KAHLER-EINSTEIN METRICS WITH POSITIVE SCALAR CURVATURE

In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has posi tive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler -Einstein metrics. Our arguments also yield that the analytic criterio n is satisfied on stable Kahler manifolds, provided that the partial C -0-estimate posed in [TG] is true.