Kaehler几何中的典则度量的形变是一个重要的研究课题。本项目主要对这个课题展开研究，主要有三个目标：（1）在容许有smoothable的Fano代数簇上证明K-稳定性蕴含Kaehler-Ricci soliton的存在性。我们计划用Aubin的经典的连续性方法来做。（2）将宋剑在平坦族上的负Kaehler-Einstein度量形变的结果推广到其度量具有锥奇性和cusp奇性的情形。（3）在（2）的基础上研究平坦族上的负Kaehler-Einstein流形特征值的收敛性，这里我们假定中心纤维是具有cusp奇性的Kaehler-Einstein流形。由于具有cusp奇性度量的流形是非紧的，因此会出现连续谱。

Deformation of canonical Kaehler metric is an important subject in Kaehler geometry. The first aim of this project is to show that K-stable implies the existence of Kaehler-Ricci soliton on a smoothable Fano variety. We plane to deal with this problem through the Aubin's continuity method. The second aim of this project is to generalize Jian Song's arguments which concern the degeneration of negative Kaehler-Einstein manifolds on a flat family. We want to establish the degeneration of negative Kaehler-Einstein metrics with conic and cusp singularities. Next, in the view of the second aim, it is necessary to investigate the convergence of eigenvalues of negative Kaehler-Einstein manifolds on a flat family, where we assume that the central fiber has a Kaehler-Einstein metric with cusp singularities. Note that a Kaehler manifold with a cusp metric is noncompact, it may appear continuous spectrum.