We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
The continuity method is used to deform the cone angle of a weak conical Kahler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldsons Openness Theorem
on deforming cone angle cite{Don} by combining it with the regularity result of Guenancia-P$breve{text{a}}$un cite{GP} and Chen-Wang cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical Kahler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a emph{weak conical Kahler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).
The existence of emph{weak conical Kahler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2pi$ is obtained by studying a smooth continuity method and a emph{local Mosers iteration} technique. In the case of negative and zero
Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the emph{rough bound} on the approximations to a emph{uniform $C^2$ bound}, thus produce a emph{weak conical Kahler-Einstein} metric. The method used here do not depend on the bound of any background conical Kahler metrics.
