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Complex geometry of the canonical bundle; Study of deformations and degenerations of complex manifolds.

Research Project

Project/Area Number 16H03929
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

Takayama Shigeharu  東京大学, 大学院数理科学研究科, 教授 (20284333)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥13,260,000 (Direct Cost: ¥10,200,000、Indirect Cost: ¥3,060,000)
Fiscal Year 2020: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2017: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2016: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Keywords複素幾何学 / 相対標準束 / 変形空間 / モジュライ空間 / ファイバー積分 / 漸近挙動
Outline of Final Research Achievements

Let f : X --> Y be a family of compact algebraic manifolds, which can be singular along the central fiber X_0. We obtained various results which connect properties of X_0, properties of general fibers X_y, and properties of Y associated with f. We also studied in details functions on Y which come from fiber integrals along the fibers of f :X --> Y. We proved a formula of the asymptotic expansion of such functions on Y. There is no development for about 40 years in the case dim Y > 1, because of technical difficulties.

Academic Significance and Societal Importance of the Research Achievements

f : X --> Y の中心ファイバー X_0 は考え方によっては無限遠に位置するファイバーと見做せる。これを有限部分に存在する一般ファイバー X_y の情報から理解できる方法を与える、という考え方は数学以外においても有用である。ファイバー積分により得られる底空間上の関数の研究は、積分の族を用いるような現象を扱う場面においてしばしば現れ、応用範囲が広い。漸近展開公式の発散項の特定、記述はその初めに問題となるような点であり基本的である。

Report

(6 results)
  • 2021 Final Research Report ( PDF )
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • Research Products

    (24 results)

All 2019 2018 2017 2016

All Journal Article (5 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (10 results) (of which Int'l Joint Research: 7 results,  Invited: 10 results) Funded Workshop (9 results)

  • [Journal Article] Moderate Degeneration of Kahler-Einstein Manifolds with Negative Ricci Curvature2019

    • Author(s)
      Takayama Shigeharu
    • Journal Title

      Publications of the Research Institute for Mathematical Sciences

      Volume: 55 Issue: 4 Pages: 779-793

    • DOI

      10.4171/prims/55-4-4

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Moderate degenerations of Ricci-flat Kahler-Einstein manifolds over higher dimensional bases2019

    • Author(s)
      Shigeharu Takayama
    • Journal Title

      J. Math. Sci. Univ. Tokyo

      Volume: 26 Pages: 335-359

    • NAID

      120006942541

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A filling-in problem and moderate degenerations of minimal algebraic varieties2019

    • Author(s)
      Takayama Shigeharu
    • Journal Title

      Algebraic Geometry

      Volume: 6 Pages: 26-49

    • DOI

      10.14231/ag-2019-002

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Positivity of twisted relative pluricanonical bundles and their direct images.2018

    • Author(s)
      Mihai Paun, Shigeharu Takayama
    • Journal Title

      J. Algebraic Geom.

      Volume: 27 Issue: 2 Pages: 211-272

    • DOI

      10.1090/jag/702

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Effective degree bounds for generalized Gauss map images2017

    • Author(s)
      G. Heier and S. Takayama
    • Journal Title

      Advanced Studies in Pure Math., Math. Soc. Japan

      Volume: 74

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Asymptotic expansions of fiber integrals and applications2019

    • Author(s)
      高山 茂晴
    • Organizer
      第15回代数・解析・幾何学セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Asymptotic expansions of fiber integrals and application2019

    • Author(s)
      Shigeharu Takayama
    • Organizer
      Birational geometry and moduli spaces
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic expansions of fiber integrals2019

    • Author(s)
      Shigeharu Takayama
    • Organizer
      Tianyuan International Workshop in Several Complex Variables
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Degeneration of Kahler-Einstein manifolds2018

    • Author(s)
      高山 茂晴
    • Organizer
      2018 International conference on complex geometry and several complex variables
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Moderate degenerations of Calabi-Yau manifolds over higher dimensional bases2017

    • Author(s)
      高山 茂晴
    • Organizer
      Analytic Methods in Algebraic Geometry Day
    • Place of Presentation
      Northwestern Univ., シカゴ市(アメリカ合衆国)
    • Year and Date
      2017-03-18
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ファイバー空間の相対標準束と充填問題2017

    • Author(s)
      高山茂晴
    • Organizer
      代数学シンポジウム
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Moderate degeneration of K\"ahler-Einstein manifolds with negative Ricci curvature2017

    • Author(s)
      高山茂晴
    • Organizer
      第23回複素幾何シンポジウム
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Moderate degeneration of K\"ahler-Einstein manifolds2017

    • Author(s)
      高山茂晴
    • Organizer
      Mini-workshop on Complex Geometry
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Positivity of the direct image of an adjoint bundle with multiplier ideal2016

    • Author(s)
      高山 茂晴
    • Organizer
      第22回複素幾何シンポジウム
    • Place of Presentation
      金沢県政記念 しいのき迎賓館(石川県, 金沢市)
    • Year and Date
      2016-11-02
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 標準束の複素幾何学2016

    • Author(s)
      高山 茂晴
    • Organizer
      日本数学会2016年度秋季総合分科会 幾何学・トポロジー分科会合同 特別講演
    • Place of Presentation
      関西大学(大阪府, 吹田市)
    • Year and Date
      2016-09-16
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Funded Workshop] 第25回複素幾何シンポジウム2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Complex Geometry 2019 Tokyo2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] 第21回多変数複素解析葉山シンポジウム2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] 第20回多変数複素解析葉山シンポジウム2018

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

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] 第19回多変数複素解析葉山シンポジウム2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] 第23回複素幾何シンポジウム2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] 第22回複素幾何シンポジウム2016

    • Place of Presentation
      金沢県政記念 しいのき迎賓館
    • Year and Date
      2016-10-31
    • Related Report
      2016 Annual Research Report
  • [Funded Workshop] 第18回多変数複素解析葉山シンポジウム2016

    • Place of Presentation
      湘南国際村センター,
    • Year and Date
      2016-07-16
    • Related Report
      2016 Annual Research Report

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Published: 2016-04-21   Modified: 2023-01-30  

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