Title: Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two
Abstract: 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 II , P IV , P III D 8 , P III D 7 , P III D 6 , P V and P VI define elliptic surfaces with the singular fiber at H=∞ of the Dynkin types 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.