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Study of harmonic bundles and related objects

Research Project

Project/Area Number 15K04843
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionKyoto University

Principal Investigator

Mochizuki Takuro  京都大学, 数理解析研究所, 教授 (10315971)

Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
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: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords調和束 / モノポール / ツイスターD加群 / Kontsevich複体 / 漸近挙動 / Dirac型特異点 / Kobayashi-Hitchin対応 / ホロノミックD加群 / Riemann-Hilbert対応 / 差分加群 / q-差分加群 / ワイルド調和束 / ヒッグス束 / ストークス構造 / フーリエ変換 / Kontsevich complex / 調和バンドル / 小林-Hitchin対応 / Riemann・Hilbert対応 / リーマン・ヒルベルト対応 / enhanced ind-sheaf / ディラク型特異点 / GKZ超幾何系 / Hitchin-WKB-問題 / asymptotic decoupling / limiting configuration
Outline of Final Research Achievements

We applied the knowledge and results in the study of harmonic bundles to solve new problems and to study new subjects. We studied the natural deformation family of
harmonic bundles on a compact Riemann surface. We particularly established the Hitchin-WKB problem, and determined the limiting configuration in the rank 2 case.
We also clarified the relation between the Kontsevich complexes and the mixed twistor D-modules associated to algebraic functions, which allows us to deduce many known results for Kontsevich complexes from a general theory of mixed twistor D-modules. Moreover, we started to study monopoles and difference modules. We established simple characterizations of Dirac type singularity of monopoles. We also established the Kobayashi-Hitchin correspondence for holomorphic bundles on non-compact Kahler manifolds with infinite volume.

Academic Significance and Societal Importance of the Research Achievements

調和束に関する以前の研究で得られていた知見を、新しい問題に適用することで興味深い進展が得られ、さらに以前の研究結果をより汎用性の高いものにすることができました。また、モノポールと差分加群の間の新しい対応を追求することで、Dirac型特異点の特徴付けや体積無限大のケーラー多様体上のKobayashi-Hitchin対応などの基礎的な意義を持つ成果が得られました。

Report

(6 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (42 results)

All 2020 2019 2018 2017 2016 2015

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Open Access: 4 results) Presentation (37 results) (of which Int'l Joint Research: 29 results,  Invited: 37 results) Book (1 results)

  • [Journal Article] Kobayashi-Hitchin correspondence for analytically stable bundles.2020

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 373 (no. 1) Issue: 1 Pages: 551-596

    • DOI

      10.1090/tran/7956

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A twistor approach to the Kontsevich complexes2017

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      manuscripta mathematica

      Volume: 2017 Issue: 1-2 Pages: 193-231

    • DOI

      10.1007/s00229-017-0989-5

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Some Characterizations of Dirac Type Singularity of Monopoles2017

    • Author(s)
      Mochizuki Takuro、Yoshino Masaki
    • Journal Title

      Communications in Mathematical Physics

      Volume: 356 Issue: 2 Pages: 613-625

    • DOI

      10.1007/s00220-017-2981-z

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2016

    • Author(s)
      望月 拓郎
    • Journal Title

      Journal of Topology

      Volume: 9 Issue: 4 Pages: 1021-1073

    • DOI

      10.1112/jtopol/jtw018

    • NAID

      120006365529

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Kobayashi-Hitchin correspondences for periodic monopoles2019

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Geometry and integrable systems
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kobayashi-Hitchin correspondences for monopoles with periodicity2019

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Algebraic analysis and geometry with a view on Higgs bundles and D-modules
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-abelian Hodge theory for Monopoles with Periodicity2019

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Resurgence in Mathematics and Physics
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some topics in Kobayashi-Hitchin correspondences2019

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Integrability, Geometry and Moduli
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 調和束・モノポール・インスタントン2019

    • Author(s)
      望月拓郎
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] The Riemann-Hilbert correspondence and Fourier transform2019

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Wild Ramification and Irregular Singularities
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Complex Geometry and Lie groups
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Toda equations and Harmonic bundles2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Higgs Bundles and Harmonic Maps of Riemann Surfaces
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Kobayashi-Hitchin correspondence for wild harmonic bundles2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Quantum Fields, Geometry and Representation Theory
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Geometry and Physics of Quantum Curves
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stokes shells and Fourier transform2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      CATS5
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stokes shells and Fourier transform2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Various Problems of Algebraic Analysis
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Recent developments in Higgs theory
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Kobayashi-Hitchin correspondences for periodic monopoles2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      A workshop on challenges at the interface of Hitchin Systems and String Theory
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Riemann-Hilbert correspondences
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 周期的モノポールと差分加群の間のKobayashi-Hitchin対応について2018

    • Author(s)
      望月拓郎
    • Organizer
      名古屋大学における多弦数理物理学セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Tata研究所におけるcolloquium
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Periodic monopoles and difference modules2018

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Complex Analytic Geometry 2018
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A Hukuhara-Levelt-Turrittin type theorem2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Complex Analygic Geometry
    • Place of Presentation
      タタ研究所(インド)
    • Year and Date
      2017-03-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mixed twistor D-modules and some examples2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      D-modules and Hodge theory
    • Place of Presentation
      東京大学Kavli IPMU
    • Year and Date
      2017-01-23
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Twistor D-modules and some examples2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Hodge Theory, Stokes Phenomenon and Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A Hukuhara-Levelt-Turrittin type theorem2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Developments of mathematics at IPMU: in honor of Kyoji Saito
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mixed twistor D-modules and some examples2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Higgs bundles and related topics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mixed twistor D-modules and some examples2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Algebraic Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Decomposition theorem for semisimple algebraic holonomic D-modules2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      プリンストン高等研究所におけるセミナー
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Wild harmonic bundles and related topics2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      プリンストン高等研究所における講義
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ヒッグス束や接続の小林-ヒッチン対応について2017

    • Author(s)
      望月拓郎
    • Organizer
      岡シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2017

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Landau-Ginzburg models and application
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Curve test for enhanced ind-sheaves and holonomic D-modules2016

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Algebraic Geometry and Integrable Systems, Kobe 2016
    • Place of Presentation
      神戸大学
    • Year and Date
      2016-12-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2016

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Workshop on New Perspectives on Moduli Spaces in Gauge Theory
    • Place of Presentation
      シンガポール国立大学(シンガポール)
    • Year and Date
      2016-08-03
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2016

    • Author(s)
      Takuro Mochizuki
    • Organizer
      離散群と双曲空間のトポロジーと解析
    • Place of Presentation
      京都大学
    • Year and Date
      2016-06-23
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2016

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Flat connections, Higgs bundles and Painleve equations
    • Place of Presentation
      国立台湾大学(台湾)
    • Year and Date
      2016-05-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2016

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Modern Interactions between Algebra, Geometry and Physics
    • Place of Presentation
      東北大学
    • Year and Date
      2016-04-20
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A twistor approach to Kontsevich complexes2015

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Mirror Symmetry and Algebraic Geometry 2015
    • Place of Presentation
      京都大学(京都)
    • Year and Date
      2015-12-07
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces2015

    • Author(s)
      Takuro Mochizuki
    • Organizer
      The 21th Symposium on Complex Geometry
    • Place of Presentation
      金沢大学サテライトプラザ3階集会室(金沢)
    • Year and Date
      2015-10-30
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Quantum $D$-modules and mixed twistor $D$-modules2015

    • Author(s)
      Takuro Mochizuki
    • Organizer
      ハイデルベルグ大学数学教室談話会
    • Place of Presentation
      ハイデルベルグ(ドイツ)
    • Year and Date
      2015-04-30
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Quantum $D$-modules and mixed twistor $D$-modules2015

    • Author(s)
      Takuro Mochizuki
    • Organizer
      国際研究集会``Mirror Symmetry, Hodge Theory and Differential Equations''
    • Place of Presentation
      オーバーヴォルヴァッファ(ドイツ)
    • Year and Date
      2015-04-21
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] Mixed Twistor D-modules2015

    • Author(s)
      Takuro Mochizuki
    • Publisher
      Springer-Verlag
    • Related Report
      2015 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2021-02-19  

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