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Research of Motivic Geometry from the viewpoint of Non-commutative algebraic geometry

Research Project

Project/Area Number 15540032
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KIMURA Shun-ichi  Hiroshima University, Graduate School of Science, Associated Professor, 大学院・理学研究科, 助教授 (10284150)

Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsMotives / finite dimensionality / Tensor Category / Bloch's Conjecture / Schur finite / Jacobian Conjecture / Alexander scheme / Motivic zeta / モチーフの有限次元性 / Bivariant theory / Jacobian conjecture / Mixed Motive / Nilpotence conjecture / Bloch conjecture / Bivariant Sheaf / Alexander Scheme / Chow群の有限次元性 / Albanese Kernel
Research Abstract

During the period of this research, it was found that the notion of finite dimensionality of motives can be greatly generalized, by Yves Andre and Bruno Kahn. The possible finite dimensionality of Chow motives was the starting point of this research. One can formulate the finite dimensionality in any tensor category, in particular in the category of Mixed motives. Unfortunately, we cannot expect that the category of mixed motives to be finite dimensional in my sense (O'Sullivan), and hence the notion of finite dimensionality should be generalized to the notion of Schur finiteness. This generalization posed major problems, for example, the problem of Schur Nilpotency.
Under this circumstances, following is the list of major results of this research. (1)Brushing up the notion of finite dimensionality of Chow motives (2)Relativization of the notion of motivic spaces (3)Positive characteristic approach to the Jacobian conjecture (4)Etaleness property of Alexander schemes (5)Chow motives are 1 dimensional if and only if they are invertible (6)Finding the Schur dimension (7)The finite dimensionality of the motives is stable under the deformation with smooth fiber (8)The relation between the finite dimensionality of motives and the rationality of Motivic Zeta function
Among this list, (7)may have a strong implication in the future. It is a joint work with Vladimir Guletskii, and the main limitation is that we can apply this result only for the family with the smooth fiber. If one can generalize this result to non-smooth fiber spaces, then that would be a breakthrough towards the proof of finite dimensionality of all Chow motives.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (12 results)

All 2005 Other

All Journal Article (12 results)

  • [Journal Article] Chow groups are finite dimensional,in some sense2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Mathematische Annalen 331

      Pages: 173-201

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Correspondences to abelian varieties II2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Hiroshima Mathematical Journal 35

      Pages: 251-261

    • NAID

      110002538254

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Chow groups are finite dimensional, in some sense2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Mathematische Annalen 331

      Pages: 173-201

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Correspondence to Abelian Varieties II2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Hiroshima Mathematical Journal 35

      Pages: 251-261

    • NAID

      110002538254

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Chow Groups are finite dimensional, in some sense2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Mathematische Annalen 331

      Pages: 173-201

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Correspondences to Abelian varieties II2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Hiroshima Mathematical Journal 35

      Pages: 251-261

    • NAID

      110002538254

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Chow groups are finite dimensional, in some sense2005

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Mathematicsche Annalen 331

      Pages: 173-201

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Positibe Characteristic Approach to Weak Kernel Conjecture

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Hiroshima Mathematical Journal To appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Positive Characteristic Approach to Weak Kernel Conjecture

    • Author(s)
      Kimura, Shun-ichi, Okuda, Shun-ichiro
    • Journal Title

      Hiroshima Mathematical Journal (to appear)

    • NAID

      110004455823

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Positive Characteristic Approach to Weak Kernel Conjecture

    • Author(s)
      Kimura, Shun-ichi, Okuda
    • Journal Title

      Hiroshima Mathematical Journal To appear

    • NAID

      110004455823

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Positive characteristic approach to weak kernel conjecture

    • Author(s)
      Kimura, Shunichi, Okuda, Shunichiro
    • Journal Title

      Hiroshima Mathematical Journal (発表予定)

    • NAID

      110004455823

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Correspondences to Abelian Varieties II

    • Author(s)
      Kimura, Shun-ichi
    • Journal Title

      Hiroshima Mathematical Journal (発表予定)

    • NAID

      110002538254

    • Related Report
      2004 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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