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

Geometric Non-linear problems and stability in invariant theory

Research Project

Project/Area Number 14204002
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology

Principal Investigator

FUTAKI Akito  Tokyo Institute of Technology, Department of Mathematics, Professor, 大学院・理工学研究科, 教授 (90143247)

Co-Investigator(Kenkyū-buntansha) ITOH Mitsuhiro  Tsukuba University, Institute of Mathematics, Professor, 大学院・数理物質科学研究科, 教授 (40015912)
MABICHI Toshiki  Osaka University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (80116102)
NAKAJIMA Hiraku  Kyoto University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (00201666)
LI Wein  Kyushu University, Department of Mathematics, Assisrant Professor, 大学院・数理科学研究院, 助教授 (60304002)
TSUJI Hajime  Sophia University, Department of Mathematics, Professor, 理工学部, 教授 (30172000)
森田 茂之  東京大学, 大学院・数理科学研究科, 教授 (70011674)
小林 亮一  名古屋大学, 大学院・多元数理研究科, 教授 (20162034)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥49,140,000 (Direct Cost: ¥37,800,000、Indirect Cost: ¥11,340,000)
Fiscal Year 2005: ¥11,570,000 (Direct Cost: ¥8,900,000、Indirect Cost: ¥2,670,000)
Fiscal Year 2004: ¥11,570,000 (Direct Cost: ¥8,900,000、Indirect Cost: ¥2,670,000)
Fiscal Year 2003: ¥11,570,000 (Direct Cost: ¥8,900,000、Indirect Cost: ¥2,670,000)
Fiscal Year 2002: ¥14,430,000 (Direct Cost: ¥11,100,000、Indirect Cost: ¥3,330,000)
Keywordsscalar curvature / K-stability / geometric invariant theory / moment map / ケーラー多様体 / トーラス埋め込み / ケーラー計量 / 同変コホモロジー / 不変コホモロジー
Research Abstract

The purpose of this research is to find a necessary and sufficient condition for the existence of Kaehler metrics of constant scalar curvature in terms of stability in geometric invariant theory. There is a description of stability expressed using moment maps on symplectic manifolds. Namely orbits having a zero of the moment map are the stable orbits. From this view point it is important to study the space of all complex structures compatible with a given symplectic form, which is an infinite dimensional symplectic manifold and of which the scalar curvature is the moment map. I developed a perturbation theory of this fact and studied the infinite dimensional symplectic manifold from perturbed structures. Secondly T.Mabuchi proved that asymptotic Chow stability is equivalent to asymptotic Hilbert scheme stability.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (33 results)

All 2005 2004 2003 Other

All Journal Article (20 results) Publications (13 results)

  • [Journal Article] Canonical metrics in K"ahler geometry and stability in GIT2005

    • Author(s)
      A.Futaki
    • Journal Title

      Seminar on Mathematical Sciences 30

      Pages: 68-77

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] The modified Yamabe problem and geometry of modified scalar curvatures2005

    • Author(s)
      M.Itoh
    • Journal Title

      J. Geom. Anal. 15

      Pages: 63-81

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II2005

    • Author(s)
      T.Mabuchi
    • Journal Title

      Invent. Math. 159

      Pages: 225-243

    • NAID

      120005986685

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. II. $K$-theoretic partition function.2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Transform. Groups 10

      Pages: 489-519

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Non-abelian zeta functions for function fields2005

    • Author(s)
      L.Weng
    • Journal Title

      Amer. J. Math. 127

      Pages: 973-1017

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Canonical motrics in K" ahler geometry and stability in GIT2005

    • Author(s)
      A.Futaki
    • Journal Title

      Seminar on Mathematical Sciences 30

      Pages: 68-77

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The modified Yamabe problem and geometry of modified scalar curvatures2005

    • Author(s)
      M.Itoh
    • Journal Title

      J.Geom.Anel. 15

      Pages: 63-81

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II2005

    • Author(s)
      T.Mabuchi
    • Journal Title

      Invent.Math. 159

      Pages: 225-243

    • NAID

      120005986685

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. II. $K$-theoretic partition function.2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Transform.Groups 10

      Pages: 489-519

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-abelian zeta functions for function fields2005

    • Author(s)
      L.Weng
    • Journal Title

      Amer.J.Math. 1272

      Pages: 973-1017

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The modified Yamabe problem and geometry of modified scalar curvatures2005

    • Author(s)
      M.Itoh
    • Journal Title

      J.Geom.Anal. 15

      Pages: 63-81

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Non-abelian zeta functions for function fields2005

    • Author(s)
      L.Weng
    • Journal Title

      Amer.J.Math. 127

      Pages: 973-1017

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Signatures of foliated surface bundles and the symplectomorphism groups of surfaces2005

    • Author(s)
      D.Kotschick, S.Morita
    • Journal Title

      Topology 44

      Pages: 131-149

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Subadjunction theorem2004

    • Author(s)
      H.Tsuji
    • Journal Title

      Adv. Stud. Pure Math. 42

      Pages: 313-318

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Subadjunction theorem2004

    • Author(s)
      H.Tsuji
    • Journal Title

      Adv.Stud.Pure Math. 42

      Pages: 313-318

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Asymptotic Chow semi-stability and integral invariants2004

    • Author(s)
      A.Futaki
    • Journal Title

      International Journal of Mathematics 15

      Pages: 967-979

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I2004

    • Author(s)
      T.Mabuchi
    • Journal Title

      Invent.Math. 159

      Pages: 225-243

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Crystal bases and two-sided cells of quantum affine algebras2004

    • Author(s)
      J.Beck, H.Nakajima
    • Journal Title

      Duke Math.J. 123

      Pages: 335-402

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Quasi-projectivity of moduli spaces of polarized varieties2004

    • Author(s)
      G.Schumacher, H.Tsuji
    • Journal Title

      Annals of Mathematics 159

      Pages: 597-639

    • Related Report
      2004 Annual Research Report
  • [Journal Article] L^2-metrics, projective flatness and families of polarized abelian varieties2003

    • Author(s)
      W.-K.To, L.Weng
    • Journal Title

      Trans, Amer.Math.Soc. 356

      Pages: 2685-2707

    • Related Report
      2004 Annual Research Report
  • [Publications] 二木昭人: "微分幾何とディラク作用素"数理科学. 490(未定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Morita: "Generators for the tautological algebra of the moduli spaces of curves"Topology. 42. 787-819 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Mabuchi: "Multipier Hermitian structures on kahler manifolds"Nagoya Math.J.. 170. 73-115 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Kobayashi: "Towards Nevalinna Theory as a geometric model for Diophantine approximation"Sugaku Expositions. 16. 39-79 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] G.Schumacher, H.Tsuji: "Quasi-projectivity of moduli spaces of polarized projective varieties"Annals of Math.. 159(未定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Nakajima: "t-analogues of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras"Representation Theory. 7. 259-274 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 二木昭人: "微分幾何講義"サイエンス社. 182 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Futaki: "Some invariant and equi variant cohomology classes of the space of kahler metrics"Proc. Japan A cad.. vol78 No.3. 27-29 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Futaki, T.Mabuchi: "Moment maps and symmetric multilinear forms associated with symplectic classes"Asian J. Math.. 6. 349-371 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Mabuchi, Y.Nakagawa: "The Bando-Calabi-Futaki character as an obstruction to semi stability"Math. Ann.. 324. 187-193 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Mabuchi: "Heat kermel estimates and the Green functiono on multiplier Hermitian maniholds"Tohoku Math. J.. 54. 261-277 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Itoh: "contact metric 5-manifolds, CR twistor spaces and integrability"J. Math. Phys.. 43. 3783-3797 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Tsuji: "Deformation invariance of pluri genera"Nagoya Math. J.. 166. 117-134 (2002)

    • Related Report
      2002 Annual Research Report

URL: 

Published: 2002-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi