| In this dissertation we prove certain deformation behaviors of Kahler-Einstein metrics with singularities. Based on smooth approximation and a priori estimates, we first give a new proof to Donaldson's Openness Theorem of deforming the cone angle of conical Kahler-Einstein metric on smooth Fano manifold, and prove a direct generalization to conical Kahler-Einstein metrics along simple normal crossing pluri-anti-canonical divisors on smooth Fano manifold. Then we use continuity method to study the deformation of weak conical Kahler-Einstein metric on Q-Fano variety. The idea of our new proof above is generalized to prove the openness part of the continuity method argument, while the closedness part directly follows from the weak compactness result of Chen-Donaldson-Sun. |
