Is it a solved problem whether every smooth hypersurface X in CP^n admits a Kaehler-Einstein metric? Of course the c_1<0 and c_1 =0 cases are old, and every Fermat hypersurface does. It seems that existing knowledge implies they all do.
For example they are all Chow-Mumford stable. G. Tian then proved that the K-energy is proper, so if there are no hol. vector fields there is a K-E metric.
I'm reading someone's paper that has a result that gives another proof. But I'm not sure if this paper is correct yet.