Splitting aspects of holomorphic distributions with locally free tangent sheaf

In this work, we mainly deal with a two-dimensional singular holomorphic distribution \mathcal{D} defined on M , where M represents a complex manifold of dimension n \geq 3 or a germ of it, whose tangent sheaf T_{\mathcal{D}}D is locally free. As is well known, when M=\mathbb{P}^{n} or M=(\mathbb{C}^{n},0) , there is a one-dimensional foliation \mathcal{G} on M tangent to \mathcal{D} , and we study whether T_{\mathcal{D}} splits starting from it. In both cases, 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}} . We introduce a concept of local division of \mathcal{D} by \mathcal{G} , exhibiting a characterization of \mathcal{S}(\mathcal{G},\mathcal{D}) , the set of points p \in M where \mathcal{G} does not locally divide \mathcal{D} at p . Furthermore, for M=\mathbb{P}^{n} , we prove that the existence of such \mathcal{H} is equivalent to \mathcal{S}(\mathcal{G},\mathcal{D})=\emptyset . Additionally, given a codimension one holomorphic foliation \mathcal{F} on \mathbb{P}^{3} with locally free tangent sheaf, we show that T_{\mathcal{F}} splits provided there exists a nonzero holomorphic vector field on \mathbb{P}^{3} tangent to \mathcal{F} . We obtain division results involving holomorphic differential forms and vector fields, and some of them could serve as alternatives to classical results coming from the De Rham–Saito division lemma, while others can be applied in situations not covered by the latter.