Takasaki’s rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations

Abstract

We propose a notion of regularisation which extends Okamoto’s construction of spaces of initial conditions for the Painlevé differential equations to the class of systems with globally finite branching about movable singularities in the sense of the algebro-Painlevé property. We illustrate this regularisation first in the case of a Hamiltonian system obtained by Takasaki as part of the Painlevé-Calogero correspondence, which is related by an algebraic transformation to the fourth Painlevé equation. Through a combination of compactification, blowups and removal of certain curves we obtain a space on which the system is everywhere either regular or regularisable by certain algebraic transformations. We provide an atlas for this space in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of the Painlevé equations on Okamoto’s spaces. We also compare the surface associated with the Takasaki system with that of the fourth Painlevé equation, showing that they are related by a combination of blowdowns and a branched double cover. We provide more examples of algebro-Painlevé equations regularised in this way and also discuss applications of this generalised construction of the space of initial conditions to the identification and classification of algebro-Painlevé equations.