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Research on non-abelian Hodge structures of algebraic varieties

Research Project

Project/Area Number 11640016
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YOKOGAWA Koji  Ochanomizu University, Faculty of Sciences, Associate Professor, 理学部, 助教授 (40240189)

Co-Investigator(Kenkyū-buntansha) OHBA Kiyoshi  Ochanomizu University, Faculty of Sciences, Research Assistant, 理学部, 助手 (80242337)
TAKEDA Yoshihumi  Nara Wemen's University, Faculty of Sciences, Associate Professor, 理学部, 助教授 (50227039)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsVecdtor bundle / Algebraic variety / Moduli / Hodge theory / Non-abelian cohomology / 正標数の代数幾何 / ホモトピー / スタック / 非可換ホッジ理論 / 非可換コホモロジ-
Research Abstract

There were the following three purposes for this project.
1) Study of the non-abelian Hodge decomposition (or filtration) in the case of positive characteristic.
2) Study of the relation between defomation theory of non-commutative schemes and non-abelian Hodge theory.
3) Study of non-abelian mixed Hodge structures of algebraic varieties.
As for the first purpose, I realized that it would be important to construct the crystellin homotopy theory using n-stacks. If such a theory is constructed, the original problem would be clear and there would be many applications. It would be an extention of the work of N. Katz on Frobenius maps and Hodge filtrations in the abelian cases. I am now trying this next big project. On the second one, after the study of non-abelian schemes I understood that it is more natural to describe the deformation parameter space as an n-stacks. Such description would be useful to study the relationship with non-abelian Hodge structures. On the third one, there was a great development by C. Simpson. He describe the non-abelian mixed Hodge structures as kinds of filtrations on non-abelian cohomology n-stacks. I studied his theory and make some explicit calculations of such filtrations on n-stacks. Moreover I worked with Y. Takeda on the pre-Tango structures on curves. The work is related on the first one. With K. Ohba, I studied on the relation between the study of gerbs (using n-stacks) and the study of cycles of moduli spaces.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] 横川光司: "Rationality of moduli of parabolic sheaves on curve"Journal of London Mathematical Society. 59. 461-478 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 大場清: "Cutting and pasting of Riemann surfaces with Abelian differentials"International Journal of Mathematics. 10. 587-617 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 大場清: "Higher cycles on the moduli space of stable curves"Journal of Mathematical Society of Japan. 52. 231-267 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 大場清: "Embedding of the moduli space of Riemann surfaces with Igeta structures into the Sato Grassmann manifold"Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics. 75-78 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 横川光司: "Pre-Tango Structures on Curves"Tohoku Mathematical Journal. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Koji Yokogawa: "Rationality of modeli of parabolic sheaves on curve"Journal of London Mathematical Society. 59. 461-478 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kiyoshi Ohba: "Cutting and pasting of Riemann surfaces with Abelian Differentials"International Journal of Mathematics. 10. 587-617 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kiyoshi Ohba: "Higher cycles on the moduli space of stable curves"Journal of Mathematical Society of Japan. 52. 231-267 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kiyoshi Ohba: "Embedding of the moduli space of Riemann surfaces with Igeta structures into the Sato Grassmann manifold"Differential Geometry and Mathematical Physics. 75-78 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Koji Yokogawa: "Pre-Tango Structures on Curves"Tohoku Mathematical Journal. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 横川光司: "Rationality of moduli of parabolic sheaves on curve"Journal of London Mathematical Society. 59. 461-478 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大場清: "Cutting and pasting of Riemann surfaces with Abelian differentials"International Journal of Mathematics. 10. 587-617 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大場清: "Higher cycles on the moduli space of stable curves"Journal of Mathematical Society of Japan. 52. 231-267 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大場清: "Embedding of the moduli space of Riemann surfaces with Igeta structures into the Sato Grassmann manifold"Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics. 75-78 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 横川光司: "Pre-Tango Structures on Curves"Tohoku Mathematical Journal. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 大場清: "Higher cycles on the moduli space of stable curves"Journal of the mathematical society of Japan. 52. 231-267 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 大場清、橋本義武: "Cutting and pasting of Riemann surfaces with Abelian differentials, I"International Journal of Mathematics. 10. 587-617 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大場清、橋本義武: "On the anti-parallel Igeta construction of Riemann surfaces"Proceedings of "Aspects of Complex Analysis, Differetial Geometry, Mathematical Physics and Applications". 60-76 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大場清: "Higher cycles on the moduli space of stable curves,"Journal of Mathematical Society of Japan.

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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