Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two
[Limite autonome des équations 4-dimensionnelles de type Painlevé et la dégénérescence des courbes de genre deux]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 845-894.

Dans des études récentes, des analogues en 4 dimensions des équations de Painlevé ont été répertoriés et il existe 40 types. Le but du présent article est de caractériser géométriquement ces 40 équations de type Painlevé. A cet effet, nous étudions la limite autonome de ces équations et la dégénérescence de leurs courbes spectrales. Les courbes spectrales sont des familles à 2 paramètres de courbes de genre deux et leur dégénérescences génériques sont d’un des types classés par Namikawa et Ueno. L’algorithme de Liu nous permet de trouver le types de dégénérescence de courbes spectrales pour nos 40 types de systèmes intégrables. Ce résultat est analogue au fait suivant ; les familles des courbes spectrales des équations de Painlevé autonomes bidimensionnelles ${P}_{I}$, ${P}_{\mathrm{II}}$, ${P}_{\mathrm{IV}}$, ${P}_{\mathrm{III}}^{{D}_{8}}$, ${P}_{\mathrm{III}}^{{D}_{7}}$, ${P}_{\mathrm{III}}^{{D}_{6}}$, ${P}_{\mathrm{V}}$ et ${P}_{\mathrm{VI}}$ définissent des surfaces elliptiques avec une fibre singulière à $H=\infty$ des types Dynkin ${E}_{8}^{\left(1\right)}$, ${E}_{7}^{\left(1\right)}$, ${E}_{6}^{\left(1\right)}$, ${D}_{8}^{\left(1\right)}$, ${D}_{7}^{\left(1\right)}$, ${D}_{6}^{\left(1\right)}$, ${D}_{5}^{\left(1\right)}$ et ${D}_{4}^{\left(1\right)}$, respectivement.

In recent studies, 4-dimensional analogs of the Painlevé equations were listed and there are 40 types. The aim of the present paper is to geometrically characterize these 40 Painlevé-type equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. The spectral curves are 2-parameter families of genus two curves and their generic degeneration are one of the types classified by Namikawa and Ueno. Liu’s algorithm enables us to find the degeneration types of the spectral curves for our 40 types of integrable systems. This result is analogous to the following fact; the families of the spectral curves of the autonomous 2-dimensional Painlevé equations ${P}_{I}$, ${P}_{\mathrm{II}}$, ${P}_{\mathrm{IV}}$, ${P}_{\mathrm{III}}^{{D}_{8}}$, ${P}_{\mathrm{III}}^{{D}_{7}}$, ${P}_{\mathrm{III}}^{{D}_{6}}$, ${P}_{\mathrm{V}}$ and ${P}_{\mathrm{VI}}$ define elliptic surfaces with the singular fiber at $H=\infty$ of the Dynkin types ${E}_{8}^{\left(1\right)}$, ${E}_{7}^{\left(1\right)}$, ${E}_{6}^{\left(1\right)}$, ${D}_{8}^{\left(1\right)}$, ${D}_{7}^{\left(1\right)}$, ${D}_{6}^{\left(1\right)}$, ${D}_{5}^{\left(1\right)}$ and ${D}_{4}^{\left(1\right)}$, respectively.

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DOI : https://doi.org/10.5802/aif.3260
Classification : 34M55,  33E17,  14H70
Mots clés : Limite autonome des équations 4-dimensionnelles de type Painlevé, dégénérescence des courbes de genre deux
@article{AIF_2019__69_2_845_0,
author = {Nakamura, Akane},
title = {Autonomous limit of the 4-dimensional Painlev\'e-type equations and degeneration of curves of genus two},
journal = {Annales de l'Institut Fourier},
pages = {845--894},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {69},
number = {2},
year = {2019},
doi = {10.5802/aif.3260},
zbl = {07067421},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.3260/}
}
Nakamura, Akane. Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 845-894. doi : 10.5802/aif.3260. http://archive.numdam.org/articles/10.5802/aif.3260/

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