Holomorphic foliations of degree two and arbitrary dimension

2022/07/26 by Maurício Corrêa, Corrêa, Maurício, Muniz, Alan · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2207.12880

openalex publication_date 2022/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a complete classification of degree-2 foliations on ℙⁿ in any dimension, assuming they are not algebraically integrable. If F is such a foliation, then either F is the linear pull-back of a degree-2 foliation by curves on 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 ℙ^n-k+2 tangent to an action of the Lie algebra \mathfrakaff(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 O_ℙⁿ(a), with a∈\0,1\. As a byproduct of our classification, we describe the geometry of Poisson structures on ℙ⁴ with generic rank two.

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