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Complex analysis on quasi-Abelian varieties

Research Project

Project/Area Number 13640199
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyushu Sangyo University

Principal Investigator

UMENO Takashi  Kyushu Sangyo University, Faculty of Engineering, Professor, 工学部, 教授 (30098769)

Co-Investigator(Kenkyū-buntansha) FURUSHIMA Mikio  Kumamoto University, Faculty of Science, Professor, 理学部, 教授 (00165482)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsquasi-Abelian variety / Abelian variety / period matrix / toroidal group / de Rham cohomology / ^^-__∂-cohomology / quasi-Abelian variety / Abelian variety / toroidal group / complex torus / periodic matrix / cohomology / Quasi-Abelian Variety / Toroidal Group / Complex Torus / Period Matrix / Cohomology
Research Abstract

1. We obtained all standard forms of period matrices for quasi-Abelian varieties. Using these forms, we got a brief proof of fibration theorems for quasi-Abelian varieties. Further, we have an another proof, which does not depend on the theory of harmonic integrals, of classical Riemann conditions for Abelian varieties.
2. The author gave a talk about the above results at the Fifth International Workshop on Real and Complex Analysis (October 2001, Hiroshima. University). He wrote the paper : Period matrices for quasi-Abelian Varieties, which includes de Rham cohomology of toroidal groups, period matrices and fibration theorems for quasi-Abelian varieties. This will be published in Japanese Journal of Mathematics vol.29-1(2003).
3. From the theorems of period matrices, we found the existence of quasi-Abelian varieties which have principal bundle structures over non-algebraic complex tori. Further we could construct many examples of toroidal groups which have no non-constant meromorphic fu … More nctions on them. This is an extension of a method of Siegel who constructed examples of complex tori which have no non-constant meromorphic functions on them. We could not construct these examples without the computers and the softwares which was brought by the Grant-in-Aid for Scientic Research. Because we needed an enormous amount of calculations of matrices including polynomials.
4. The author gave an lecture about the theory of period matrices as an invited talk at the conference of the mathematical society of Japan, March 2002. Further, he gave a talk about these examples obtained by the theorems of period matrices at the Sixth International Workshop on Real and Complex Analysis (December 2002, Hiroshima University). By these results, We think that the main aims of the research was obtained.
5. As above, we use computers as important tools for a study of mathematics. The method of using computers for mathematics which we have developed will be applicable for mathematical educations. The author gave a talk about a computerization of mathematical education at the conference of information processing education (October 2002, Tokyo University). We think these methods were also obtained by the research. Less

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] Takashi Umeno: "On period matrices for quasi-Abelian varieties"Proceedings of the the Fifth International Workshop on Real and Complex Analysis, Hiroshima University. 92-97 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "トロイダル群-∂^^-コホモロジーと準アーベル多様体-"日本数学会2002年度年会 函数論分科会(特別講演)講演アブストラクト. 59-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "数学教育の情報化"平成14年度情報処理教育研究集会講演論文集,文科省・東京大学. 663-665 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "On quasi-Abelian varieties over non-algebraic complex tori"Proceedings of the Sixth International Workshop on Real and Complex Analysis, Hiroshima_University. 145-149 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "Period matrices for quasi-Abelian varieties"Japanese Journal of Mathematics(to be published). 29・1. (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "On period matrices for quasi-Abelian varieties"Proceedings of the Fifth International Workshop on Real and Complex Analysis. 92-97 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "Toroidal Groups -^^-__∂-cohomology and quasi-Abelian varieties -"Abstracts of talks at the conference of the mathematical society of Japan. 59-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "Computerization of mathematical education"Collected papers of talks at the conference of information processing education, Tokyo University. 663-665 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "On quasi-Abelian varieties over non-algebraic complex tori"Proceedings of the Sixth International Workshop on Real and Complex Analysis. 145-149 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takashi Umeno: "Period matrices for quasi-Abelian varieties"Japanese Journal of Mathematics. 29-1, (to be published). (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 梅野高司: "数学教育の情報化"平成14年度情報処理教育研究集会講演論文集,文科省・東京大学. 663-665 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takashi Umeno: "On quasi-Abelian varieties over non-algebraic complex tori"Proceedings of the Workshop on Real and Complex Analysis, Hiroshima University. 145-149 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takashi Umeno: "Period matrices for quasi-Abelian varieties (to be published)"Japanese Journal of Mathematics. 29・1. (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takashi Umeno: "On period matrices for quasi-Abelian varieties"Proceedings of the Workshop on Real and Complex Analysis, Hiroshima University. 92-97 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takashi Umeno: "トロイダル群…∂コホモロジーと準アーベル多様体…"日本数学会2002年度年会 函数論分科会講演アブストラクト(特別講演). (2002)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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