Let $X$ be a fake projective plane, namely, a compact complex surface with $$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi conjecture implies that $X$ is a compact quotient of the unit ball $\mathbb{B}^2 \subset \mathbb{C}^2$, hence its cotangent bundle $\Omega_X$ is ample, too.
By Dolbeault isomorphism one has $0=h^1(X, \, \mathcal{O}_X)=h^0(X, \, \Omega_X)$, namely, the cotangent bundle of $X$ has no global sections at all. On the other hand, by general results on ample vector bundles, there exists $N \geq 2$ such that the symmetric power $S^N \Omega_X$ is generated by global sections for all $n \geq N$.
Question. Is an explicit value of $N$ known, at least for some particular fake projective plane?
