Geometry of Ricci solitons on complex manifolds

复流形上 Ricci 孤子的几何

• 批准号: 20244005
• 负责人: MABUCHI Toshiki
• 金额: $ 24.79万
• 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: 2008
• 资助国家: 日本
• 起止时间: 2008-04-08 至 2013-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20244005/
• 关键词: 複素多様体; Ricci soliton; K-安定性; Donaldson-Tian-Yau予想; テスト配位; 佐々木アインシュタイン計量; 偏極射影代数多様体; トーリック微分幾何; 複素解析幾何; モンジュ=アンペール方程式

(1) In a joint work (Tohoku Math. J., 65, 2013, 243-252) with Y. Nakagawa, we generalized Sakane-Koiso's construction of Kaehler-Einstein metrics to the Kaehler-Ricci soliton case where the Futaki invariant is non-vanishing. In this case, we obtain Sasaki-Einstein metrics in place of Kaehler-Einstein metrics.(2) For the Kaehler-Einstein metric on the blowing-up of the complex projective plane at 3 non-colinear points, its detailed description was obtained by asymptotic expansion of the solution of a hyperbolic affine sphere equation on a bounded domain in the real 2-plane (AMS/IP Stud. Adv. Math. 48, 219-229).(3) As to the Donaldson-Tian-Yau Conjecture, we proved: i) Asymptotic relative Chow stability implies the existence of a sequence of polybalanced metrics (Osaka J. Math. 48, 2011, 845-856); ii) strong relative K-stability implies asymptotic relative Chow stability (joint work with Y.Nitta).

(1)在联合工作（Tohoku Math. J.，65，2013，243-252）与Y. Nakagawa的方法，我们将Sakane-Koiso构造的Kaehler-Einstein度量推广到Futaki不变量不为零的Kaehler-Ricci孤子情形。在这种情况下，我们得到Sasaki-Einstein度量代替Kaehler-Einstein度量。(2)对于复射影平面在3个非共线点上爆破的Kaehler-Einstein度量，其详细描述是通过在真实的2-平面中的有界区域上的双曲仿射球方程的解的渐近展开获得的（AMS/IP Stud.Adv.Math.48，219-229）。(3)关于Donaldson-Tian-Yau猜想，我们证明了：i）渐近相对Chow稳定性蕴含着多平衡度量序列的存在性（Osaka J. Math. 48，2011，845-856）; ii）强相对K-稳定性蕴含着渐近相对Chow稳定性（与Y.Nitta的联合工作）。

项目成果

K-polystability for CSC polarization

CSC 极化的 K 多稳定性

• 发表时间: 2008
• 作者: Wu Y;Ghosh S;Nishi Y;Yanase T;Nawata H;Hu Y;T. Mabuchi
• 通讯作者: T. Mabuchi

An energy-theoretic apprcech to the Hitchin-Kobayashi correspondence for manifolds, II

流形 Hitchin-Kobayashi 对应关系的能量理论方法，II

• 发表时间: 2009
• 期刊: Osaka J.Math. 46
• 作者: H. Okamoto;T. Sakajo;and M. Wunsch;山口健治;Takashi Tsuboi;T.Mabuchi
• 通讯作者: T.Mabuchi

Stabitity and extremal metrics for projoctive bundles

投影束的稳定性和极值度量

• 发表时间: 2009
• 作者: S.A Kim;T.Sugawa;T.Mabuchi
• 通讯作者: T.Mabuchi

Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities

• DOI: 10.2969/jmsj/06431005
• 发表时间: 2009-06
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: R. Goto

Donaldson-Tian-Yau's conjecture and its extremal Kahler version

唐纳森-天-丘猜想及其极值卡勒版本

• 发表时间: 2010
• 作者: M. Mizuno;T. Ogawa;T.Mabuchi
• 通讯作者: T.Mabuchi