Let $\mathcal F$ be an ample subsheaf of $T\_{\mathbb P^n}$. Is it actually locally free? If not, is there a counterexample?



ample subsheaf contained in the tangent bundle of projective space
Yes, $\mathcal{F}$ is locally free. See e.g. Corollary A.11 in [this paper](http://arxiv.org/pdf/0705.4602v2.pdf) of Aprodu, Kebekus, and Peternell.


