• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

1999 Fiscal Year Final Research Report Summary

Geometry of twistor space

Research Project

Project/Area Number 10440020
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

FUJIKI Akira  Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80027383)

Co-Investigator(Kenkyū-buntansha) NAMIKAWA Yoshinori  Graduate School of Science, Osaka University Associated Professor, 大学院・理学研究科, 助教授 (80228080)
MIYANISHI Masayoshi  Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80025311)
SAKANE Yusuke  Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (00089872)
MABUCHI Toshiki  Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80116102)
GOTO Ryushi  Graduate School of Science, Osaka University Lectwer, 大学院・理学研究科, 講師 (30252571)
Project Period (FY) 1998 – 1999
Keywordsself-dual manifold / twistor space / group action / Joyce conjecture / complex manifold / toric surface / algebraic dimension / elliptic fiber space
Research Abstract

1. We have obtained an affimative answer to a conjugate of Joyce to the effect that a simply connected and complete self-dual manifold which admits a smooth action of a 2-torus is diffeomorphic to a connected sum mCP (2) of m copies of complex projective plane and its self-dual structure is isomorphic to one of the examples constructed by Joyce himself in 1995. Our method was to use the twistor space associated to the given self-dual manifold. In fact, more generally, without simple-connectivity assumption we have classified compact self-dual manifolds which admit an action of a two torus. As a byproduct of these investigation we could determine very precise structure as a complex manifold of the twistor space associated to Joyce self-dual metric. In particular we have shown that there exists a nice bimeromorphic model of the twistor space which is realized as a fiber space of torus surfaces and that this latter structure is completely determined by the invariant of the original smooth action of the torus on mCP (2). The twistor space is a Moishezon manifold and it is interesting to construct a natural birational projecitve model.
2. We have studied the deformation space of Joyce twistor space and have shown that its Kuranishi space is nonsingular. Furthermore, using such deformation we have given, for every integer m【greater than or equal】4 the first examples of self-dual structure of positive type on mCP (2) whose twistor space has algebraic dimension two. This result determines distribution of algebraic dimensions of twistor spaces associated to mCP (2). Furthermore, in the case of 4CP (2) we have given a detailed description of the twistor space as a general elliptic fiber space and then by taking its branched covering constructed a interesting family of compact complex manifolds whose canonical bundle is trivial.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Akira Fujiki: "Algebraic reduction of twistor space of Hopf surface"Osaka Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nobuhiro Honda: "Donaidson-Friedman Construction and deformations of a triple of compact complex spaces"Osaka Journal of Mathematics. 36・3. 641-672 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayoshi Miyanishi: "Invariant subvarieties of low codimension in the affine space"Tohoku Mathematics Journal. 52. 11-27 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yusuke Sakane: "Homogeneous Einstein metrics on flag manifolds"Lobachevskii Journal of Mathematics. 4. 71-87 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yasuke Ohyama: "Hypergeometric functions and non-associative algebras"Proceedings of Moonshine Workshop 1999. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ikuo Satake: "Flat Structure and the prepotential for the elliptic root system"Progress in Mathematics. 160. 427-452 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayoshi Miyanishi: "Open algebraic surfaces"American Mathematical Society (to be published).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akira Fujiki: "Algebraic reduction of twistor space of Hopt surfaces"Osaka Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nobuhiro Honda: "Donaldson-Friedman construction and deformations of a triple of compact complex spaces"Osaka Journal of Mathematics. 36-3. 641-672 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masayoshi Miganishi: "Invariant subspaces of low codimensions in the affine space"Tohaku Mathematical Journal. 52. 11-27 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yusuke Sakane: "Homogeneous Einstein metrics on flag manifolds"Lobachevshi Jowsnal of Mathematics. 4. 71-87 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yousuke Ohyama: "Hypergeometric funotims and non-associative algeliras"Roceedings of Moonshine Workshop. (to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ikuo Satahe: "Flat structure and the prepotential for the elliptic root system"Progress in Mathematics. 160. 427-452 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masayoshi Miyanishi: "Open algehraic surfaces"(to be published) American Mathematical Society.

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2001-10-23  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi