Holomorphic foliations of degree two and arbitrary dimension
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915738442268672 |
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| author | Corrêa, Maurício Muniz, Alan |
| author_facet | Corrêa, Maurício Muniz, Alan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12880 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Holomorphic foliations of degree two and arbitrary dimension Corrêa, Maurício Muniz, Alan Algebraic Geometry Complex Variables Dynamical Systems Symplectic Geometry 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. |
| title | Holomorphic foliations of degree two and arbitrary dimension |
| topic | Algebraic Geometry Complex Variables Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2207.12880 |