Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

Cousin, Gaël

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 699-737.

On peut construire facilement des exemples de connexions plates de rang $2$ sur ${\mathbb{P}}^2$ comme tirés en arrière de connexions sur ${\mathbb{P}}^1$. On donne un exemple de connexion qui ne peut être obtenue de cette manière. Cet exemple est construit à partir d’une solution algébrique de l’équation de Painlevé VI. On en déduit un feuilletage modulaire. La preuve de ce fait repose sur la classification des feuilletages sur les surfaces projectives par leurs dimensions de Kodaira, fruit du travail de Brunella, McQuillan et Mendes. On décrit ensuite le feuilletage dual. Par une analyse fine de monodromie, on voit que notre surface bifeuilletée est revêtue par la surface modulaire de Hilbert construite en faisant agir ${\mathrm{PSL}}_2\left(\mathbb{Z}\left[\sqrt{3}\right]\right)$ sur le bidisque.

One can easily give examples of rank $2$ flat connections over ${\mathbb{P}}^2$ by rational pull-back of connections over ${\mathbb{P}}^1$. We give an example of a connection that can not occur in this way; this example is constructed from an algebraic solution of Painlevé VI equation. From this example we deduce a Hilbert modular foliation. The proof of this relies on the classification of foliations on projective surfaces due to Brunella, Mc Quillan and Mendes. Then, we get the dual foliation and, by a precise monodromy analysis, we see that our twice foliated surface is covered by the classical Hilbert modular surface constructed from the action of ${\mathrm{PSL}}_2\left(\mathbb{Z}\left[\sqrt{3}\right]\right)$ on the bidisc.

Classification : 34M56, 14G35, 37F75, 34M35
Mot clés : feuilletages holomorphes, dimension de Kodaira, surfaces modulaires de Hilbert, connexions plates, équation de Painlevé VI.
Keywords: holomorphic foliations, Kodaira dimension, Hilbert modular surfaces, flat connections, Painlevé VI equation.

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