite-Einstein計量 / 平坦線束 / Hoermander評価 / 正則ベクトル束
Outline of Research at the Start

解析学的な非線形偏微分方程式の解であるエルミート・アインシュタイン計量の存在と代数的な安定性が同値であることが知られており,小林-ヒッチン対応と呼ばれている.解析と代数という,本質的に異なる分野から定まる2つの概念が同値であるという驚くべき定理だが,証明は高度に技術的であることが知られていた.本研究の目標は,これら2つの概念がどのように関係し合うのかを,変分法という解析的手法とQuotスキームという代数的な物体を用いて,幾何学的直観に訴える形で理解することである.

Outline of Final Research Achievements

This project concerned the Kobayashi-Hitchin correspondence for holomorphic vector bundles over a smooth complex projective variety. In place of the previous method using the theory of nonlinear partial differential equations, I analysed the asymptotic behaviour of an energy functional called the Donaldson functional, and employed the Quot-schemes in algebraic geometry, to clarify the relationship between differential and algebraic geometry from a variational point of view. I also worked on many related topics.

Academic Significance and Societal Importance of the Research Achievements

小林-Hitchin対応は,Hermite-Einstein計量と呼ばれる非線形偏微分方程式の解の存在が代数幾何学的な安定性条件と同値であることを主張する非常に重要な定理である.本研究では,このような定理がなぜ成立するのかを,変分法の観点からより幾何学的直感に訴える形で理解するために重要な結果を示した.特に,Fubini-Study計量のQuotスキーム極限を定義することにより,極限Donaldson汎関数の漸近挙動を記述する代数的不変量を求めることに成功した.

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (44 results)

All 2024 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (6 results) Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (32 results) (of which Int'l Joint Research: 15 results,  Invited: 27 results)

  • [Int'l Joint Research] University of Michigan(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Universite du Quebec a Montreal(カナダ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Universite du Quebec a Montreal(カナダ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Tongji University(中国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Universite du Quebec a Montreal(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Universite du Quebec a Montreal(カナダ)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Balanced Metrics for Extremal Kaehler Metrics and Fano Manifolds2024

    • Author(s)
      Yoshinori Hashimoto
    • Journal Title

      In: Hirachi, K., Ohsawa, T., Takayama, S., Kamimoto, J. (eds) The Bergman Kernel and Related Topics. HSSCV 2022. Springer Proceedings in Mathematics & Statistics

      Volume: 447 Pages: 169-188

    • DOI

      10.1007/978-981-99-9506-6_5

    • ISBN
      9789819995059, 9789819995066
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Anticanonically balanced metrics and the Hilbert--Mumford criterion for the $$\delta _m$$-invariant of Fujita--Odaka2023

    • Author(s)
      Yoshinori Hashimoto
    • Journal Title

      Annals of Global Analysis and Geometry

      Volume: 64 Issue: 2 Pages: 1-40

    • DOI

      10.1007/s10455-023-09911-2

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Quot-Scheme Limit of Fubini--Study Metrics and Its Applications to Balanced Metrics2023

    • Author(s)
      Yoshinori Hashimoto and Julien Keller
    • Journal Title

      In: Cheltsov, I., Chen, X., Katzarkov, L., Park, J. (eds) Birational Geometry, Kähler–Einstein Metrics and Degenerations. Springer Proceedings in Mathematics & Statistics

      Volume: 409 Pages: 281-312

    • DOI

      10.1007/978-3-031-17859-7_14

    • ISBN
      9783031178580, 9783031178597
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Expected Centre of Mass of the Random Kodaira Embedding2022

    • Author(s)
      Yoshinori Hashimoto
    • Journal Title

      The Journal of Geometric Analysis

      Volume: 32 Issue: 4

    • DOI

      10.1007/s12220-021-00778-y

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles2021

    • Author(s)
      Yoshinori Hashimoto, Julien Keller
    • Journal Title

      Epijournal de Geometrie Algebrique

      Volume: 5

    • DOI

      10.46298/epiga.2022.6577

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] 小林-Hitchin対応への変分法的アプローチに対するQuotスキームの応用2020

    • Author(s)
      橋本義規
    • Journal Title

      第67回 幾何学シンポジウム 予稿集

      Volume: I Pages: 145-154

    • Related Report
      2020 Research-status Report
  • [Presentation] 非自明平坦線束に対する一様Hoermander評価2023

    • Author(s)
      橋本義規
    • Organizer
      大阪大学幾何セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Uniform Hoermander estimates for flat nontrivial line bundles2023

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      UQAM Geometry and Topology Seminar
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Uniform Hoermander estimates for flat holomorphic line bundles2023

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 7th Workshop "Complex Geometry and Lie Groups"
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Uniform Hoermander estimates for flat holomorphic line bundles2023

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Pacific Rim Complex and Symplectic Geometry Conference 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Recent developments on constant scalar curvature Kaehler metrics with cone singularities along a divisor2023

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 8th China-Japan Geometry Conference
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 因子に沿って錐的特異点をもつ定スカラー曲率Kahler計量についての最近の進展2023

    • Author(s)
      橋本義規
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Uniform Hormander estimates for flat nontrivial line bundles2023

    • Author(s)
      橋本義規
    • Organizer
      OCAMI微分幾何セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] Uniform Hormander estimates for flat holomorphic line bundles2023

    • Author(s)
      橋本義規
    • Organizer
      1day workshop on dynamical systems and complex geometry
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Coupled Ding stability and related topics2022

    • Author(s)
      橋本義規
    • Organizer
      The 28th Symposium on Complex Geometry
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 相対満渕汎関数についての簡単な観察2022

    • Author(s)
      橋本義規
    • Organizer
      専門家向け勉強会「ケーラー多様体上の標準計量とその周辺 3」
    • Related Report
      2022 Research-status Report
  • [Presentation] 因子に沿って錐的特異点をもつ定スカラー曲率Kaehler計量について2022

    • Author(s)
      橋本義規
    • Organizer
      OCAMI複素解析セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] 標準計量,幾何学的不変式論,ベルグマン核2022

    • Author(s)
      橋本義規
    • Organizer
      大阪公立大学数学研究所談話会,
    • Related Report
      2022 Research-status Report
  • [Presentation] Some recent results on constant scalar curvature Kaehler metrics with cone singularities2022

    • Author(s)
      橋本義規
    • Organizer
      CCG workshop "Complex Analytic Geometry"
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Donaldson's quantisation: extremal Kahler metrics and Fano manifolds2022

    • Author(s)
      橋本義規
    • Organizer
      HAYAMA Symposium on Complex Analysis in Several Variables XXIII
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Anticanonically balanced metrics and the Hilbert-Mumford criterion2022

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Minicourse at Yau Mathematical Sciences Center, Tsinghua University
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Expected centre of mass of the random Kodaira embedding2021

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 27th Symposium on Complex Geometry
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Anticanonically balanced metrics and the Hilbert-Mumford criterion for the delta_m invariant of Fujita-Odaka2021

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 6th China-Japan Geometry Conference
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 藤田-尾高のdelta_m不変量のHilbert-Mumford型基準とanticanonically balanced計量2021

    • Author(s)
      橋本義規
    • Organizer
      阪大オンライン代数幾何学セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Expected centre of mass of the random Kodaira embedding2021

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Seminaire Analyse, IRMA, Universite de Strasbourg
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ランダム小平埋め込みの質量中心の期待値2021

    • Author(s)
      橋本義規
    • Organizer
      日本数学会 2021 年度年会(一般公演)
    • Related Report
      2020 Research-status Report
  • [Presentation] 小林-Hitchin対応への変分法的アプローチに対するQuotスキームの応用2020

    • Author(s)
      橋本義規
    • Organizer
      第67回 幾何学シンポジウム(基調講演)
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Stability2020

    • Author(s)
      橋本義規
    • Organizer
      Maths Seminar at Shanghai
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Applications of the Quot-scheme limit to variational aspects of the Hermitian-Einstein metric2020

    • Author(s)
      橋本義規
    • Organizer
      複素解析幾何セミナー(東大数理)
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Expected centre of mass of the random Kodaira embedding2020

    • Author(s)
      橋本義規
    • Organizer
      Tokyo Tech - Uppsala University 7th Joint Symposium
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ランダム小平埋め込みの質量中心の期待値2020

    • Author(s)
      橋本義規
    • Organizer
      幾何学セミナー(九州大学)
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Variational aspects of the Hermitian-Einstein metrics and the Quot-Scheme limit2020

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Seminar at Xiamen University
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Introduction to Geometric Invariant Theory2020

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 4th CAGWS Complex Algebraic Geometry Winter School
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Variational aspects of the Kobayashi-Hitchin correspondence and the Quot-Scheme limit2019

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      The 3rd Symposium in Geometry and Differential Equations
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Variational aspects of the Kobayashi-Hitchin correspondence and the Quot-Scheme limit2019

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Trends in Modern Geometry 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Variational aspects of the Kobayashi-Hitchin correspondence and the Quot-scheme limit2019

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Younger generations in Algebraic and Complex geometry VI
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] ariational aspects of the Hermitian-Einstein metrics and the Quot-Scheme limit2019

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Seminar at Tongji University
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Canonical metrics, Geometric Invariant Theory, and the Bergman kernel2019

    • Author(s)
      Yoshinori Hashimoto
    • Organizer
      Colloquium at ShanghaiTech University
    • Related Report
      2019 Research-status Report
    • Invited

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Published: 2019-04-18   Modified: 2025-01-30  

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