Holomorphic foliations of degree two and arbitrary dimension

Abstract

We prove a complete classification of degree-$2$ foliations on ${\mathbb{P}}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a degree-$2$ foliation by curves on ${\mathbb{P}}^{n-k+1}$, or a logarithmic foliation of type $(1^{n-k+1},2)$, or a logarithmic foliation of type $(1^{n-k+3})$, or the linear pull-back of a degree-$2$ foliation of dimension $2$ on ${\mathbb{P}}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. Meanwhile, we prove that any $2$-dimensional foliation tangent to a global vector field must satisfy that its tangent sheaf is either not locally free or has a direct summand isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^n}(a)$, with $a\in \{0,1\}$. As a byproduct of our classification, we describe the geometry of Poisson structures on ${\mathbb{P}}^4$ with generic rank two.

23 pages; Comments welcome!