The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, for $G\subset Aut(X)$ a compact subgroup of the automorphism group, one defines $\alpha_G(X)$ considering only $G$-invariant divisors. The alpha invariant has a corresponding analytic definition involving complex singularity exponents of singular hermitian metrics [2, Appendix].

Tian introduced this invariant and proved that the lower bound $\alpha_G(X)>\frac{n}{n+1}$ implies the existence of a Kähler-Einstein metric [1] (in fact, even today it is one of the few sufficient conditions which is checkable in practice). I'd like to know if this theorem is sharp? That is:

Question: Are there examples Fano manifolds such that $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

I'd also be interested in any partial results in the positive direction.

An example I know of with $\alpha(X)=\frac{n}{n+1}$ is a del Pezzo surface of degree $4$ (this is due to Cheltsov [3]), however by Tian's classification of Kähler-Einstein metrics on del Pezzo surfaces [4], such surfaces are known to admit Kähler-Einstein metrics.


[1] G. Tian. On Kahler-Einstein metrics on certain K ̈ahler manifolds with $c_1(M)>0$.

[2] I. Cheltsov, C. Shramov, Appendix by J. P. Demailly. Log canonical thresholds of smooth Fano threefolds.

[3] I. Cheltsov. Log canonical thresholds of del Pezzo surfaces.

[4] Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class.

  • $\begingroup$ Can you please give a reference for this statement? Thanks a lot. $\endgroup$
    – ACL
    Aug 14 '13 at 17:41
  • 1
    $\begingroup$ No problem, I'll add some references. $\endgroup$ Aug 14 '13 at 18:00

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


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.


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

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    $\begingroup$ If you do not care about G-invariant alpha-invariants, then it is easy to constructs many examples. $\endgroup$ Aug 15 '13 at 7:09
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    $\begingroup$ I don't think this answers the question, which was for examples where $\alpha = \frac n {n+1}$ and no Kähler-Einstein metric exists. $\endgroup$ Aug 15 '13 at 7:15
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    $\begingroup$ Sorry Vanya, perhaps I should have been more clear. Dan is right, what I'd like is an example where $\alpha_G(X)=\frac{n}{n+1}$ but where no Kähler-Einstein metric exists. I'll edit the question to make it clearer. Thanks for the answer, though. $\endgroup$ Aug 15 '13 at 10:26

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