Geometric Aspects of Painlev\'e Equations

TL;DR

This paper reviews the geometric structure of Painlevé equations, focusing on their classification via rational surfaces, and details their properties, symmetries, solutions, and Lax pairs within this geometric framework.

Contribution

It systematically describes the geometric classification of Painlevé equations, including discrete cases, using rational surfaces, Picard lattices, and root systems, and compiles comprehensive data on their properties.

Findings

Classification of Painlevé equations via rational surfaces.

Explicit descriptions of symmetries and solutions.

Collection of data on equations, configurations, and Lax pairs.

Abstract

In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $\mathbb{P}^1\times\mathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.