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, Serie 5, Volume 12 (2013) no. 1, p. 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.

Published online : 2019-02-21
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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