Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two

Abstract

Higher dimensional analogs of the Painlevé equations have been proposed from various aspects. In recent studies, 4-dimensional analogs of the Painlevé equations were classified into 40 types. The aim of the present paper is to geometrically characterize these 40 types of equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. We obtain two functionally independent conserved quantities $H_1$ and $H_2$ for each system. We construct fibrations whose fiber at a general point $h_i$ is the spectral curve of the system with $H_i=h_i$ for $i=1,2$. The singular fibers at $H_i=\infty$ are one of the degenerate curves of genus 2 classified by Namikawa and Ueno. Liu's algorithm enables us to give degeneration type of spectral curves for our 40 types of integrable systems. This result is analogous to the following observation; spectral curve fibrations of the autonomous 2-dimensional Painlevé equations $P_{\mathrm{I}}$, $P_{\mathrm{II}}$, $P_{\mathrm{IV}}$, $P_{\mathrm{III}}^{D_8}$, $P_{\mathrm{III}}^{D_7}$, $P_{\mathrm{III}}^{D_6}$, $P_V$ and $P_{\mathrm{VI}}$ are elliptic surfaces with the singular fiber at $H=\infty$ of Dynkin type $E_8^{(1)}$, $E_7^{(1)}$, $E_6^{(1)}$, $D_8^{(1)}$, $D_7^{(1)}$, $D_6^{(1)}$, $D_5^{(1)}$ and $D_4^{(1)}$, respectively.

Amended to generic degeneration. This paper will appear at Annales de l'Institut Fourier

Cited by in corpus (5)

• Quantum Curve and the First Painlevé Equation
• Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems
• Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation
• Uniqueness of polarization for the autonomous 4-dimensional Painlevé-type systems
• Weights, Kovalevskaya exponents and the Painlevé property