极值 Kahler 度量是复几何领域中的一个重要课题，也是目前国际上复几何和代数几何中主流数学家所关注的热点问题。尽管取得了很大的进展，但人们对于极值 Kahler度量的曲率行为的认识依然非常有限，以及如何验证一个复流形的一致 K- 稳定性，还没有有效的方法。本项目拟在前期工作的基础上，研究复流形上的极值 Kahler 度量的曲率性质和发展证明一致 K-稳定性的行之有效方法。本项目具体的研究内容：1、复 toric 流形和复 toric orbifold 上的极值 Kahler 度量的曲率性质;2、复 toric 流形和复 toric orbifold上的上一致 K-稳定性的证明算法。本项目希望在对上述问题的研究过程中取得实质性进展的同时，发展关于这类问题的新的一些研究方法和技巧，以此来丰富和发展复几何、代数几何。

The extremal Kahler metrics is a very important branch in the research areas of complex geometry and algebraic geometry. However, the research on the curvature property of extremal Kahler metrics has only a few progresses. And the check of uniformly K-stability of a complex manifold is still a difficult problem due to lack of effective methods. Based on the previous studies in this field, this proposal studies the curvature property of extremal Kahler metrics and explore the effective methods to check the stability. In the project, the specific studies are:1、the scalar curvature property of extremal Kahler metrics of complex toric manifold s and complex toric orbifolds ；2、Investigate the effective algorithms and methods to check the uniformly K-stability .While substantial progress on above researching issues, this project aims to find some novel ideas about the methods and techniques to study these issues. Consequently enrich and develop the theory of complex geometry and algebraic geometry.