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Mixed motivic sheaves, category theory, and study of cycle complexes

Research Project

Project/Area Number 21340002
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

HANAMURA Masaki  東北大学, 大学院・理学研究科, 教授 (60189587)

Co-Investigator(Renkei-kenkyūsha) KIMURA Shunichi  広島大学, 大学院・理学研究科, 教授 (10284150)
ISHIDA Masanori  東北大学, 大学院・理学研究科, 教授 (30124548)
YUKIE Akihiko  京都大学, 大学院・理学研究科, 教授 (20312548)
Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥6,630,000 (Direct Cost: ¥5,100,000、Indirect Cost: ¥1,530,000)
Fiscal Year 2011: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2010: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2009: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Keywords代数幾何学 / モティーフ理論 / 三角圏 / 代数的サイクル / モティーフ / Chow群 / 代数サイクル / コホモロジー / チャウ群
Research Abstract

Quasi DG category, a generalization of DG category given by the principal researcher, is a notion that appears in the theory of mixed motivic sheaves. We developed the basic theory of quasi DG categories. The mail features are basic axioms of a quasi DG category, the notion of C-diagrams with values in a quasi DG category, additivity of the function complexes, the construction of the quasi DG category of C-diagrams, and the proof that the homotopy category of the quasi DG category of C-diagrams has the structure of a triangulated category.In particular, when the function complex (depending on n objects) that constitutes part of the notion, is additive in each variable, the same property was proven to hold for the function complex for C-diagrams.
We described the Chow cohomology of an algebraic surface using its resolution of singularities. In particular, we gave a condition for Chow cohomology and homology to coincide.
We showed the blow-up formula for the motives and higher Chow groups of a quasi-projective variety.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (20 results)

All 2013 2012 2011 2010 2009 Other

All Journal Article (10 results) (of which Peer Reviewed: 9 results) Presentation (10 results) (of which Invited: 2 results)

  • [Journal Article] 相対的な代数的対応と 混合モティーフ層2013

    • Author(s)
      Masaki Hanamura
    • Journal Title

      数理研講究

      Volume: 録別冊

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Chow cohomology groups of algebraic surfaces2013

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Mathematical Res. Letters

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Chow cohomology groups of algebraic surfaces2013

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Mathematical Research Letters

      Volume: 未定

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Relative algebraic correspondences and mixed motivic sheaves2012

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Proc. Japan Academy

      Volume: 88 Pages: 121-126

    • NAID

      40019463899

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Relative algebraic correspondences and mixed motivic sheaves2012

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Proceedings of the Japan Academy

      Volume: 88 Pages: 121-126

    • NAID

      40019463899

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-ups and mixed motives2011

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Tohoku Math. J

      Volume: 63 Pages: 751-773

    • NAID

      110008764869

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Blow-ups and mixed motives2011

    • Author(s)
      Masaki Hanamura
    • Journal Title

      Tohoku Math.J.

      Volume: 63 Pages: 751-774

    • NAID

      110008764869

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 混合モティーフの理論と応用2010

    • Author(s)
      花村昌樹
    • Journal Title

      数学

      Volume: 62巻2号 Pages: 182-193

    • NAID

      130004558907

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] モティーラ理論2009

    • Author(s)
      花村昌樹
    • Journal Title

      数理科学(別冊)

      Pages: 123-129

    • Related Report
      2009 Annual Research Report
  • [Journal Article] 混合モティーフの理論と応用

    • Author(s)
      花村昌樹
    • Journal Title

      数学 未定(採用決定)

    • NAID

      130004558907

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Quasi DG categories and mixed motivic sheaves2013

    • Author(s)
      Masaki Hanamura
    • Organizer
      代数幾何解析セミナー
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2013-02-19
    • Related Report
      2011 Final Research Report
  • [Presentation] Mixed Tate categories and the bar complex2012

    • Author(s)
      Masaki Hanamura
    • Organizer
      周期積分と超幾何 関数
    • Place of Presentation
      玉原セミナーハウス
    • Year and Date
      2012-09-06
    • Related Report
      2011 Final Research Report
  • [Presentation] Hodge realization of mixed motives2012

    • Author(s)
      Masaki Hanamura
    • Organizer
      研究集会「モティーフ理 論と代数幾何」
    • Place of Presentation
      東北大学
    • Year and Date
      2012-03-09
    • Related Report
      2011 Final Research Report
  • [Presentation] Relative algebraic correspondences and mixed motivic sheaves2011

    • Author(s)
      Masaki Hanamura
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2011-11-28
    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Presentation] DG categories and mixed motivic sheave2011

    • Author(s)
      Masaki Hanamura
    • Organizer
      東北復旦代数幾何 シンポジウム
    • Place of Presentation
      復旦大 学(中国)
    • Year and Date
      2011-06-15
    • Related Report
      2011 Final Research Report
  • [Presentation] Mixed Tate motifs and etale cohomology2010

    • Author(s)
      Masaki Hanamura
    • Organizer
      研究集会「周期積分と モチーフ」
    • Place of Presentation
      東京大学
    • Year and Date
      2010-02-20
    • Related Report
      2011 Final Research Report
  • [Presentation] Mixed Tate motifs and cohomology realizations2009

    • Author(s)
      Masaki Hanamura
    • Organizer
      東北復丹旦大 学代数幾何学シンポジウム
    • Place of Presentation
      東北大学
    • Year and Date
      2009-11-26
    • Related Report
      2011 Final Research Report
  • [Presentation] Mixed Tate motifs and cohomology realizations2009

    • Author(s)
      花村昌樹
    • Organizer
      東北復旦大学代数幾何学シンポジウム
    • Place of Presentation
      東北大学
    • Year and Date
      2009-11-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] Quasi DG categories and mixed motivic sheaves

    • Author(s)
      Masaki Hanamura
    • Organizer
      Seminar on algebra, geometry and analysis
    • Place of Presentation
      Kagoshima
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Mixed Tate categories and bar complex

    • Author(s)
      Masaki Hanamura
    • Organizer
      Semina at Tanbara
    • Place of Presentation
      Tanbara seminar house
    • Related Report
      2011 Annual Research Report
    • Invited

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Published: 2009-04-01   Modified: 2016-04-21  

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