A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations

Cerveau, Dominique; Lins Neto, Alcides

A structural theorem for codimension-one foliations on

ℙ^{n}

n \geq 3

, with an application to degree-three foliations

Cerveau, Dominique; Lins Neto, Alcides

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 1, p. 1-41

Abstract

Let $ℱ$ be a codimension-one foliation on $ℙ^{n}$ : for each point $p \in ℙ^{n}$ we define $𝒥 (ℱ, p)$ as the order of the first non-zero jet $j_{p}^{k} (ω)$ of a holomorphic 1-form $ω$ defining $ℱ$ at $p$ . The singular set of $ℱ$ is $sing (ℱ) = {p \in ℙ^{n} | 𝒥 (ℱ, p) \geq 1}$ . We prove (main Theorem 1.2) that a foliation $ℱ$ satisfying $𝒥 (ℱ, p) \leq 1$ for all $p \in ℙ^{n}$ has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on $ℙ^{n}$ , with $n \geq 3$ , is either the pull-back of a foliation on $ℙ^{2}$ , or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.

Published online : 2019-02-21

MR 3088436 | Zbl 1267.32030

Classification: 37F75, 34M45

@article{ASNSP_2013_5_12_1_1_0,
     author = {Cerveau, Dominique and Lins Neto, Alcides},
     title = {A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {1},
     year = {2013},
     pages = {1-41},
     zbl = {1267.32030},
     mrnumber = {3088436},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_1_1_0}
}

Cerveau, Dominique; Lins Neto, Alcides. A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 1, pp. 1-41. http://www.numdam.org/item/ASNSP_2013_5_12_1_1_0/

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