Kähler-Einstein metrics with positive scalar curvature

Abstract.

In this paper, we prove that the existence of Kähler-Einstein metrics implies the stability of the underlying Kähler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kähler manifold admits Kähler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kähler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kähler manifolds, provided that the partial C ⁰-estimate posed in [T6] is true.