Logarithmic forms and singular projective foliations

Gargiulo Acea, Javier

Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 171-203.

Dans cet article nous étudions des $q$-formes logaritmiques polynomiales sur un espace projectif et nous caractérisons celles qui définissent des feuilletages singuliers de codimension $q$. Notre principal résultat est la preuve algébrique de leur stabilité infinitésimale lorsque $q=2$ avec quelques hypothèses supplémentaires sur leurs degrés. Nous donnons des nouvelles composantes irréductibles des espaces de modules des feuilletages projectifs de codimension deux et de degré quelconque, et nous montrons que ces composantes sont génériquement réduites selon leur structure naturelle de schéma. Notre méthode est basée sur le calcul explicite de l’espace tangent de Zariski de l’espace de modules en une forme logarithmique générique. Nous posons aussi les bases pour l’extension de nos résultats de stabilité au cas général $q\ge 2$

In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure. Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given generic logarithmic form. Furthermore, we lay the groundwork for an extension of our stability results to the general case $q\ge 2$.

Classification : 14D20, 37F75, 14B10, 32S65
Keywords: logarithmic forms, singular projective foliations, moduli spaces.
Mot clés : formes logarithmiques, feuilletages projectifs singuliers, espaces de modules.

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