Higher codimensional foliations and Kupka singularities

We consider holomorphic foliations of dimension \(k>1\) and codimension \(\geq 1\) in the projective space \(\mathbb{P}^n\), with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the...

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Bibliographic Details
Published in:arXiv.org 2014-08
Main Authors: Corrêa, Maurício, Calvo-Andrade, Omegar, Fernández-Pérez, Arturo
Format: Article
Language:English
Subjects:
Eigenvalues
Integers
Singularities
Online Access:Get full text
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Summary:We consider holomorphic foliations of dimension \(k>1\) and codimension \(\geq 1\) in the projective space \(\mathbb{P}^n\), with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the fibers of a rational fibration. As a corollary, if \(\mathcal{F}\) is a foliation such that \(dim(\mathcal{F})\geq cod(\mathcal{F})+2\) and has transversal type diagonal with different eigenvalues, then the Kupka component \(K\) is a complete intersection and we get the same conclusion. The same conclusion holds if the Kupka set is a complete intersection and has radial transversal type. Finally, as an application, we find a normal form for non integrable codimension one distributions on \(\mathbb{P}^{n}\).
ISSN:2331-8422
DOI:10.48550/arxiv.1408.7020