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Hyper Kahler manifolds

Research Project

Project/Area Number 11640076
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka University

Principal Investigator

GOTO Ryushi  Osaka University, Graduate School of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (30252571)

Co-Investigator(Kenkyū-buntansha) NAMIKAWA Yoshinori  Osaka University, Graduate School of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (80228080)
MABUCHI Toshiki  Osaka University, Graduate School of Mathematics, Professor, 大学院・理学研究科, 教授 (80116102)
FUJIKI Akira  Osaka University, Graduate School of Mathematics, Professor, 大学院・理学研究科, 教授 (80027383)
OHYAMA Yosuke  Osaka University, Graduate School of Mathematics, Lecturer, 大学院・理学研究科, 講師 (10221839)
永友 清和  大阪大学, 大学院・理学研究科, 助教授 (90172543)
大野 浩司  大阪大学, 大学院・理学研究科, 助手 (20252570)
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: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsHyper Kahler manifolds / Calabi-Yau manifolds / G_2 manifolds / Spin(7) manifolds / Calabi-Yau多様体 / G_2-多様体 / Spin(7)-多様体 / モノポール / G2多様体 / ミラー対称性
Research Abstract

Let X be a compact Riemannian manifold with vanishing Ricci curvature. Then the list of holonomy group of X includes four interesting classes of the holonomy groups: SU(n), Sp(m), G_2 and Spin(7). The Lie group SU(n) arises as the holonomy group of Calabi-Yau manifolds and Sp(m) is the holonomy group of hyper Kahler manifolds. G_2 and Spin(7) occur as the holonomy groups of 7 and 8 dimensional manifolds respectively. There are many intriguing common properties between these four geometries. The author's research is based on the study of hyper Kahler manifolds, from which he obtains some ideas with the potential to unify moduli spaces results of these different kinds of geometric structures. His main results are the followings:
(1) One of the most remarkable common property is smoothness of the deformation spaces of these geometric structures. The author show that these deformations can be regarded as deformations of special kind of differential forms and he constructs a new kind of deformations theory of these differential forms. Then an obstruction of the deformation is given by the certain exact forms. Hence he shows that the obstruction vanishes in terms of cohomological (topological) argument. This result can be considered as a natural generalization of Kodaira-Spencer theory.
(2) He also constructs the moduli space and shows that the local Torelli's type theorem holds in these cases.
(3) As an application he obtain the smooth moduli space of Calabi-Yau structures and Special Lagrangian submanifolds.

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] 藤木 明: "Topology of Compact Self-dual manifolds"Journal of Math. Soc. Japan. 54. (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 並木 良典: "Deformation theory of Singular Sympletu n-folds"Math. Annalen. 319. 597-623 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 並木 良典: "Extension of 2-forms and Symplatic Varieties"J. Reine Angew Math. 539. 123-147 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 満渕 俊樹: "Vectur field energies and Critical metac on Kahler mfds"Nagoya Math. J. 162. 41-63 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 大山 陽介: "Differential equations for modular forms with level three"Funkeial, E Kvac. 44. 377-389 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Fujiki: "Topology of compact self-dual manifolds"Journal of Math. Soc. Japan. 54. (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Namikawa: "Deformation theory of Singular symplectic n-folds"Math. Anallen.. 319. 597-623 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Namikawa: "Extension of 2-forms and Symplectic varieties"J. Reine Angew. Math.. 1539. 123-147 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Mabuchi: "Vector field energies and Critical metrics on Kahler manifolds"Nagoya Math. J.. 162. 41-63 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Ohyama: "Differential equations for modular forms with level three"Funkcial Ekvac.. 44. 377-389 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 藤木明: "Topology of compact Self-dual manifolds…"Journal of Math.Soc.Japan. 54. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 並河良典: "Deformation theory of Singular Symplectic n-folds"Math.Annalen. 319. 597-623 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 並河良典: "Extension of 2-forms and Symplector Varieties"J.Reine.Angew.Math. 539. 123-147 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 満渕俊樹: "Vector field energies and critical metrics on Kahler mtds"Nagoya Math.J. 162. 41-63 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大山陽介: "Differential equations for modular forms with level three"Funkeial.Ekvac. 44. 377-389 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 藤木明: "Algebraic dimension of tuistor spaces of Hopf Surfaces"Osaka J.Math. 37. 847-858 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 藤木明: "Compact Self-dual manifolds with torus Actions"to appear in Journal of Differenal Geometry.

    • Related Report
      2000 Annual Research Report
  • [Publications] 満渕俊樹: "kahler-Einstein metrics for manifolds with non- Vanishing Futaki characters"to appear in Tohoku math J.

    • Related Report
      2000 Annual Research Report
  • [Publications] 満渕俊樹: "Vector field energies and Critical metrics of Kahler manifolds"to appear in Nagoya math J..

    • Related Report
      2000 Annual Research Report

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

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