Unfoldings and deformations of rational and logarithmic foliations
Annales de l'Institut Fourier, Volume 66 (2016) no. 4, p. 1583-1613

We study codimension one foliations in projective space n over by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type.

For a differential form ω defining a codimension one foliation, we present a graded module 𝕌(ω), related to the first order unfoldings of ω. If ω is a generic form of rational or logarithmic type, as a first application of the construction of 𝕌(ω), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of ω, in terms of a Hilbert polynomial associated to 𝕌(ω).

We review the notion of regularity of ω in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, ω is regular if and only if every unfolding is trivial up to isomorphism.

Nous étudions des feuilletages de codimension un dans l’espace projectif n sur en regardant leurs perturbations du premier ordre : déploiements et déformations. Nous prêtons une attention particulière aux feuilletages rationnels et logarithmiques.

Pour une forme différentielle ω définissant un feuilletage de codimension un, nous présentons un module gradué 𝕌(ω), lié aux déploiements du premier ordre de ω. Si ω est une forme générique de type rationnel ou logarithmique, comme une première application de la construction de 𝕌(ω), nous classifions les déformations du premier ordre qui apparaissent à partir des déploiements du premier order. Ensuite, nous comptons le nombre de points isolés dans l’ensemble singulier de ω, en termes d’un polynôme de Hilbert associé à 𝕌(ω).

Nous revoyons la notion de régularité de ω en termes d’un complexe long de modules gradués que nous introduisons dans ce travail. Nous utilisons ce complexe pour prouver que, pour des feuilletages rationnels et logarithmiques génériques, ω est régulièr si et seulement si tout déploiement est trivial modulo isomorphisme.

Received : 2015-01-27
Accepted : 2015-10-09
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3044
Classification:  37F75,  14D20,  14B10
Keywords: foliations, codimension one, unfoldings, deformations
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     author = {Molinuevo, Ariel},
     title = {Unfoldings and deformations of rational and logarithmic foliations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     pages = {1583-1613},
     doi = {10.5802/aif.3044},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_4_1583_0}
}
Molinuevo, Ariel. Unfoldings and deformations of rational and logarithmic foliations. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1583-1613. doi : 10.5802/aif.3044. http://www.numdam.org/item/AIF_2016__66_4_1583_0/

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