Splitting aspects of holomorphic distributions with locally free tangent sheaf

Abstract: In this work, we mainly deal with a two-dimensional singular holomorphic distribution $\mathcal{D}$ defined on $M$, in the two situations $M=\mathbb{P}^n$ or $M=(\mathbb{C}^n,0)$, tangent to a one-dimensional foliation $\mathcal{G}$ on $M$, and whose tangent sheaf $T_{\mathcal{D}}$ is locally free. We provide sufficient conditions on $\mathcal{G}$ so that there is another one-dimensional foliation $\mathcal{H}$ on $M$ tangent to $\mathcal{D}$, such that their respective tangent sheaves satisfy the splitting relation $T_{\mathcal{D}}=T_{\mathcal{G}} \oplus T_{\mathcal{H}}$. As an application, we show that if $\mathcal{F}$ is a codimension one holomorphic foliation on $\mathbb{P}^3$ with locally free tangent sheaf and tangent to a nontrivial holomorphic vector field on $\mathbb{P}^3$, then $T_{\mathcal{F}}$ splits. Some division results for vector fields and differential forms are also obtained.