Bound on the (anticanonical) degree of toric Fano varieties

Question

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ is at most $C^n (n+1)^n$?

This would follow from the weakening of Ehrhart's conjecture that I proposed in Reference request: Ehrhart's conjecture on the geometry of numbers (or at least this is what I understand from reading page 6 of the paper of Nill and Paffenholtz https://arxiv.org/abs/0905.2054).

I wonder:

1. Is this known?

2. Is this interesting?

My knowledge of algebraic geometry is pitiful so please do not be offended if the question is really naive. I'm just trying to see what possible interesting consequences the weakened conjecture could have.

Answer 1

I think the answer to this is no. In fact according to [1] there is no universal polynomial bound on the $n$-th root of $c_1(X)^n$ as X runs over all toric Fanos of dimension n (this is referenced to Debarre but I am afraid I do not have this source with me at the moment).

By contrast the purpose of [1] is to prove that such a bound does exist if you restrict to Kahler-Einstein metrics.

[1] The projective space has maximal volume among all toric Kähler-Einstein manifolds Robert J. Berman, Bo Berndtsson http://arxiv.org/abs/1112.4445

Answer 2

The counterexample of Debarre goes as follows: Consider the projective bundle $$\pi:X=\mathbb{P}(O_{\mathbb{P}^s}^{\oplus r}\oplus O_{\mathbb{P}^s}(a))\to P^s,$$ which is a smooth toric variety. Here $$-K_X\sim (r+1)O_X(1)+(s+1-a)\pi^*H$$ where $H$ is a hyperplane in $\mathbb{P}^s$. If $0\le a\le s$, this is also a Fano variety, as one can check using Kleiman's criterion. Also, using the relation $O_X(1)^{\cdot(r+1)}=a\pi^*H \cdot O_X(1)^{\cdot r}$, Debarre finds after setting $a=n-r$, that $$(-K_X)^n=(-K_X)^{r+s}\ge (r+1)^n (n-r)^{n-r}> \left(\frac{3n^2}{10\log n}\right)^n$$ In particular, the $n$-th root is unbounded as $n\to \infty$.