Unfoldings and deformations of rational and logarithmic foliations

Molinuevo, Ariel

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1583-1613.

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

Pour une forme différentielle $\omega$ définissant un feuilletage de codimension un, nous présentons un module gradué $𝕌\left(\omega \right)$, lié aux déploiements du premier ordre de $\omega$. Si $\omega$ est une forme générique de type rationnel ou logarithmique, comme une première application de la construction de $𝕌\left(\omega \right)$, 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 $\omega$, en termes d’un polynôme de Hilbert associé à $𝕌\left(\omega \right)$.

Nous revoyons la notion de régularité de $\omega$ 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, $\omega$ est régulièr si et seulement si tout déploiement est trivial modulo isomorphisme.

We study codimension one foliations in projective space ${\mathbb{P}}^n$ over $ℂ$ 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 $\omega$ defining a codimension one foliation, we present a graded module $𝕌\left(\omega \right)$, related to the first order unfoldings of $\omega$. If $\omega$ is a generic form of rational or logarithmic type, as a first application of the construction of $𝕌\left(\omega \right)$, 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 $\omega$, in terms of a Hilbert polynomial associated to $𝕌\left(\omega \right)$.

We review the notion of regularity of $\omega$ 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, $\omega$ is regular if and only if every unfolding is trivial up to isomorphism.

Classification : 37F75, 14D20, 14B10
Keywords: foliations, codimension one, unfoldings, deformations
Mot clés : feuilletages, codimension un, déploiements, déformations

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