Study of Kahler Ricci flows

• Project/Area Number: 24540092
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Sophia University
• Principal Investigator: TSUJI Hajime 上智大学, 理工学部, 教授 (30172000)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: ケーラー・リッチ流 / ベルグマン核 / 随伴直線束 / 多重劣調和関数 / ケーラー多様体 / 極小モデル / フランス / パリ / 極値的体積形式 / 標準測度 / 半正値性 / 国際研究交流 / ケーラー・アインシュタイン計量 / 複素多様体

Outline of Final Research Achievements

We study the family of fiberwise Kaehler-Ricci flows on a smooth projective family of projective varieties with pseudoeffective canonical classes. We have proven that if the initial form is a Kaehler form on the total space, the resulting family of Kahler-Ricci flows preserves the semipositivity. The method is the approximation of the Kaehler-Ricci flows in terms of the dynamical system of Bergman kernels and then apply the logarithmic plurisubharmonic variation properties of Bergman kernels due to B. Berndtsson. This resul can be viewed as a refinement of the semiposity of the direct image of pluri adjoint line bundles.

This reserch is a joint project with S. Boucksom in Ecole Polytechnic in Paris.

Research Products

• [Journal Article] On the extremal measure on a complex manifold (2015)
  • Author(s): H. Tsuji
  • Journal Title: Progress in Mathematics Volume: 308 Pages: 159-175
  • Peer Reviewed
• [Journal Article] Some dynamical systems of extremal measures (2015)
  • Author(s): H.Tsuji
  • Journal Title: Proc. of KSCV 10 Volume: 未定
  • Peer Reviewed
• [Presentation] Invariance of plurigenera in K\"ahler case (2015)
  • Author(s): H. Tsuji
  • Organizer: Colloquium talk
  • Place of Presentation: Ecole polytechnique (Paris)
  • Year and Date: 2015-03-12
• [Presentation] Stablity of pluricanonical systems of K\"ahler fibrations (2014)
  • Author(s): H. Tsuji
  • Organizer: 複素幾何学菅平シンポジウム
  • Place of Presentation: 菅平高原 プチホテルゾンタック
  • Year and Date: 2014-11-07
• [Presentation] Some dynamical systems of extremal measures (2014)
  • Author(s): H. Tsuji
  • Organizer: KCSV 10
  • Place of Presentation: 慶州 コロンホテル
  • Year and Date: 2014-08-10
  • Invited
• [Presentation] A new invaraiant measure in complex geometry (2014)
  • Author(s): Ｈａｊｉｍｅ Ｔｓｕｊｉ
  • Organizer: The Shoshichi .Kobayashi Memorial Symposium
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Dynamical systems of extremal measures (2013)
  • Author(s): Hajime Tsuji
  • Organizer: Extremal Kaeher metrics
  • Place of Presentation: CRM (Montreal, Canada)
  • Invited
• [Presentation] Semipositivity of invariant measures (2013)
  • Author(s): Hajime Tsuji
  • Organizer: Rational Points, Rational Curves and Entire Holomorphic Curves on Algebraic Varieties
  • Place of Presentation: CRM (Motreal Canada)
  • Invited
• [Presentation] Constructions of canonical singular hermitian metrics on relative adjoint bundles (2013)
  • Author(s): Hajime Tsuji
  • Organizer: KIAS Complex Geometry Symposium
  • Place of Presentation: KIAS Seoul
  • Invited
• [Presentation] A Schwarz type lemma for canonical measure on a family of projective manifolds with nonnegative Kodaira dimension (2013)
  • Author(s): Hajime TSUJI
  • Organizer: CRM workshop Extremal Kaehler Metrics
  • Place of Presentation: CRM Motreal Canada
  • Invited
• [Presentation] Schawarz type lemma for canonical measures
  • Author(s): 辻 元
  • Organizer: 複素幾何菅平シンポジウム
  • Place of Presentation: 長野県菅平高原
  • Invited
• [Book] 複素多様体論講義 (2013)
  • Author(s): 辻 元
  • Total Pages: 230
  • Publisher: サイエンス社
• [Book] 複素多様体論講義 (2012)
  • Author(s): 辻 元
  • Total Pages: 207
  • Publisher: サイエンス社