Localization of Ricci form and the existence of an anti-canonical divisor on asymptotically Chow stable Fano manifolds

• Project/Area Number: 23654025
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: KOBAYASHI Ryoichi 名古屋大学, 多元数理科学研究科, 教授 (20162034)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 複素モンジュアンペール方程式 / ケーラー計量の空間 / モンジュアンペールエネルギー汎関数 / 修正リッチ流の離散化 / ケーラー計量の力学系 / 定スカラー曲率ケーラー計量 / CSCK計量 / 修正ケーラーリッチ流 / モンジュアンペール方程式 / 離散化 / 修正リッチイテレーション / 代数的極小曲面のガウス写像 / 放物型変換が惹き起こす局所化 / 周期条件 / リッチ平坦完備ケーラー計量 / スカラー平坦完備ケーラー計量 / MA方程式の解の無限遠での漸近挙動 / ケーラーリッチ流 / ケーラーリッチフロー / スカラー曲率 / ハミルトン変形

Outline of Final Research Achievements

cscK metric is not directly expressed as a solution of a Monge-Ampere equation. However, cscK metric is a critical point of the Monge-Ampere energy functional on the space of Kaehler metrics. It is expected that the study of csck metric is deeply connected to that of Monge-Ampere equations. I discovered a dynamical system on the space of Kahler metrics which discretizes the modified KRF (the evolution equation obtained by replacing -Ric(ω)+ω in the RHS of KRF by -Ric(ω)+HRic(ω), where H means the harmonic part). This dynamical system consists of iteration of the following procedure. First we solve a Poisson equation for u so that the trace of Ric(ω)+ddc u becomes constant. Next we solve the Monge-Ampere equation Ric(ω(φ))=Ric(ω)+ddc u for φ to get a Kaehler metric whose Ricci form coincides with Ric(ω)+ddc u. This procedure constitutes the basic step of out dynamical system. The iteration of this procedure is our dynaical system. This dynakical system fixes cdcK metric.

Research Products

• [Presentation] Metrization of Osserman’s theory on the Gauss map of algebraic minimal surfaces (2015)
  • Author(s): Ryoichi Kobayashi
  • Organizer: Distribution of Rational Points and Rational Curves in Algebraic Varieties
  • Place of Presentation: BIRS, Banff, Canada
  • Year and Date: 2015-03-15 – 2015-03-20
  • Invited
• [Presentation] 満淵汎関数 (2015)
  • Author(s): 小林亮一
  • Organizer: 満渕俊樹教授退職記念シンポジウム
  • Place of Presentation: 大阪大学
  • Year and Date: 2015-03-12 – 2015-03-13
  • Invited
• [Presentation] Localization arising from iteration of parabolic translation and collective Cohn- Vossen inequality (2014)
  • Author(s): Ryoichi Kobayashi
  • Organizer: 5-th International Congference on Differential Geometry and Analysis,
  • Place of Presentation: Karatsu
  • Year and Date: 2014-05-31 – 2014-06-04
  • Invited
• [Presentation] Localization via parabolic translation and collective Cohn-Vossen inequality (2014)
  • Author(s): Ryoichi Kobayashi
  • Organizer: 5th international workshop on difgferential geometry and analysis
  • Place of Presentation: 唐津 虹の松原ホテル
  • Invited
• [Presentation] Relations among various invariants of algebraic minimal surfaces (2013)
  • Author(s): Ryoichi Kobayashi
  • Organizer: 19-th international Symposium on Complex Geometry
  • Place of Presentation: 菅平
  • Invited
• [Presentation] On Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2013)
  • Author(s): Ryoichi Kobayashi
  • Organizer: GeoQuant
  • Place of Presentation: ESI, Wien
  • Invited
• [Presentation] Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2013)
  • Author(s): Ryoichi Kobayashi
  • Organizer: OCAMI-TIMS Joint International Workshop on Diff. Geom. and Geom. Anal.
  • Place of Presentation: 大阪市立大学
  • Invited
• [Presentation] Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: 葉山多変数関数論国際シンポジウム
  • Place of Presentation: 葉山
  • Invited
• [Presentation] Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: 菅平複素幾何国際シンポジウム
  • Place of Presentation: 菅平
  • Invited
• [Presentation] Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: China-Japan Geometry Conference
  • Place of Presentation: 四川大学
  • Invited
• [Presentation] Hamiltonian volume minimizing property of $U(1)^n$-orbits in $P^n(C)$ (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: Conference M.Schlichenmaier's 60th birthday
  • Place of Presentation: Univ. Luxembourg
  • Invited
• [Presentation] Nevanlinna theory from the view point of lemma on logarithmic derivative (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: Topology of Algebraic Varieties
  • Place of Presentation: CRM, Univ. Montreal
  • Invited
• [Presentation] Hamiltonian K\"ahler-Ricci flow, self-similar solutions and a generalization of Futaki's invariant (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: Topology of Algebraic Varieties
  • Place of Presentation: CRM, Univ. Montreal
  • Invited
• [Presentation] Global Hamiltonian Stability of Maximal Torus Orbits in the Complex Projective Space (2012)
  • Author(s): Ryoichi Kobayashi
  • Organizer: Several Complex Variables
  • Place of Presentation: Steklov Mathematical Institute
• [Presentation] Singular Perturbation of Calabi-Yau Theorem and Conical Metrics of Constant Positive Curvature on Compact Riemann Surfaces (2011)
  • Author(s): Ryoichi Kobayashi
  • Organizer: The 7-th International Symposium on Complex Geometry
  • Place of Presentation: Sugadaira
• [Book] リッチフローと幾何化予想 (2013)
  • Author(s): 小林亮一
  • Total Pages: 350
  • Publisher: 培風館