Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

This question was answered negatively by Kento Fujita today (at least when $G$ is trivial).

Theorem (Fujita): If $\alpha(X,-K_X)=\frac{n}{n+1}$, then $X$ is K-stable and hence admits a Kähler-Einstein metric.

I see. THis is more subtle. There is no known example. I think it will be impossible or very hard to create one. Vanya

No, this is not sharp. General smooth cubic surface with Eckardt point is an example. Then Aut=Z_2, \alpha_G=2/3 and KE metric exists. If you want very non sharp example, use Kollar's paper http://arxiv.org/abs/math/0507289 Du Val del Pezzo surfaces with A1 and A2 singularities are KE. But their \alpha-invariants are small. See paper of Park and Won: Log canonical thresholds on del Pezzo surfaces of degree >=2, Nagoya Math. J. 200 (2010), 1-26. Vanya