Holomorphic foliations of degree two and arbitrary dimension
Maurício Corrêa, Alan Muniz
TL;DR
This work delivers a complete classification of degree-$2$ holomorphic foliations on ${ mf P}^n$ in arbitrary dimension once non-algebraically-integrable is assumed, showing that general members arise as linear pullbacks, logarithmic foliations of specific pole-types, or Affine-Lie foliations tied to ${ m aff}(\\mathbb{C})$ actions. The authors develop a robust framework around subfoliations and stability to constrain possible tangent-sheaf structures, proving that 2-dimensional foliations with global vector fields must have tangent sheaves that either split or are non-locally-free, and they classify 3-dimensional nonstable cases into pullbacks, logarithmic, affine-Lie, or stable-subfoliation scenarios. A central technical achievement is the detailed description of Affine-Lie components via partitions and explicit first integrals, which together with Log and LPB families yield a comprehensive global picture. The paper also develops connections to Poisson geometry, deriving structural consequences for Poisson bivectors on ${ mf P}^4$ with generic rank $2$ and showing that the Poisson divisor geometry largely aligns with the foliation classification. Overall, the results extend known degree-$2$ classifications to higher dimension, illuminate the moduli of such foliations, and provide explicit geometric and algebraic descriptions that facilitate further study of their deformations and related Poisson structures.
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.