Generalized Matsushima's theorem and Kahler-Einstein cone metrics

In this paper, we first prove a generalized Matsushima's theorem, i.e. the automorphism group is reductive on a Fano manifold admitting Kahler-Einstein metrics with cone singularities along a smooth divisor. Note that the divisor in our paper is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kahler-Einstein cone metrics by combining the reductivity of the automorphism group with the continuity method, the approximation trick and the bifurcation technique. Moreover, our method provides an existence theorem of Kahler-Einstein cone metrics with respect to conic Ding functional.