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代数多様体の連接層の導来圏の生成系および次元に関する研究

Research Project

Project/Area Number 17J00857
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Algebra
Research InstitutionWaseda University

Principal Investigator

原 和平  早稲田大学, 基幹理工学研究科, 特別研究員(PD)

Project Period (FY) 2017-04-26 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2019: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2018: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2017: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords導来圏 / 弱Fano束 / 傾斜ベクトル束 / 非可換クレパント解消 / フロップ / 変形理論 / モジュライ
Outline of Annual Research Achievements

昨年から引き続き,Grassmann多様体の余接束上の傾斜ベクトル則について考察した.昨年度発見したGr(2,4)の余接束上の傾斜ベクトル束を多少修正することで,期待する性質を満たす傾斜ベクトル束を構成することに成功した.この傾斜ベクトル束を用いて既存のD^b(T^*Gr(2,4))の導来自己同値を構成しようと試みたが,これに関して研究を進展させることは出来なかった.
一方,福岡氏,石川氏と共同で3次元del Pezzo多様体上の弱Fano束の分類を研究した.結果,石川氏による次数3のdel Pezzo上の弱Fano束の分類結果を拡張して,3次元Picard数1のdel Pezzo多様体上の階数2弱Fano束を分類することが出来た.特に,次数5の3次元del Pezzo多様体の場合に,この多様体が傾斜ベクトル束を持つことに着目し,傾斜ベクトル束が誘導する導来同値で射影分解に対応する完全列を書き下す形で,弱Fano束を分類した.他にも,Picard数1の3次元Fano多様体上でc_1が反標準束のクラスと一致するnef束が,次数1のdel Pezzoの場合を除いて大域生成であることや,次数4の場合にSerre構成によってc_1が偶数の階数2弱Fano束と対応する非特異楕円曲線を具体的に構成する等,この分野において有効と思われる結果を多く得ることが出来た.
他にも,海外での研究集会に積極的に参加し,自身の研究について発信した.

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。

Report

(3 results)
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • Research Products

    (10 results)

All 2019 2018 2017

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

  • [Journal Article] Strong full exceptional collections on certain toric varieties with Picard number three via mutations2017

    • Author(s)
      Hara Wahei
    • Journal Title

      Matematiche (Catania)

      Volume: 72 Pages: 3-24

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops2017

    • Author(s)
      Hara Wahei
    • Journal Title

      Advances in Mathematics

      Volume: 318 Pages: 355-410

    • DOI

      10.1016/j.aim.2017.08.010

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Presentation] Derived equivalence for the simple K-equivalence of type G_22019

    • Author(s)
      Wahei Hara
    • Organizer
      International workshop on derived categories and related topics
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Deformation of tilting-type derived equivalences for crepant resolutions2019

    • Author(s)
      Wahei Hara
    • Organizer
      the Japanese-European Symposium on Symplectic Varieties and Moduli spaces
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Derived equivalence for the simple K-equivalence of type G_22019

    • Author(s)
      Wahei Hara
    • Organizer
      Interaction between Algebraic Geometry and QFT
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mukai flops and P-twists from non-commutative point of view2018

    • Author(s)
      Wahei Hara
    • Organizer
      Intersections between Algebraic geometry and non-commutative algebras
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mukai flops and P-twists from non-commutative point of view2018

    • Author(s)
      Wahei Hara
    • Organizer
      城崎代数幾何学シンポジウム
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On derived equivalences for Abuaf’s 5-dimensional flop2018

    • Author(s)
      Wahei Hara
    • Organizer
      McKay correspondence and noncommutative algebra
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mukai flops and P-twists via non-commutative crepant resolutions2017

    • Author(s)
      Hara Wahei
    • Organizer
      Matrix fac- torizations and related topics, International Center for Materials Science
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mukai flops and P-twists via non-commutative crepant resolutions2017

    • Author(s)
      Hara Wahei
    • Organizer
      Symplectic geometry and Representation theory
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited

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Published: 2017-05-25   Modified: 2024-03-26  

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