Splitting aspects of holomorphic distributions with locally free tangent sheaf

TL;DR

This paper investigates conditions under which a two-dimensional singular holomorphic distribution with a locally free tangent sheaf can be decomposed into a direct sum of one-dimensional foliations, with applications to foliations on projective space.

Contribution

It provides new sufficient conditions for splitting of tangent sheaves of holomorphic distributions and applies these to specific cases on projective space.

Findings

Splitting of tangent sheaves under certain conditions

Application to foliations on ^3 with tangent vector fields

Division results for vector fields and differential forms

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.