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Berkovich analytic space, tropical geometry, and algebraic/arithmetic dynamics

Research Project

Project/Area Number 18H01114
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionDoshisha University

Principal Investigator

Kawaguchi Shu  同志社大学, 理工学部, 教授 (20324600)

Co-Investigator(Kenkyū-buntansha) 山木 壱彦  筑波大学, 数理物質系, 教授 (80402973)
Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥11,570,000 (Direct Cost: ¥8,900,000、Indirect Cost: ¥2,670,000)
Fiscal Year 2022: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2021: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2020: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2019: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
KeywordsBerkovich 解析空間 / トロピカル幾何 / 代数・数論力学系 / 非アルキメデス幾何 / 解析的捩率 / Berkovich解析空間
Outline of Final Research Achievements

To an algebraic variety defined over a complete non-Archimedean value field, one can attach an analytic space in the sense of Berkovich. Fixing a model over the valuation ring, this analytic space contains a polyhedral complex, called the skeleton associated with the model, which preserves important information about the original variety. With Kazuhiko Yamaki, we have studied faithful tropicalizations associated to the linear system of a divisor. The published papers treat the case of curves and adjoint bundles on smooth projective varieties. In this direction, we study the cases of tropical toric varieties and tropical abelian varieties. In algebraic/arithmetic dynamics, with Liang-Chung Hsia, we study when two sections of a one-parameter family of Henon maps have infinitely many points over which the sections give periodic points. With Shigeru Mukai and Ken-Ichi Yoshikawa, we study explicit relations of the difference of elliptic j-functions and Borcherd's Phi function.

Academic Significance and Societal Importance of the Research Achievements

代数幾何で多項式の共通零点で表される図形である代数多様体を扱う.一方,トロピカル幾何は,代数幾何,数論幾何,組合せ論,数理物理など多くの分野とかかわっている.代数多様体が非アルキメデス付値体で定義された場合,直線束の切断の付値写像により多面体的多様体ができる一方,代数多様体の付値環上のモデルにより付随する解析空間にも多面体的多様体ができる.大雑把にいって,両者が一致するときに,トロピカル化は忠実とよばれ,多くの研究がされている.本研究では,直線束に付随する忠実トロピカル化がいつできるかを,曲線の場合と一般の代数多様体の場合,さらにトーリック多様体とアーベル多様体のときに詳しく調べている.

Report

(6 results)
  • 2023 Final Research Report ( PDF )
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (13 results)

All 2024 2023 2022 2021 2019 2018 Other

All Int'l Joint Research (1 results) Journal Article (3 results) (of which Peer Reviewed: 3 results) Presentation (8 results) (of which Int'l Joint Research: 2 results,  Invited: 8 results) Funded Workshop (1 results)

  • [Int'l Joint Research] National Taiwan Normal University(台湾)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Effective faithful tropicalizations associated to linear systems on curves2021

    • Author(s)
      Kawaguchi Shu、Yamaki Kazuhiko
    • Journal Title

      Memoirs of the American Mathematical Society

      Volume: 270 Issue: 1323 Pages: 1-110

    • DOI

      10.1090/memo/1323

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Effective Faithful Tropicalizations Associated to Adjoint Linear Systems2018

    • Author(s)
      Kawaguchi Shu、Yamaki Kazuhiko
    • Journal Title

      International Mathematics Research Notices

      Volume: - Issue: 19 Pages: 6089-6112

    • DOI

      10.1093/imrn/rnx302

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Resultants and the Borcherds Φ-function2018

    • Author(s)
      Shu Kawaguchi, Shigeru Mukai, Ken-Ichi Yoshikawa
    • Journal Title

      American Journal of Mathematics

      Volume: 140 Issue: 6 Pages: 1471-1519

    • DOI

      10.1353/ajm.2018.0045

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] 代数多様体の忠実トロピカル化とトロピカル多様体の忠実埋め込み2023

    • Author(s)
      川口周
    • Organizer
      湯布院代数幾何ワークショップ
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] 直線束に付随した忠実トロピカル化とトロピカル多様体の忠実埋め込み2022

    • Author(s)
      川口 周
    • Organizer
      2022早稲田整数論研究集会
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] j-invariant and Borcherds Phi-function2021

    • Author(s)
      Shu Kawaguchi
    • Organizer
      Recent Developments in Algebraic Geometry, Arithmetic and Dynamics Part 1, National University of Singapore
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Andre-Oort予想の最近の進展(企画サーベイ)2021

    • Author(s)
      川口 周
    • Organizer
      代数幾何学城崎シンポジウム
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] 数論力学系における高さ関数2021

    • Author(s)
      川口周
    • Organizer
      日本数学会2021年度年会 企画特別講演
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] j-invariant and Borcherds Phi-function2019

    • Author(s)
      Shu Kawaguchi
    • Organizer
      Number Theory Seminar, Cambridge University
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Some arithmetic properties of one-parameter families of Henon maps2019

    • Author(s)
      Shu Kawaguchi
    • Organizer
      Vietnam-USA Joint Mathamatical Meeting, Special Session on Complex Geometry and Dynamical Systems
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Heights and periodic points for one-parameter families of Henon maps2018

    • Author(s)
      Shu Kawaguchi
    • Organizer
      Intercity Seminar on Arakelov Geometry 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Workshop on Nonarchimedean Geometry and Related Fields2024

    • Related Report
      2022 Annual Research Report

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Published: 2018-04-23   Modified: 2025-01-30  

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