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arXiv: 1201.2755

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

Gaël Cousin 1

(2012-01-12)

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.

1:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Research team: Géométrie analytique

Subject

:

Mathematics/Complex Variables Mathematics/Algebraic Geometry

Keyword(s): feuilletages holomorphes – dimension de Kodaira – surfaces modulaires de Hilbert – connexions plates – équation de Painlevé VI

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From: Gaël Cousin <>

Submitted on: Monday, 30 January 2012 12:32:00

Updated on: Monday, 30 January 2012 14:38:02