Moduli spaces for linear differential equations and the Painlevé equations
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, p. 2611-2667

A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map RH:, where is a moduli space of connections and , the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of RH (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces . The induced Painlevé equations are computed explicitly. Except for the Painlevé VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces , which are families of affine cubic surfaces, related to Okamoto–Painlevé pairs. A weak and a strong form of the Riemann–Hilbert problem is treated. Our paper extends the fundamental work of Jimbo–Miwa–Ueno and is related to recent work on Painlevé equations.

Une construction systématique des familles isomonodromiques de connections de rang 2 sur la sphère de Riemann est obtenue de l’application analytique de Riemann–Hilbert RH:, où est un espace de modules de connections et est un espace de modules pour les données analytiques (i.e., la monodromie usuelle, les matrices de Stokes et les “links”). La condition que les fibres de RH (i.e., les familles isomonodromiques) sont de dimension un mène à dix espaces de modules . L’équation induite de Painlevé est calculée explicitement. À l’exception du cas Painlevé VI, les familles ont des singularités irrégulières. Utilisant la classification des singularités irrégulières, on obtient les espaces comme familles explicites de surfaces affines cubiques liées aux pairs de Okamoto–Painlevé. Une forme faible et une forme forte du problème de Riemann–Hilbert sont démontrées. Notre article est une extension du travail fondamental de Jimbo-Miwa-Ueno et est en relation avec des travaux récents sur les équations de Painlevé.

DOI : https://doi.org/10.5802/aif.2502
Classification:  14D20,  14D25,  34M55,  58F05
Keywords: Moduli space for linear connections, irregular singularities, Stokes matrices, monodromy spaces, isomonodromic deformations, Painlevé equations
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     author = {van der Put, Marius and Saito, Masa-Hiko},
     title = {Moduli spaces for linear differential equations and the Painlev\'e equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     pages = {2611-2667},
     doi = {10.5802/aif.2502},
     mrnumber = {2649335},
     zbl = {1189.14021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_7_2611_0}
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van der Put, Marius; Saito, Masa-Hiko. Moduli spaces for linear differential equations and the Painlevé equations. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2611-2667. doi : 10.5802/aif.2502. http://www.numdam.org/item/AIF_2009__59_7_2611_0/

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