Let $\mathcal{F}$ be a saturated coherent subsheaf of $T\mathbb{CP}^n$. In particular, $\mathcal{F} \subset T\mathbb{CP}^n$ is a holomorphic vector subbundle outside a subset $Z \subset \mathbb{CP}^n$ of complex codimension $2$. Assume that $c\_1(\mathcal{F}) = 0$.


**Question:** Does there exists a non-zero holomorphic vector field $X \in H^0(T\mathbb{CP}^n)$ which is tangent to $\mathcal{F}$ at every point?


By being tangent I mean that the sheaf generated by the vector field $\mathcal{O}\_{\mathbb{CP}^n} \cdot X$ is contained in $\mathcal{F}$. Another equivalent formulation of the Question (as I believe it follows from Hartog's) is the following.


Does the vector bundle $\mathcal{F}|\_{\mathbb{CP}^n \setminus Z}$ admits a non-zero holomorphic section?



Vector fields tangent to distributions with zero first Chern class
Not in general.


Let $F \in H^0(\mathbb P^3, \mathcal O\_{\mathbb P^3}(3))$ be a general cubic form and let $H \in H^0(\mathbb P^3, \mathcal O\_{\mathbb P^3}(1))$ be a linear form. The kernel of
$\omega = 3FdH - HdF \in H^0(\mathbb P^3, \Omega^1\_{\mathbb P^3}(4))$ is a saturated subsheaf $T\_{\mathcal F}$ of $T\_{\mathbb P^3}$ with trivial determinant and $H^0(\mathbb P^3, T\_{\mathcal F})=0$. Indeed, a non-zero section of $T\_{\mathcal F}$ would determine a non-zero vector field on the cubic surface $\{F=0\}$, but we know that smooth cubic surfaces carry no holomorphic vector field.


