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Transformation groups and geometry

Research Project

Project/Area Number 61460001
Research Category

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

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionUniversity of Tokyo, Faculty of Science, Department of Mathematics

Principal Investigator

HATTORI Akio  University of Tokyo, Dept. of Mathematics, 理学部, 教授 (80011469)

Co-Investigator(Kenkyū-buntansha) FURUTA Mikio  University of Tokyo Dept. of Mathematics, 理学部, 助手 (50181459)
UE Masaaki  University of Tokyo, Dept. of Mathematics, 理学部, 助手 (80134443)
KAWAMATA Yujiro  University of Tokyo, Dept. of Mathematics, 理学部, 助教授 (90126037)
MATSUMOTO Yukio  University of Tokyo, Dept. of Mathematics, 理学部, 助教授 (20011637)
OCHIAI Takusiro  University of Tokyo, Dept. of Mathematics, 理学部, 教授 (90028241)
Project Period (FY) 1986 – 1987
Project Status Completed (Fiscal Year 1987)
Budget Amount *help
¥6,000,000 (Direct Cost: ¥6,000,000)
Fiscal Year 1987: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1986: ¥3,300,000 (Direct Cost: ¥3,300,000)
KeywordsSymplectic manifold / Moment map / Group action / Eixed point / Self-dual connections / Moduli space / Sain manifold / モジュライ空間 / シンプレクティック多様体
Research Abstract

1. Symplectic manifolds and group actions. Symplectic manifolds acted on by the group S^1 and admitting a moment map were mainly investigated. (1) In case the symplectec structure determines a complex line bundle it.was proved that there was a close relation between the weight determined by the line bundle at each fixed point and the value of moment map taken at the same point (Hattori). (2) If all the fixed points are isolated and the action is semi-free then the S^1-manifold coincides essentially with a product of 2-spheres (Hattori).(3) A complete classification of 4-dimensional symplectic S^1-manifolds admitting moment map was derived (Hattori).
2. Moduli spaces of self-dual donnections. (1) The topology of the moduli space of SU(2)-instantons with instanton number 2 was determined (Hattori). (2) The sectional curvature of the natural Riemannian metric on the moduli space of SU(2)-instantons with instanton number 1 were calculated (Matsumoto). (3) Group actions on the moduli spaces of instantons were investigated by Furuta. As an spplication Furuta proved that the Euler number of the moduli space of instanton number l was the number of positive divisors of l (4) As an application of the topology of moduli spaces Furuta proved that homology cobordism group of homology 3-spheres contained an infinite product of infinite cyclic groups.
3. Miscellaneous. (1) Characterization of spheres and projective spaces by project transformation group (Ochiai). (2) Important result on the existence of minimal models for 3-folds (Kawamata). (3) Study on diffeomorphism classification of elliptic surfaces (Matsumoto and Ue). Here the notion of torus fibration introduced by Matsumoto was effectively used. (4) A vanishing theorem of A-genus on span manifolds with zero scalar curvature and non-amenable fundamental group (Ono).

Report

(2 results)
  • 1987 Final Research Report Summary
  • 1986 Annual Research Report

Research Products

(14 results)

All Other

All Publications

  • [Publications] 服部晶夫: Springer tecture Notes in Math.1217. 115-122 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 落合卓四郎: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33. 233-246 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 松本幸夫: Topology. 25. 549-563 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 川又雄二郎: Ann. of Math.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 上正明: Invent. Math.84. 633-643 (1986)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 古田幹雄: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34. 275-297 (1987)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Akio HATTORI: "Almost complex S1-actions on cohomology complex projective spaces" Springer,Lecture Notes in Math.1217. 115-122 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Takushiro OCHIAI (with T. Nagano): "On compact-Riemannian manifolds admitting essential projective transformations" J. Fac. Sci Univ. Tokyo Sect. IA Math.33. 233-246 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Yukio MATSUMOTO: "Diffeomorphism types of elliptic surfaces" Topology. 25. 549-563 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Yujiro KAWAMATA: "The crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces" Ann. of math.to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Masaaki UE: "On the diffeomorphism types of elliptic surfaces with" Invent. Math.84. 633-643 (1986)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 服部晶夫: Sprinqer Lecture Notes in Math.1217. 115-122 (1986)

    • Related Report
      1986 Annual Research Report
  • [Publications] 落合卓四郎,長野正: J.of Fac.Sci.Univ.of Tokyo Sect.IA Math.33. 233-246 (1986)

    • Related Report
      1986 Annual Research Report
  • [Publications] 松本幸夫: Topoloqy. 25. 549-563 (1986)

    • Related Report
      1986 Annual Research Report

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Published: 1987-03-30   Modified: 2016-04-21  

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