Geometry of Ricci solitons on complex manifolds

2010 Fiscal Year Self-evaluation Report

• Project/Area Number: 20244005
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: MABUCHI Toshiki Osaka University, 大学院・理学研究科, 教授 (80116102)
• Project Period (FY): 2008-04-08 – 2013-03-31
• Keywords: 複素多様体

Research Abstract

この研究は,Kahler-Ricci solitonに関する以下の課題に取り組み,その解決をはかることを目指す
(1)二木-小野-Wang がtoric 幾何の立場から佐々木多様体に関連した結果として得たものを,もう少し広い枠組み,すなわちKahler-Ricci solitonをもつFano多様体に関する結果として再構成する
(2)Donaldson-Tian-Yau予想を考察するとともに,AdS/CFT 対応についての種々の予想を考察する
(3)複素射影平面の3点blowing-up上のKahler-Einstein計量や,2点blowing-up上のKahler-Ricci solitonのexplicitな表示を求めたい
このうち(1) については,研究代表者および研究分担者である中川(金沢大)が研究チームを作って解決にあたり,(2)については,研究代表者および平成22年度まで研究分担者であった後藤(阪大)が研究チームを作って解決にあたり,(3)については,研究代表者と研究分担者の佐々木(神戸大)および長谷川(新潟大)研究チームを作って解決にあたる計画である

Research Products

• [Journal Article] Asymptotics of polybalanced metrics under relative stability constraints (2011)
  • Author(s): T.Mabuchi
  • Journal Title: Osaka J.Math.to appear 48 to appear
  • Peer Reviewed
• [Journal Article] Poisson structures and generalized Kahler submanifolds (2009)
  • Author(s): R.Goto
  • Journal Title: J.Math.Soc.Japan 61 Pages: 107-132
  • Peer Reviewed
• [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds II (2009)
  • Author(s): T.Mabuchi
  • Journal Title: Osaka J.Math. 46 Pages: 115-139
  • Peer Reviewed
• [Journal Article] Asymptotic structures in the geometry of stability and extremal metrics, in "Handbook of Geom.Anal.No.1" Internat.Press (2008)
  • Author(s): T.Mabuchi
  • Journal Title: Adv.Lect.in Math. 7 Pages: 277-300
  • Peer Reviewed
• [Journal Article] Chow-stability and Hilbertstability in Mumford's Geometric invariant theory (2008)
  • Author(s): T.Mabuchi
  • Journal Title: Osaka J.Math. 45 Pages: 833-846
  • Peer Reviewed
• [Presentation] On the existence problem of extremal Kahler metrics (2011)
  • Author(s): T.Mabuchi
  • Organizer: Workshop on extremal metrics
  • Place of Presentation: Stony Brook University, USA.
  • Year and Date: 2011-03-24
• [Presentation] Donaldson-Tian-Yau's conjecture and its extremal Kahler version (2010)
  • Author(s): T.Mabuchi
  • Organizer: ICM satellite conference (Geometric topology and Riemannian geometry)
  • Place of Presentation: Bangalore, India.
  • Year and Date: 2010-08-12
• [Presentation] Stability and extremal metrics for projective bundles (2009)
  • Author(s): T.Mabuchi
  • Organizer: 1^<st> PRIMA Congress
  • Place of Presentation: University of New South Wales, Sydney.
  • Year and Date: 2009-07-07
• [Presentation] Characterization of K-polystability by an asymptotic expansion of the Chow norm (2008)
  • Author(s): T.Mabuchi
  • Organizer: Perspectives in Geometric Analysis
  • Place of Presentation: Beijing International Research Center.
  • Year and Date: 2008-10-31
• [Presentation] K-polystability for CSC polarization (2008)
  • Author(s): T.Mabuchi
  • Organizer: Ricci curvature and complex geometry
  • Place of Presentation: Inst.Henri Poincare, Paris.
  • Year and Date: 2008-06-06