An example of a modular foliation deducted from an algebraic solution of Painlevé VI equation

Gaël Cousin[1]

  • [1] IMPA, Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brésil

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 699-737
  • ISSN: 0373-0956

Abstract

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One can easily give examples of rank 2 flat connections over ℙ 2 by rational pull-back of connections over 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 PSL 2 ( ℤ [ 3 ] ) on the bidisc.

How to cite

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Cousin, Gaël. "Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI." Annales de l’institut Fourier 64.2 (2014): 699-737. <http://eudml.org/doc/275557>.

@article{Cousin2014,
abstract = {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 $\mathrm\{PSL\}_2(\mathbb\{Z\}[\sqrt\{3\}])$ sur le bidisque.},
affiliation = {IMPA, Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brésil},
author = {Cousin, Gaël},
journal = {Annales de l’institut Fourier},
keywords = {holomorphic foliations; Kodaira dimension; Hilbert modular surfaces; flat connections; Painlevé VI equation},
language = {fre},
number = {2},
pages = {699-737},
publisher = {Association des Annales de l’institut Fourier},
title = {Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI},
url = {http://eudml.org/doc/275557},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Cousin, Gaël
TI - Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 699
EP - 737
AB - 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 $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ sur le bidisque.
LA - fre
KW - holomorphic foliations; Kodaira dimension; Hilbert modular surfaces; flat connections; Painlevé VI equation
UR - http://eudml.org/doc/275557
ER -

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