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Algebraic-geometrical and arithmetical study of a quotient space of a Riemannian symmetric space by an arithmetic group

Research Project

Project/Area Number 60540038
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionKyoto University

Principal Investigator

FUJIKI Akira  Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (80027383)

Co-Investigator(Kenkyū-buntansha) KATO Shinichi  Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (90114438)
SAITO Hiroshi  Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (20025464)
AKIBA Tomoharu  Kyoto University, Yoshida College, Professor, 教養部, 教授 (60027670)
TAKEUCHI Akira  Kyoto University, Yoshida College, Professor, 教養部, 教授 (40026761)
FUJI'IE Tatsuo  Kyoto University, Yoshida College, Professor, 教養部, 教授 (10026734)
Project Period (FY) 1985 – 1986
Project Status Completed (Fiscal Year 1986)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1986: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1985: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsHermitian symmetric space / arithmetic subgroup / moduli space / 複素トーラス
Research Abstract

Quotinets of pseudo-hermitian symmetric sapces by certain arithmetic groups are realized often as a moduli space of various types of polarized compact Kahler manifolds, and through this fact their structures are clarified. From this point of view we have obtained the following results.
1. (1) For a general complex tori, we have classifieid the possible types of its rational endomorphism rings, and the corresponding pseudo-hermitian symmetric space. (2) We have obtained the complete classification of finite automorphism groups of complex tori of dimension two, which reflects the structure of the singularity of our quotient space. Especially, we have obtained explicit description of the moduli space of complex tori with fixed abstract group as automorphism group, as a quotient of a pseudo-hermitian space by arithmetic group. (3) We have found that the spaces in (2) are closely related with certain root lattices.
2. (1) As a general structure theorem for compact Kahler symplectic manifolds we have obtained an analogy of Lefschetz-Hodge decomposition theorem, and some remarkable property of a natural 2n-form on the second cohomology group. (2) As an application of (1) we have shown the smoothness of the local moduli space of sufh manifolds; this shows that their moduli space is essentially realized as an open subset of hermitian symmetric space of type IV.
3. In general, using the notion of extremal metrics due to Calabi, we have introduced the notion of stability in analogy with that in algebraic geometry, and obtained an entirely new formulation of the moduli theory. There seems much to be developped in the future in this theory.

Report

(1 results)
  • 1986 Final Research Report Summary
  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] 藤木明: Advanced Studie in Pure Mathematics. 10. 105-165 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] 藤家龍雄: Journal of Mathematics Kyoto University. 26. 95-99 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Fujiki, Akira: "On the de Rham cohomology group of a compact Kahler symplectic manifold" Advanced Studie in Pure Mathematics. 10. 105-165 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Fujiki, Akira: "Finite automorphism groups of complex tori of dimension two"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Fuji'ie, Tatsuo: "Local normality of a meromorphic function and a Picard type theorem" Journal of Mathematics Kyoto University. 26. 95-99 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Kato, Shinichi: "A remark on Maass wave forms attached to real quadratic fields"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Saito, Hitoshi: "Elliptic units and a Kummer's criterion for imaginary quadratic fields"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary
  • [Publications] Saito, Hiroshi: On missing trace values for the eta multipliers,

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1986 Final Research Report Summary

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Published: 1987-03-31   Modified: 2016-04-21  

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