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2019 Fiscal Year Annual Research Report

代数多様体の双有理自己射の多面的研究

Research Project

Project/Area Number 15H03611
Research InstitutionThe University of Tokyo

Principal Investigator

小木曽 啓示  東京大学, 大学院数理科学研究科, 教授 (40224133)

Co-Investigator(Kenkyū-buntansha) 高木 俊輔  東京大学, 大学院数理科学研究科, 教授 (40380670)
權業 善範  東京大学, 大学院数理科学研究科, 准教授 (70634210)
Project Period (FY) 2015-04-01 – 2020-03-31
Keywords代数多様体の自己同型群 / ワイルドな自己同型
Outline of Annual Research Achievements

昨年までの研究の継続研究に加え、当該研究テーマに密接にかかわるが、次年度以降の新規課題に向けた今までの研究とは異なる新しい試みとして、原始的自己同型群を許容する代数多様体の分類問題に、シンガポール国立大学のDe-Qi Zhang氏と共同研究の形で着手し、一定の成果を得た。以下より具体的に述べる。

Z. Reichstein、 D. Rogalski、J. J. Zhangは、複素射影代数多様体のwildと呼ばれる自己同型(自分自身と空集合以外に安定な代数的部分集合を有しない自己同型)の概念を導入し、$2$次元以下の場合にはwildな自己同型をもつ多様体はアーベル多様体に限ることを示し、高次元においてもアーベル多様体に限ると予想した。De-Qi Zhang教授と共同で、この予想に関し次を得た:$3$次元以下であれば、アーベル多様体あるいはいくつかの特別な性質をもつ$3$次元カラビ・ヤウ多様体に限る。また、いずれの場合にもwildな自己同型のエントロピーは零になる。更に、$2$次元以下の場合に、曲面の分類に依存しない簡単な別証明も与えた。ここまでの成果は、論文「K. Oguiso, D.-Q. Zhang, Wild automorphisms of projective varieties, the maps which have no invariant proper subsets」(arXiv:2002.04437)にまとめ公表するとともに、国際研究集会「Degenerations, algebraic surfaces and related topics」(2019年12月神戸大学)での招待講演においても講演の形で公表した。

Research Progress Status

令和元年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和元年度が最終年度であるため、記入しない。

  • Research Products

    (15 results)

All 2020 2019 Other

All Int'l Joint Research (4 results) Journal Article (5 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 5 results) Presentation (5 results) (of which Int'l Joint Research: 5 results,  Invited: 5 results) Funded Workshop (1 results)

  • [Int'l Joint Research] 国立シンガポール大学(シンガポール)

    • Country Name
      SINGAPORE
    • Counterpart Institution
      国立シンガポール大学
  • [Int'l Joint Research] 天津大学(中国)

    • Country Name
      CHINA
    • Counterpart Institution
      天津大学
  • [Int'l Joint Research] Dusseldorf大学/Bayreuth大学(ドイツ)

    • Country Name
      GERMANY
    • Counterpart Institution
      Dusseldorf大学/Bayreuth大学
  • [Int'l Joint Research] KIAS(韓国)

    • Country Name
      KOREA (REP. OF KOREA)
    • Counterpart Institution
      KIAS
  • [Journal Article] Nef line bundles on Calabi-Yau threefolds, I2020

    • Author(s)
      V. Lazic, K. Oguiso, Th. Peternell
    • Journal Title

      Int. Math. Res. Not.

      Volume: - Pages: -

    • DOI

      https://doi.org/10.1093/imrn/rnx191

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Automorphism groups of smooth quintic threefolds2019

    • Author(s)
      K. Oguiso, X. Yu
    • Journal Title

      Asian J. Math.

      Volume: 23 Pages: 201--256

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A few explicit examples of complex dynamics of inertia groups on surfaces--a question of Professor Igor Dolgachev2019

    • Author(s)
      K. Oguiso
    • Journal Title

      Transform. Groups

      Volume: 24 Pages: 545--561

    • Peer Reviewed
  • [Journal Article] A surface with discrete and nonfinitely generated automorphism group2019

    • Author(s)
      T.-C. Dinh, K. Oguiso
    • Journal Title

      Duke Math. J.

      Volume: 168 Pages: 941--966

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Pisot units, Salem numbers, and higher dimensional projective manifolds with primitive automorphisms of positive entropy2019

    • Author(s)
      K. Oguiso
    • Journal Title

      Int. Math. Res. Not.

      Volume: 5 Pages: 1373--1400

    • Peer Reviewed
  • [Presentation] Projective rational manifolds with non-finitely generated discrete automorphism group and infinitely many real forms2020

    • Author(s)
      K. Oguiso
    • Organizer
      Complex Dynamics, CIRM, Luminy, France
    • Int'l Joint Research / Invited
  • [Presentation] Finite generation problem of the discrete automorphism group of a smooth projective variety2019

    • Author(s)
      K. Oguiso
    • Organizer
      Lecture Series in Algebraic Geometry, Morningside Center of Mathematics, Beijing, China.
    • Int'l Joint Research / Invited
  • [Presentation] A surface in odd characteristic with discrete and non-finitely generated automorphism group2019

    • Author(s)
      K. Oguiso
    • Organizer
      the Stefan Banach International Mathematical Center, Warsaw, Poland.
    • Int'l Joint Research / Invited
  • [Presentation] Coble's question and complex dynamics of inertia groups on K3 surfaces2019

    • Author(s)
      K. Oguiso
    • Organizer
      Birational Geometry and Fano varieties dedicated to V. Iskovskikh, Steklov Mathematical Institute, Moscow, Russia
    • Int'l Joint Research / Invited
  • [Presentation] Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups2019

    • Author(s)
      K. Oguiso
    • Organizer
      Algebraic, Complex and Arithmetic Dynamics, Simons Symposia, Schloss Elmau, Germany.
    • Int'l Joint Research / Invited
  • [Funded Workshop] A few days conference on algebraic geometry, arithmetic and complex dynamics2019

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Published: 2021-01-27  

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