A structural theorem for codimension-one foliations on n , n3, with an application to degree-three foliations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 1-41.

Let be a codimension-one foliation on n : for each point p n we define 𝒥(,p) as the order of the first non-zero jet j p k (ω) of a holomorphic 1-form ω defining at p. The singular set of is sing()={p n |𝒥(,p)1}. We prove (main Theorem 1.2) that a foliation satisfying 𝒥(,p)1 for all p n has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on n , with n3, is either the pull-back of a foliation on 2 , or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.

Publié le :
Classification : 37F75, 34M45
Cerveau, Dominique 1 ; Lins Neto, Alcides 2

1 Institut Universitaire de France & IRMAR Campus de Beaulieu 35042, Rennes Cedex, France
2 Instituto de Matemática Pura e Aplicada Estrada Dona Castorina 110 Rio de Janeiro, Brasil
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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, Série 5, Tome 12 (2013) no. 1, pp. 1-41. http://archive.numdam.org/item/ASNSP_2013_5_12_1_1_0/

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