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Study on D-modules with irregular singular points and geometric monodromies

Research Project

Project/Area Number 17H02848
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTohoku University (2020-2022)
University of Tsukuba (2017-2019)

Principal Investigator

Takeuchi Kiyoshi  東北大学, 理学研究科, 教授 (70281160)

Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥17,420,000 (Direct Cost: ¥13,400,000、Indirect Cost: ¥4,020,000)
Fiscal Year 2021: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2020: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2019: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2018: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2017: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
KeywordsD加群 / 不確定特異点 / 特異点理論 / 有理型関数 / フーリエ変換 / Bernstein-佐藤多項式 / モノドロミー / D-加群 / 超幾何関数 / 代数解析
Outline of Final Research Achievements

We clarified the basic properties of the Fourier transforms of irregular holonomic D-modules. In particular, we described explicitly their singular sets and showed that the irregularities (exponential factors) along them can be described by the characteristic varieties and the exponential factors of the original holonomic D-modules. These results largely extend those obtained by Brylinski about 30 years ago. In the course of our research, we studied also the singularities of meromorphic functions at infinity and at their points of indeterminacy. As a byproduct, we found also Bernstein-Sato polynomials for meromorphic functions and studied their basic properties.

Academic Significance and Societal Importance of the Research Achievements

望月とKedlayaの理論により、近年不確定特異点を持つホロノミーD加群の理論は劇的な
進展を遂げた。我々の研究は、それをホロノミーD加群のフーリエ変換の基本的な性質の解明という、代数解析学における長年の未解決問題に応用したものである。またこの研究の過程において、有理型関数の無限遠点や不確定点における特異性について基礎的な研究を行った。これらは非常に自然な数学的対象であるが、なぜかこれまで研究されなかったものであり、今後自然科学の多方面に応用が期待できる。

Report

(6 results)
  • 2022 Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • Research Products

    (32 results)

All 2023 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (10 results) Journal Article (12 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 12 results) Presentation (8 results) (of which Int'l Joint Research: 7 results,  Invited: 8 results) Book (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] HSE University(ロシア連邦)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] University of Nice(フランス)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] VAST(ベトナム)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] Higher school of economics(ロシア連邦)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Nice university(フランス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] VAST(ベトナム)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] ニース大学(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Higher school of economics(ロシア連邦)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] ニース大学(フランス)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Higher school of economics(ロシア連邦)

    • Related Report
      2017 Annual Research Report
  • [Journal Article] On the monodromy conjecture for non-degenerate hypersurfaces2023

    • Author(s)
      A. Esterov, A. Lemahieu and K. Takeuchi
    • Journal Title

      J. of European Mathematical Society

      Volume: 24 Pages: 3873-3949

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Meromorphic nearby cycle functors and monodromies of meromorphic functions (with Appendix by T. Saito)2023

    • Author(s)
      T.T. Nguyen and K. Takeuchi
    • Journal Title

      Revista Matematica Complutense

      Volume: -

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the monodromies and the limit mixed Hodge structures of families of algebraic varieties2023

    • Author(s)
      T. Saito and K. Takeuchi
    • Journal Title

      Michigan Math. Journal

      Volume: -

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a Bernstein-Sato polynomial of a meromorphic function2023

    • Author(s)
      K. Takeuchi
    • Journal Title

      Nagoya Math. Journal

      Volume: -

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the monodromies and the limit mixed Hodge structures of families of algebraic varieties2022

    • Author(s)
      T. Saito and K. Takeuchi
    • Journal Title

      Michigan Math. J.

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the monodromy conjecture for non-degenerate hypersurfaces2022

    • Author(s)
      A. Esterov, A. Lemahieu and K. Takeuchi
    • Journal Title

      J. of European Mathematical Society

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The bifurcation set of a rational function via Newton polytopes2021

    • Author(s)
      T.T. Nguyen, T. Saito and K. Takeuchi
    • Journal Title

      Math. Zeitschrift

      Volume: 298 Pages: 899-916

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Bifurcation values of polynomial functions and perverse sheaves2020

    • Author(s)
      K. Takeuchi
    • Journal Title

      Annales de l'Institut Fourier

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On some topological properties of Fourier transforms of regular holonomic D-modules2020

    • Author(s)
      Y. Ito and K. Takeuchi
    • Journal Title

      Canadian Math. Bulletin

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On irregularities of Fourier transforms of regular holonomic D-modules2020

    • Author(s)
      Y. Ito and K. Takeuchi
    • Journal Title

      Advances in Math.

      Volume: 366

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On vanishing theorems for local systems associated to Laurent polynomials2018

    • Author(s)
      A. Esterov and K. Takeuchi
    • Journal Title

      Nagoya Math. Journal

      Volume: 231 Pages: 1-22

    • DOI

      10.1017/nmj.2017.8

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets2017

    • Author(s)
      Yuichi Ike, Yutaka Matsui and Kiyoshi Takeuchi
    • Journal Title

      Int. Math. Res. Not.

      Volume: 印刷中 Issue: 15 Pages: 4852-4898

    • DOI

      10.1093/imrn/rnx030

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] Fourier transforms of holonomic D-modules and irregular characteristic cycles2023

    • Author(s)
      K. Takeuchi
    • Organizer
      研究集会 ``Singularities and Algebraic Geometry", Khanh Hoa 大学(ベトナム、ニャチャン)
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Exponential factors and Fourier transforms of D-modules2019

    • Author(s)
      K. Takeuchi
    • Organizer
      AGA Porto 2019, Higgs bundles and D-modules
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Meromorphic nearby cycle functors and monodromies of meromorphic functions2019

    • Author(s)
      K. Takeuchi
    • Organizer
      Non-isolated singularities and derived geometry
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On irregularities of Fourier transforms of regular holonomic D-modules2019

    • Author(s)
      K. Takeuchi
    • Organizer
      27th ICFIDCAA Krasnoyarsk
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On irregularities of Fourier transforms regular holonomic D-modules2018

    • Author(s)
      竹内潔
    • Organizer
      第14回代数解析幾何学セミナー
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On irregularities of Fourier transforms of regular holonomic D-modules2018

    • Author(s)
      竹内潔
    • Organizer
      Riemann-Hilbert correspondences
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the monodromies and the limit mixed Hodge structures of families of algebraic varieties2017

    • Author(s)
      竹内潔
    • Organizer
      Singularity theory conference
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the monodromy conjecture for non-degenerate hypersurfaces2017

    • Author(s)
      竹内潔
    • Organizer
      3rd PRIMA congress in Oaxaca
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] D加群2017

    • Author(s)
      竹内潔
    • Total Pages
      324
    • Publisher
      共立出版
    • ISBN
      9784320112056
    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Hypergeometric functions and mirror symmetry2018

    • Related Report
      2018 Annual Research Report

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Published: 2017-04-28   Modified: 2024-01-30  

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