具有高对称性流形上极值Kaehler度量的研究

The study of extremal Kaehler metrics is one of the central problems in Kaehler geometry. It is conjectured the existence of extremal Kaehler metrics are related to the stability theory in geometric invariant theory. The problem would always be simplified if the manifolds have large symmetries. The problem of the existence of extremal Kaehler metrics on toric manifolds and homogeneous toric bundles can be boiled down to solving the (generalized) Abreu PDEs in Delzant polytopes. This project intends to carry out the following studies on the basis of previous work. 1. The study of K-stabilities of toric manifolds and homogeneous toric bundles. In this part we try to compare the concepts about various K-stabilities of toric manifolds and homogeneous toric bundles, and give a sufficient condition of K-stability of toric manifolds and homogeneous toric bundles by the combinatorial properties of Delzant polytopes, and then give new examples of Kaehler manifolds which have and does not have extremal Kaehler metrics. 2. The study of the regularity theory of high dimensional Abreu equations and generalized Abreu equations. 3. The study of the scalar curvature equation of group compactifications.

极值Kaehler度量的研究是Kaehler几何的核心问题之一。极值Kaehler度量的存在性猜想与几何不变量理论中的稳定性理论有关。如果流形具有好的对称性常常会简化问题。toric流形和齐性toric丛上极值Kaehler度量的存在性问题可以归结为求解Delzant多面体上的（广义）Abreu方程。本项目拟在前期工作的基础上开展以下方面的研究：1. toric流形和齐性toric丛上K-稳定性的研究。这一方面我们试图得到toric流形和齐性toric丛上各种K-稳定性之间的关系，并且通过Delzant多面体的组合性质给出其对应的toric流形和齐性toric丛K-稳定的一个充分条件，从而给出新的具有极值Kaehler度量和不存在极值Kaehler度量的Kaehler流形的例子。2. 高维Abreu方程和广义Abreu方程的正则性的研究。3. 对约化复李群紧化空间的数量曲率方程进行研究。

极值Kaehler度量的研究是Kaehler几何的核心问题之一。极值Kaehler度量的存在性猜想与几何不变量理论中的稳定性理论有关。toric流形和齐性toric丛上极值Kaehler度量的存在性问题可以归结为求解Delzant多面体上的（广义）Abreu方程。本项目在前期工作的基础上和合作者开展以下方面的研究：1. 证明了toric流形上预定数量曲率度量的存在性与滤子意义下的稳定性是等价的。2. 证明了齐性toric丛上预定数量曲率度量的存在性与滤子意义下的稳定性是等价的。3. 证明了不稳定toric流形的最优不稳定化子由filtration给出。其结果被菲尔兹奖得主Donaldson在文章“Extremal Kähler metrics and convex analysis”中引用。