10
$\begingroup$

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.

References:

[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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Ruadhaí Dervan Aug 14 '13 at 18:00
7
$\begingroup$

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

$\endgroup$
8
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ If you do not care about G-invariant alpha-invariants, then it is easy to constructs many examples. $\endgroup$ – Ivan Cheltsov Aug 15 '13 at 7:09
  • 5
    $\begingroup$ I don't think this answers the question, which was for examples where $\alpha = \frac n {n+1}$ and no Kähler-Einstein metric exists. $\endgroup$ – Dan Petersen Aug 15 '13 at 7:15
  • 2
    $\begingroup$ 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. $\endgroup$ – Ruadhaí Dervan Aug 15 '13 at 10:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.