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2013 Fiscal Year Final Research Report

Destabilizing objects and multiplier ideal sheaves

Research Project

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Project/Area Number 23654023
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionThe University of Tokyo (2012-2013)
Tokyo Institute of Technology (2011)

Principal Investigator

FUTAKI Akito  東京大学, 数理(科)学研究科(研究院), 教授 (90143247)

Project Period (FY) 2011 – 2013
Keywordsアインシュタイン計量 / ケーラー多様体 / K-安定性 / 乗数イデアル層 / リッチ・ソリトン
Research Abstract

The multiplier ideal sheaves appear when the Monge-Ampere equation can not be solved. On the other hand the Futaki invariant is an obstruction to the existence of Kahler-Einstein metrics, and the existence of Kahler-Einstein metrics is reduced to solving the Monge-Ampere equation. It is therefore expected to have some relationship between the multiplier ideal sheaves and the Futaki invariant. Yuji Sano and I obtained results in this direction. As a similar research in the same context, Yuji Sano and I obtained a universal lower bound of the diameter of compact shrinking Ricci solitons, which are the self-similar solutions to the Ricci flow. This lower bound was further improved in a joint work with Hai-Zhong Li and Xiang-Dong Li.

  • Research Products

    (19 results)

All 2013 2012 2011 Other

All Journal Article (7 results) (of which Peer Reviewed: 7 results) Presentation (11 results) Remarks (1 results)

  • [Journal Article] An integral invariant from the view point of locally conformally K¥"ahler geometry2013

    • Author(s)
      A. Futaki, K. Hattori and L. Ornea
    • Journal Title

      Manuscripta Math

      Volume: 140 Pages: 1-12

    • DOI

      10.1007/s00229-011-0527-9

    • Peer Reviewed
  • [Journal Article] On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons2013

    • Author(s)
      A. Futaki, H.Z. Li and X.D. Li
    • Journal Title

      Ann. Glob. Anal. Geom

      Volume: 44 Pages: 105-114

    • DOI

      10.1007/s10455-012-9358-5

    • Peer Reviewed
  • [Journal Article] Lower diameter bounds for compact shrinking Ricci solitons2013

    • Author(s)
      A. Futaki and Y. Sano
    • Journal Title

      Asian J. Math

      Volume: 17, No.1 Pages: 17-31

    • Peer Reviewed
  • [Journal Article] Einstein 計量と GIT 安定性II2012

    • Author(s)
      二木昭人
    • Journal Title

      日本数学会誌「数学」

      Volume: 64巻 Pages: 113-130

    • Peer Reviewed
  • [Journal Article] Multiplier ideal sheaves and geometric problems, Variational Problems in Differential Geometry (Eds. R. Bielawski, K. Houston and M. Speight)2011

    • Author(s)
      A. Futaki and Y. Sano
    • Journal Title

      LMS Lecture Notes series

      Volume: 394 Pages: 68-93

    • Peer Reviewed
  • [Journal Article] Einstein metrics and GIT stability2011

    • Author(s)
      A. Futaki and H. Ono
    • Journal Title

      Sugaku Expositions

      Volume: 24 Pages: 93-122

    • Peer Reviewed
  • [Journal Article] Multiplier ideal sheaves and integral invariants on toric Fano manifolds2011

    • Author(s)
      A. Futaki and Y. Sano
    • Journal Title

      Mathematische Annalen

      Volume: 350 Pages: 245-267

    • DOI

      10.1007/s00208-010-0556-9

    • Peer Reviewed
  • [Presentation] Lower diameter bound for compact shrinking solitons, Extremal K¥"ahler Metrics2013

    • Author(s)
      A. Futaki
    • Organizer
      Centre de recherches math¥'ematiques
    • Place of Presentation
      Universit¥'e de Montr¥'eal
    • Year and Date
      20130526-0601
  • [Presentation] On the book "Transformation groups in Differential Geometry". Geometry and Analysis on Manifolds2013

    • Author(s)
      A. Futaki
    • Organizer
      A Memorial Symposium for Professor Shoshichi Kobayashi--
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      20130522-25
  • [Presentation] K¥"ahler-Einstein metrics and K-stablity2013

    • Author(s)
      A. Futaki
    • Organizer
      Special Seminar
    • Place of Presentation
      University of Bucharest
    • Year and Date
      2013-06-15
  • [Presentation] ケーラー・アインシュタイン計量とK安定性2013

    • Author(s)
      二木昭人
    • Organizer
      熊本大学談話会
    • Year and Date
      2013-04-24
  • [Presentation] Special Lagrangian submanifolds and Lagrangian self-shrinkers in toric Calabi-Yau cones2012

    • Author(s)
      A. Futaki
    • Organizer
      Conformal and K¥"ahler geometry
    • Place of Presentation
      Institut Henri Poincare, Paris, France
    • Year and Date
      2012-12-12
  • [Presentation] 複素微分幾何に現れる積分不変量について2012

    • Author(s)
      二木昭人
    • Organizer
      東京大学談話会
    • Year and Date
      2012-11-16
  • [Presentation] コンパクト多様体上の縮小勾配リッチソリトンの直径について2012

    • Author(s)
      二木昭人
    • Organizer
      広島幾何学研究集会
    • Year and Date
      2012-10-05
  • [Presentation] K¥"ahler-Einstein 計量と GIT安定性2012

    • Author(s)
      二木昭人
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-29
  • [Presentation] Special Lagrangian submanifolds and Lagrangian self-shrinkers in toric Calabi-Yau cones2012

    • Author(s)
      A. Futaki
    • Organizer
      Geometry Seminar
    • Place of Presentation
      University of Hong Kong
    • Year and Date
      2012-03-13
  • [Presentation] Extremal K¥"ahler metrics and GIT stability, Mathematical Science Center2011

    • Author(s)
      A. Futaki
    • Organizer
      Fall Program 2011
    • Place of Presentation
      Tsinghua University, China
    • Year and Date
      20110906-29
  • [Presentation] Integral invariants in complex differential geometry2011

    • Author(s)
      A. Futaki
    • Organizer
      Mathematics Colloquium
    • Place of Presentation
      University of Freiburg, Germany
    • Year and Date
      2011-07-07
  • [Remarks]

    • URL

      http://www.ms.u-tokyo.ac.jp/~afutaki/welcome-jtodai.html

URL: 

Published: 2015-06-25  

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