[Une nouvelle famille à deux paramètres de déformations isomonodromiques sur la sphère à cinq trous]
The object of this paper is to describe an explicit two-parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and three tangent lines. By restricting them to generic lines we get an algebraic family of isomonodromic deformations of the five-punctured sphere. This yields new algebraic solutions of a Garnier system. Finally, we use the associated Riccati one-forms to construct and prove the integrability (in the transversally projective sense) of a subfamily of Lotka-Volterra foliations.
Le but de cet article est de décrire une famille explicite à deux paramètres de connexions logarithmiques plates au dessus du plan projectif complexe. Ces connexions sont à monodromie diédrale et leur lieu polaire est une quintique prescrite, composée d'une conique et de trois droites tangentes. Par restriction aux droites génériques, on obtient alors une famille algébrique de déformations isomonodromiques de la sphère à cinq trous. Ceci livre de nouvelles solutions algébriques d'un système de Garnier. Enfin, nous utilisons les formes de Riccati associées à ces connexions pour construire et montrer l'intégrabilité (au sens transversalement projectif) d'une sous-famille de feuilletages de Lotka-Volterra.
DOI : 10.24033/bsmf.2716
Keywords: Isomonodromic deformations, Garnier systems, logarithmic flat connections, transversally projective foliations, Lotka-Volterra systems, Riccati forms, dihedral monodromy groups.
Mots-clés : Déformations isomonodromiques, systèmes de Garnier, connexions logarithmiques plates, feuilletages transversalement projectifs, systèmes de Lotka-Volterra, formes de Riccati, groupes de monodromie diédraux.
@article{BSMF_2016__144_2_339_0,
author = {Girand, Arnaud},
title = {A new two-parameter family of isomonodromic deformations over the five punctured sphere},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {339--368},
year = {2016},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {144},
number = {2},
doi = {10.24033/bsmf.2716},
mrnumber = {3499084},
zbl = {1344.14011},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2716/}
}
TY - JOUR AU - Girand, Arnaud TI - A new two-parameter family of isomonodromic deformations over the five punctured sphere JO - Bulletin de la Société Mathématique de France PY - 2016 SP - 339 EP - 368 VL - 144 IS - 2 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2716/ DO - 10.24033/bsmf.2716 LA - en ID - BSMF_2016__144_2_339_0 ER -
%0 Journal Article %A Girand, Arnaud %T A new two-parameter family of isomonodromic deformations over the five punctured sphere %J Bulletin de la Société Mathématique de France %D 2016 %P 339-368 %V 144 %N 2 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2716/ %R 10.24033/bsmf.2716 %G en %F BSMF_2016__144_2_339_0
Girand, Arnaud. A new two-parameter family of isomonodromic deformations over the five punctured sphere. Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 2, pp. 339-368. doi: 10.24033/bsmf.2716
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