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

Gaël Cousin (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ on the bidisc.

Mots clés: équation de Painlevé VI; connexions plates; surfaces modulaires de Hilbert; feuilletages holomorphes; dimension de Kodaira

Domaines: Variables complexes [math.CV]; Géométrie algébrique [math.AG]

Gaël Cousin. Un exemple de feuilletage modulaire déduit d'une solution algébrique de l'équation de Painlevé VI. Annales de l'Institut Fourier, 2014, 64 (2), pp.699-737. ⟨10.5802/aif.2863⟩. ⟨hal-00659358v3⟩