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 $M$, where $M$ represents a complex manifold of dimension $n\ge 3$ 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 $\mathcal{S}(\mathcal{G},\mathcal{D})=\varnothing$. 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.