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Study on the dimension of the global sections of adjoint bundles for polarized manifolds via their invariants

Research Project

Project/Area Number 16K05103
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionKochi University

Principal Investigator

Yoshiaki Fukuma  高知大学, 教育研究部自然科学系理工学部門, 教授 (20301319)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords代数学 / 代数幾何学 / 偏極多様体 / 豊富な因子 / 随伴束 / nefかつbigな因子 / 断面不変量 / Okounkov体 / Δ-種数 / 豊富なベクトル束 / 射影多様体
Outline of Final Research Achievements

Let X be a smooth projective variety defined over the field of complex numbers and let L be an ample Cartier divisor on X. Then the pair (X,L) is called a polarized manifold. We conducted studies on the dimension of the global sections of adjoint bundles K+tL, where K denotes the canonical divisor of X and t is a positive integer. In particular, we studied the following conjecture and its related topics: Let (X,L) be an n-dimensional polarized manifold. If K+(n-1)L is nef, then the dimension of the global sections of K+(n-1)L is positive. Consequently we obtained that the above conjecture is true for some special cases. We also were able to achieve some research results about the dimension of the global sections of adjoint bundles and invariants of polarized manifolds.

Academic Significance and Societal Importance of the Research Achievements

偏極多様体の随伴束は, 射影多様体の研究においていろいろな場面で使われており, 射影多様体の分類や高次元代数幾何学の研究においてとても重要な役割を果たしている. 偏極多様体の随伴束の持つ性質に関する研究については, 例えば基点自由性に関する研究があり, いわゆる藤田予想といわれる予想の解決に向けた研究成果が高次元代数多様体論の研究に大きな役割を果たしていることを考えると, 随伴束の大域切断のなす次元に関する研究が今後の代数幾何学、特に偏極多様体のさらなる研究に活かされていくことが大いに期待される.

Report

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

    (13 results)

All 2021 2020 2019 2018 2017

All Journal Article (7 results) (of which Peer Reviewed: 7 results,  Open Access: 5 results) Presentation (6 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results)

  • [Journal Article] On invariants of polynomial functions, II2021

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Algebra and Discrete Mathematics

      Volume: 31 Issue: 1 Pages: 71-83

    • DOI

      10.12958/adm1319

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A generalization of Δ-genus for big divisors on projective varieties2020

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Communications in Algebra

      Volume: 48 Issue: 1 Pages: 168-184

    • DOI

      10.1080/00927872.2019.1635609

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the dimension of the global sections of K_{X}+4L for polarized 5-folds (X, L)2019

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      ANNALI DELL'UNIVERSITA' DI FERRARA

      Volume: 65 Issue: 2 Pages: 231-240

    • DOI

      10.1007/s11565-019-00322-5

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Classification of bi-polarized 3-folds (X, L_{1}, L_{2}) with h^{0}(K_{X}+L_{1}+L_{2})=12018

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 48 Issue: 2 Pages: 159-170

    • DOI

      10.32917/hmj/1533088829

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on the dimension of global sections of adjoint bundles for polarized 4-folds2018

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Proceedings of the Japan Academy. Series A, Mathematical Sciences

      Volume: 94 Issue: 5 Pages: 53-58

    • DOI

      10.3792/pjaa.94.53

    • NAID

      40021548036

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] New invariants of ample vector bundles over smooth projective varieties2018

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Rendiconti di Matematica e delle sue Applicazioni. Serie VII

      Volume: 39 Pages: 97-131

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on classification of generalized polarized manifolds by the Cr-sectional Hodge number of type (1,1) and the Cr-sectional Betti number.2017

    • Author(s)
      Yoshiaki Fukuma
    • Journal Title

      Revue Roumaine de Mathematiques Pures et Appliquees

      Volume: 62 Pages: 529-535

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] On the dimension of the global sections of adjoint bundles for polarized manifolds2019

    • Author(s)
      Yoshiaki Fukuma
    • Organizer
      Algebraic surfaces and related topics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 偏極多様体の随伴束の大域切断のなす次元について~予想・問題と現在までの結果2019

    • Author(s)
      福間 慶明
    • Organizer
      日本数学会 中国・四国支部例会
    • Related Report
      2018 Research-status Report
  • [Presentation] 5次元偏極多様体の随伴束の大域切断のなす次元に関する考察2019

    • Author(s)
      福間 慶明
    • Organizer
      日本数学会 年会
    • Related Report
      2018 Research-status Report
  • [Presentation] 偏極多様体の随伴束の大域切断のなす次元について2018

    • Author(s)
      福間 慶明
    • Organizer
      日本数学会年会
    • Related Report
      2017 Research-status Report
  • [Presentation] 偏極多様体の随伴束の大域切断のなす次元に関する諸問題について2017

    • Author(s)
      福間 慶明
    • Organizer
      九州大学代数幾何学セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] 偏極多様体の随伴束の大域切断のなす次元に関する話題2017

    • Author(s)
      福間 慶明
    • Organizer
      代数学ミニシンポジウム2017(倉敷)
    • Related Report
      2017 Research-status Report

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Published: 2016-04-21   Modified: 2022-01-27  

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