黎曼流形的收敛是大范围微分几何研究中的重要工具，极限空间的奇点结构蕴含了流形序列的丰富信息。里奇曲率有下界流形的紧性理论近年来有很大发展，这些结果在几何问题上有很多应用。Fano流形上著名的Yau-Tian-Donaldson猜想联系了凯勒-爱因斯坦度量的存在性和K-稳定性。紧性理论在猜想的解决中发挥了重要作用，其中的部分C^0估计的证明就依赖于对极限空间的奇点结构的刻画。我们在最近的工作中将凯勒-爱因斯坦度量的存在性推广到了一类代数簇上，深化了度量存在与稳定性的关系。在紧性理论以及凯勒几何的这些最新进展之上，本项目将研究在Gromov-Hausdorff拓扑下相关几何量的连续性，例如里奇曲率有界流形序列曲率的平方积分的连续性；带锥奇点的里奇流的紧性理论；以及奇异代数簇上度量的存在性，例如一般K稳定代数簇上的凯勒-爱因斯坦度量，达到最大里奇曲率下界的带锥奇点度量等的存在性等。

Gromov-Hausdorff convergence is the basic tool for the study in the area of global diffetential geometry, and the structure of the singular set in the limit space contains abundant information of the sequence of manifold. The theory on the limit space of manifolds with Ricci curvature bounded from below has a great development in recent years. These results have many applications in geometric problems. The famous Yau-Tian-Donaldson conjecture for Fano manifolds connectes the existence of Kahler-Einstein metrics and K-polystable. The compactness theory play a great role in the solution and the proof of partial C^0 estimate depends on the structure of the limit space. In recent work, we extend the existence of Kahler-Einstein metrics to a class of singular varieties. This deepens the connections of existence of metrics and the algebraic stability. Based on the development of compactness theory and Kahler geometry, in this program we will study the continuity of some geometric quantities in the Gromov-Hausdorff topology, such as the integral of the square of the curvature of a sequence of manifold with bounded Ricci curvature; the compactness theory of the conic Ricci flow; and the existence of metric on singular varieties, such as the existence of Kahler-Einstein metric on the K-polystable varieties and the conic Kahler-Einstein metric whose angle achieves the greatest Ricci lower bound.