Un exemple de feuilletage modulaire d\'eduit d'une solution alg\'ebrique de l'\'equation de Painlev\'e VI

摘要

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 ofa connection that can not occur in this way; this example is constructed froman algebraic solution of Painlev\'e VI equation. From this example we deduce aHilbert modular foliation. The proof of this relies on the classification offoliations 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 ourtwice foliated surface is covered by the classical Hilbert modular surfaceconstructed from the action of $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ on thebidisc.