counterexample to the Chern number inequality on Fano manifold

Question

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$

My question is whether there exists a Fano manifold which does not satisfy this inequality (of course in this case this Fano manifold can not have any K-E metric). I calculated some examples which have been famously known that they don't support K-E metrics and find out that all of them still satisfy this inequality.

Thanks in advance.

Answer 1

I think a counterexample is given by $\mathbb{P}_{P}(\mathcal{O}_{P}\oplus \mathcal{O}_{P}(n-1))$ for $n\geq 4$, with $P:=\mathbb{P}^{n-1}$. The computation is a bit long but here are the main steps (I hope I didn't make mistakes). Let $h\in H^2(X,\mathbb{Z})$ be the class of the tautological bundle, and $f$ the pull back of the class of $\mathcal{O}_{P}(1)$. Standard computations give $$c_1(X)=2h+f\qquad c_2(X)= 2n\,hf-\binom{n}{2}f^2\, \quad\mbox{ hence }$$ $$\Delta :=2(n+1)c_2-nc_1^2=4n\,hf - n^3f^2\ .$$ I want to prove $\Delta .c_1^{n-2}<0$. This is a sum of terms $\Delta .h^pf^{n-2-p}$ with positive coefficients. Note that $h^2=(n-1)hf$ (Chern class relation), so $h^pf^{n-2-p}=(n-1)^{p-1}hf^{n-3}$ for $p\geq 1$. Then $$\Delta .hf^{n-3}=4n(n-1)-n^3\qquad \Delta .f^{n-2}=4n\ .$$ For $n\geq 4$ all the terms are highly negative except $\Delta .f^{n-2}$, one sees immediately that the sum is negative.