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Study of complex analysis from a point of view of parabolic geometry

Research Project

Project/Area Number 18340036
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionThe University of Tokyo

Principal Investigator

HIRACHI Kengo  The University of Tokyo, 大学院・数理科学研究科, 教授 (60218790)

Co-Investigator(Kenkyū-buntansha) KOMATSU Gen  大阪大学, 大学院・理学研究科, 准教授 (60108446)
TAKAYAMA Shigeharu  東京大学, 大学院・数理科学研究科, 准教授 (20284333)
Research Collaborator ROBIN Graham  ワシントン州立大学, 数学科, 教授
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥8,730,000 (Direct Cost: ¥7,200,000、Indirect Cost: ¥1,530,000)
Fiscal Year 2009: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2008: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2007: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsCR幾何 / 共形幾何 / セゲー核 / アインシュタイン計量 / 放物型幾何 / アンビエント計量 / ベルグマン核 / Q-曲率 / ケーラー計量 / 積分不変量 / 放物型不変式論 / 放物型幾何学
Research Abstract

We have given a complete construction of scalar conformal invariants of dimension less than 9 or degree less than 3 ; this was a unsolved problem for even dimensions and is expected to have application to the construction of CR invariants. This result is a consequence of the jet isomorphism theorem that gives a decomposition of the jet space of conformal structures as irreducible tensor spaces ; the isomorphism is defined via the curvature of the ambient metric, a Ricci-flat Lorentz metric associated to conformal structure. On CR geometry, we have formulated an invariant theory that can be applied to describe the Szego kernel.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (27 results)

All 2009 2008 2007 2006 Other

All Journal Article (11 results) (of which Peer Reviewed: 6 results) Presentation (15 results) Remarks (1 results)

  • [Journal Article] Ambient metric const-ruction of CR invariant differential operators, in "Symmetries and Overdeter-mined Systems of Partial Differential Equations2008

    • Author(s)
      Kengo Hirachi
    • Journal Title

      The IMA volumes in mathematics and its applications 144

      Pages: 61-76

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Inhomogeneous Ambient Metrics, in "Symmetries and Overdetermined Systems of Partial Differential Equations2008

    • Author(s)
      C. Robin Graham, Kengo Hirachi
    • Journal Title

      The IMA volumes in mathematics and its applications 144

      Pages: 403-420

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Ambient metric construction of CR invariant differential operators, in Symmetries and Overdetermined Systems of Partial Differential Equations2008

    • Author(s)
      K. Hirachi
    • Journal Title

      The IMA volumes in mathematics and its applications 144

      Pages: 61-76

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Inhomogeneous Ambient Metrics, in Symmetries and Overdetermined Systems of Partial Differential Equations2008

    • Author(s)
      C.R. Graham and K. Hirachi
    • Journal Title

      The IMA volumes in mathematics and its applications 144

      Pages: 403-420

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Inhomogeneous ambient metrics2008

    • Author(s)
      Graham, C.R. and Hirachi, K.
    • Journal Title

      IMA Vol. Math. Appl. 144

      Pages: 403-420

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ambient metric construction of CR invariant differential operators2008

    • Author(s)
      Hirachi, K.
    • Journal Title

      IMA Vol. Math. Apol. 144

      Pages: 61-75

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ambient metric construction of CR invariant differential operators2008

    • Author(s)
      平地健吾
    • Journal Title

      The IMA vol. in math. and its applications 144

      Pages: 61-76

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic singularity of the Szego kernel and a global invar-iant of strictly pseudoconvex domains2006

    • Author(s)
      Kengo Hirachi
    • Journal Title

      Ann.of Math. 163

      Pages: 499-515

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains2006

    • Author(s)
      K. Hirachi
    • Journal Title

      Ann. Of Math 163

      Pages: 499-515

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Logarithemic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains2006

    • Author(s)
      平地 健吾
    • Journal Title

      Annals of Mathematics 163

      Pages: 389-405

    • Related Report
      2006 Annual Research Report
  • [Journal Article] セゲー核の対数項とRamadanov予想

    • Author(s)
      平地健吾
    • Journal Title

      京都大学数理解析研究所講究録 (掲載確定)

    • Related Report
      2009 Annual Research Report
  • [Presentation] Integral Kahler invariants and the singularity of the Szego kernel2009

    • Author(s)
      Kengo Hirachi
    • Organizer
      Programme for the d-bar Neumann problem
    • Place of Presentation
      ESI (Austria)
    • Year and Date
      2009-12-15
    • Related Report
      2009 Final Research Report
  • [Presentation] Integral Kahler invariants and the singularity of theSzego kernel2009

    • Author(s)
      K.Hirachi
    • Organizer
      Programme for the d-bar Neumann problem
    • Place of Presentation
      ESI(オーストリア)
    • Year and Date
      2009-12-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] セゲー核の対数項とRamadanov予想2009

    • Author(s)
      K.Hirachi
    • Organizer
      ポテンシャル論とべルグマン核
    • Place of Presentation
      京都大学
    • Year and Date
      2009-12-02
    • Related Report
      2009 Annual Research Report
  • [Presentation] Quadratic invariants of conformal structures2009

    • Author(s)
      K.Hirachi
    • Organizer
      Geometry and PDE Seminar
    • Place of Presentation
      ワシントン大学(米国)
    • Year and Date
      2009-10-28
    • Related Report
      2009 Annual Research Report
  • [Presentation] Fefferman-Graham's ambient metric and scalar invariants of conformal structures2009

    • Author(s)
      Kengo Hirachi
    • Organizer
      PDE Seminar
    • Place of Presentation
      MIT (USA)
    • Year and Date
      2009-03-18
    • Related Report
      2009 Final Research Report
  • [Presentation] Scalar Invariants for Even Dimensional Conformal Structures2009

    • Author(s)
      Kengo Hirachi
    • Organizer
      Geometric PDE seminar, Institute for Advanced Study
    • Place of Presentation
      Princeton (USA)
    • Year and Date
      2009-02-03
    • Related Report
      2009 Final Research Report
  • [Presentation] Scalar Invariants for Even Dimensional Conformal Structures2009

    • Author(s)
      Kengo Hirachi
    • Organizer
      Geometric PDE seminar, Institute for Advanced Study, Princeton
    • Place of Presentation
      USA
    • Year and Date
      2009-02-03
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Scalar Invariants for Even Dimensional Conformal Structures2009

    • Author(s)
      平地健吾
    • Organizer
      Geometric PDE seminar
    • Place of Presentation
      Institute for A6anced Study, Princeton, USA
    • Year and Date
      2009-02-03
    • Related Report
      2008 Annual Research Report
  • [Presentation] 佐々木多様体上のセゲー核とQ-曲率2008

    • Author(s)
      平地健吾
    • Organizer
      葉層の微分幾何とベルグマン核
    • Place of Presentation
      京都大学
    • Year and Date
      2008-12-15
    • Related Report
      2007 Annual Research Report
  • [Presentation] Scalar invariants for even dimensional conformal structures2008

    • Author(s)
      平地健吾
    • Organizer
      Parabolic Geometry and PDE
    • Place of Presentation
      University of Auckland, NewZealand
    • Year and Date
      2008-08-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] Ambient realization of the conformal deformation complex2008

    • Author(s)
      Kengo Hirachi
    • Organizer
      Conformal Geometry : invariant theory and the variational method
    • Place of Presentation
      Roscoff (France)
    • Year and Date
      2008-07-13
    • Related Report
      2009 Final Research Report
  • [Presentation] Ambient realization of the conformal deformation complex2008

    • Author(s)
      平地健吾
    • Organizer
      Conformal Geometry : invariant theory and the variational method
    • Place of Presentation
      Roscoff, France.
    • Year and Date
      2008-07-01
    • Related Report
      2008 Annual Research Report
  • [Presentation] Q-curvature in CR geometry2007

    • Author(s)
      Kengo Hirachi
    • Organizer
      Midwest Geometry Conference
    • Place of Presentation
      University of Iowa (USA)
    • Year and Date
      2007-05-19
    • Related Report
      2009 Final Research Report
  • [Presentation] Jet isomotphism theorem in conformal geometry2007

    • Author(s)
      Kengo Hirachi
    • Organizer
      Asymptotic structures in geometric analysis
    • Place of Presentation
      Stanford Univ. (USA)
    • Year and Date
      2007-03-17
    • Related Report
      2009 Final Research Report
  • [Presentation] The ambient metric to all orders in even dimensions2006

    • Author(s)
      Kengo Hirachi
    • Organizer
      8th Pacific Rim Geometry Conference
    • Place of Presentation
      South Durras (Australia)
    • Year and Date
      2006-12-13
    • Related Report
      2009 Final Research Report
  • [Remarks]

    • URL

      http://www.ms.u-tokyo.ac.jp/~hirachi

    • Related Report
      2009 Final Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

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