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Semi-positive holomorphic line bundles and complex dynamics

Research Project

Project/Area Number 20K14313
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)
Osaka City University (2020-2021)

Principal Investigator

Koike Takayuki  大阪公立大学, 大学院理学研究科, 准教授 (30784706)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2021: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords半正正則直線束 / 正則葉層構造 / 半正直線束 / 上田理論 / K3曲面 / 平坦直線束 / 正則直線束 / レビ平坦 / エルミート計量 / 部分多様体近傍
Outline of Research at the Start

一年目には, 形式化原理及び複素解析幾何学に於けるL2理論の進展について, その最新の情報の収集に努める. 同時に形式化原理に於ける代数的手法についての習得にも務める.
二年目には, 【研究手法1】に基づいた研究をまず進める.
三年目には, 【研究手法2】とこれまでの研究成果とを組み合わせることで研究を遂行する.

Outline of Final Research Achievements

We have succeeded in determining the geometric structure of complex manifolds associated with semi-positive line bundles and obtaining its applications by using a method based on a technique completely different from conventional dynamical approaches. We mainly deal with dynamical properties (especially linearization problems) on a neighborhood of submanifolds based on the semi-positivity of line bundles. First we obtained some results by applying techniques from the theory of several complex functions. By additionally applying differential geometrical techniques on holomorphic foliations from the viewpoint of algebraic geometry, we succeeded in obtaining an affirmative answer to the conjecture we have posed as a goal of this program, and also in obtaining its applications especially on complex surfaces. Concurrently, in collaboration with Takato Uehara at Okayama University, we have solved the realizability problem of projective K3 surfaces by our gluing construction.

Academic Significance and Societal Importance of the Research Achievements

本研究では, 複素多様体の複素解析幾何学的構造の解明を行った. 複素多様体は局所的に複素数によってパラメータ付けられる対象であり, 多項式たちの共通ゼロ点集合の様な非常に基礎的かつ重要な幾何学的対象である.私の研究成果では, 複素多様体研究に於いて金字塔ともいえる小平の埋め込み定理の深化にあたる結果を得ている. この成果は複素多様体の解析的・幾何学的構造の詳細を明らかにするものであり, 複素多様体が登場する数学, 延いては関連する数理科学全般に於ける学術的意義は大きい. また本研究成果により新たな射影的K3曲面の構成方法が判明したことに対しては, 数理物理学的応用が大いに期待される.

Report

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

    (27 results)

All 2023 2022 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Open Access: 1 results) Presentation (18 results) (of which Int'l Joint Research: 8 results,  Invited: 14 results) Remarks (2 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Universite Cote d'Azur(フランス)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Universite Cote d'Azur(フランス)

    • Related Report
      2022 Research-status Report
  • [Journal Article] A gluing construction of projective K3 surfaces2022

    • Author(s)
      Koike Takayuki、Uehara Takato
    • Journal Title

      Epijournal de Geometrie Algebrique

      Volume: Volume 6 Pages: 1-15

    • DOI

      10.46298/epiga.2022.volume6.8504

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Linearization of transition functions of a semi-positive line bundle along a certain submanifold2021

    • Author(s)
      T. Koike
    • Journal Title

      Ann. Inst. Fourier (Grenoble)

      Volume: 71 Issue: 5 Pages: 2237-2271

    • DOI

      10.5802/aif.3439

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle2021

    • Author(s)
      T. Koike
    • Journal Title

      Math. Ann.

      Volume: online Issue: 1-2 Pages: 291-313

    • DOI

      10.1007/s00208-021-02199-2

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On minimal singular metrics of line bundles whose stable base loci admit holomorphic tubular neighborhoods2020

    • Author(s)
      G. Hosono, T. Koike
    • Journal Title

      Ann. Fac. Sci. Toulouse Math. (6)

      Volume: Volume XXIX Facicle 1 Pages: 149-175

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Holomorphic foliation associated with a semi-positive class of numerical dimension one2023

    • Author(s)
      T. Koike
    • Organizer
      SCV, CR geometry and Dynamics
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Holomorphic foliation associated with a semi-positive class of numerical dimension one2023

    • Author(s)
      T. Koike
    • Organizer
      HAYAMA Symposium on Complex Analysis in Several Variables XXIV
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Delbar cohomology of the complement of a semi-positive anticanonical divisor of a compact surface2023

    • Author(s)
      T. Koike
    • Organizer
      葉層構造論シンポジウム
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Delbar cohomology of the complement of a semi-positive anticanonical divisor of a compact surface2023

    • Author(s)
      T. Koike
    • Organizer
      複素幾何における葉層と力学系の諸問題
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Delbar cohomology of the complement of a semi-positive anticanonical divisor of a compact surface2023

    • Author(s)
      T. Koike
    • Organizer
      多変数関数論冬セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Delbar cohomology of the complement of a semi-positive anticanonical divisor of a compact surface2023

    • Author(s)
      T. Koike
    • Organizer
      日本数学会年会
    • Related Report
      2023 Annual Research Report
  • [Presentation] A gluing construction of projective K3 surfaces2023

    • Author(s)
      T. Koike
    • Organizer
      K3, Enriques Surfaces, and Related Topics
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Holomorphic foliation associated with a semi-positive class of numerical dimension one2022

    • Author(s)
      T. Koike
    • Organizer
      Complex Geometry and Dynamical Systems
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Projective K3 surfaces which contain Levi-flat hypersurfaces2022

    • Author(s)
      T. Koike
    • Organizer
      Complex Analytic Geometry
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Projective K3 surfaces which contain Levi-flat hypersurfaces2022

    • Author(s)
      T. Koike
    • Organizer
      葉層構造論シンポジウム
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Semipositive line bundles and holomorphic foliations2021

    • Author(s)
      T. Koike
    • Organizer
      葉層構造シンポジウム
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Semipositive line bundles and holomorphic foliations2021

    • Author(s)
      T. Koike
    • Organizer
      Dynamics, SCV and CR geometry
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Holomorphic foliation associated with a semi-positive class of numerical dimension one2021

    • Author(s)
      T. Koike
    • Organizer
      2021年度多変数関数論冬セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Holomorphic foliation associated with a semi-positive class of numerical dimension one2021

    • Author(s)
      T. Koike
    • Organizer
      日本数学会年会
    • Related Report
      2021 Research-status Report
  • [Presentation] Linearization of transition functions of a semi-positive line bundle along a certain submanifold2021

    • Author(s)
      T. Koike
    • Organizer
      Grauert theory and recent complex geometry
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] 半正直線束の変換関数の固定部分近傍における線形化について2021

    • Author(s)
      小池貴之
    • Organizer
      日本数学会年会函数論分科会
    • Related Report
      2020 Research-status Report
  • [Presentation] On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle2020

    • Author(s)
      T. Koike
    • Organizer
      複素幾何シンポジウム(金沢) 2020
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 半正直線束の変換関数の固定部分近傍における線形化について2020

    • Author(s)
      小池貴之
    • Organizer
      第63回 函数論シンポジウム
    • Related Report
      2020 Research-status Report
    • Invited
  • [Remarks] https://tkoike.com/

    • Related Report
      2022 Research-status Report
  • [Remarks] ウェブサイト

    • URL

      https://tkoike.com/

    • Related Report
      2021 Research-status Report 2020 Research-status Report
  • [Funded Workshop] 1day workshop on dynamical systems and complex geometry2023

    • Related Report
      2022 Research-status Report

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Published: 2020-04-28   Modified: 2025-01-30  

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