Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?

Question

This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along a smooth plane cubic or not. I believe that the answer shoud be "no" by Matsushima's criterion saying that if the automorphism group of the variety is not reductive then there is no such metric.

I was thinking that the following might be a (sketch of) proof: The automorphisms of $\mathbb{P}^3$ fixing a plane (say $x_3=0$) are of the form $$\begin{pmatrix} * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ 0 & 0 & 0 & * \end{pmatrix} $$
This group has projective dimension 12 and since 9 points determines a cubic in $\mathbb{P}^2$ we have that the automorphism group of the blow-up is 3-dimensional. On the other hand, the 3-dimensional unipotent subgroup $$\begin{pmatrix} 1 & 0 & 0 & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ acts trivially on the plane $x_3=0$, so in particular fixes the 9 points determining the cubic and hence lifts to the automorphism group of the blow-up. Since they have the same dimension (but the latter might be non connected), the automorphism group if a finite extension of a unipotent group, hence not reductive.

Am I right? Sorry for being sketchy! Thanks a lot in advance for any comment, I just started to introduce myself to the subject.

PS: By the way, does the fact that $\operatorname{Aut}(X)$ is reductive implies that the connected component of the identity $\operatorname{Aut}^0(X)$ is reductive as well?

Answer 1

No, there are no Kähler—Einstein metrics on the blow-up of $\mathbb{P}^3$ along a plane cubic.

Let $X$ be the blow-up of $\mathbb{P}^3$ along a plane cubic. Then the alpha-invariant $\alpha(X)=1/4$ by Cheltsov—Shramov[1]. On the other hand, the only K-semistable smooth Fano 3-fold with $\alpha=1/4$ is $\mathbb{P}^3$ by my result[2]. So $X$ could not be K-semistable, hence not KE.

There is also another earlier proof by Kento Fujita[3], which shows that the blow-up of $\mathbb{P}^3$ along a plane cubic (which is referred to as No. 28 in Table 2 of Mori--Mukai) is not K-semistable.

Using the same methods you can show that the blow-up of $\mathbb{P}^3$ along a conic or a line is not K-semistable.

References:

[1] Bottom line, Page 952, Log canonical thresholds of smooth Fano threefolds, Russian Math. Surveys 63:5 859--958

[2] K-semistable Fano manifolds with the smallest alpha invariant, Internat. J. Math. 28 (2017), no. 6, 1750044, 9pp

[3] On K-stability and the volume functions of Q-Fano varieties, Proc. LMS 113.5 (2016): 541--582