4
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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

$\endgroup$

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.