Existence Problem of Fxtremal Metran and Degeneration of Balanced Metrics

极值Metran的存在问题和平衡度量的退化

• 批准号: 16204006
• 负责人: MABUCHI Toshiki
• 金额: $ 17.8万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16204006/
• 关键词: geometry; extremal metric; stability; balanced metric; degeneration; constant scalar curvature; test configuration; obstruction; Test configuration; Hitchin-Kobayashi対応; 定スカラー曲率ケーラー計量; 漸近安定性; Chow-mumford安定性; Hilbert-mumford安定性; extremal-Kah; geometry; extremal metric; stability; balanced metric; degeneration; constant scalar curvature; test configuration; obstruction; Test configuration; Hitchin-Kobayashi対応; 定スカラー曲率ケーラー計量; 漸近安定性; Chow-mumford安定性; Hilbert-mumford安定性; extremal-Kah

(1) Related to the existence problem of extremal metrics, we studied various kinds of stabilities for manifolds For instance, we succeeded in showing that Chow-Mumford stability and Hilbert-Mumford stability are asymptotically equivalent (Chow-Mumford stability implies HiItert-Mumford stability by the work of Fogarty, while nothing was known even asymptotically about its converse).(2) Associated to the Monge-Ampere equation for the Kahler-Einstein metric on an Einstein toric surface, we have a hyperbolic affine sphere equation with Dirichlet condition defined on a bounded convex C^<2> domain in R^<2>, and for the solution of the equation, we obtain a very explicit asymptotic expansion along the boundary.(3) It is known that the anticanonical bundle of a toric or Kahler Einstein Fano manifold admits a Ricci-flat Kahler metric inducing a Sasaki-Einstein metric on a suitable quotient. We studied analogous examples for anticanonical bundles of Fano manifolds with Kahler-Ricci solitons.

(1) Related to the existence problem of extremal metrics, we studied various kinds of stabilities for manifolds For instance, we succeeded in showing that Chow-Mumford stability and Hilbert-Mumford stability are asymptotically equivalent (Chow-Mumford stability implies HiItert-Mumford stability by the work of Fogarty, while nothing was known even asymptotically about its converse).(2) Associated to the Monge-Ampere Kahler-Einstein度量的方程式在爱因斯坦复曲面上，我们具有在r^<2>中定义的dirichlet条件的双曲线仿射球方程，在r^<2>中定义了dirichlet条件，对于方程式解决方案，我们的方程式是在方程式的解决方程，我们获得了一个非常明显的沿边界的范围或buns的buns。歧管承认ricci-flat kahler度量标准在合适的商上诱导sasaki-einstein度量。我们研究了与Kahler-Icci Solitons的Fano歧管的抗议捆绑包相似的例子。