Uniqueness of polarization for the autonomous 4-dimensional Painlev\'e-type systems

TL;DR

This paper proves that autonomous 4-dimensional Painlevé-type systems have a unique polarization on their spectral curve Jacobian, allowing for precise identification of spectral curves and boundary components in their Liouville tori.

Contribution

It establishes the uniqueness of polarization for spectral curves in 4D Painlevé-type systems, enabling spectral curve identification via Torelli's theorem.

Findings

Jacobian of spectral curve has a unique polarization

Spectral curve can be uniquely identified from the Jacobian

Irreducible genus two boundary components are distinguishable

Abstract

We prove that for any autonomous 4-dimensional integral system of Painlev\'e type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus two component of the boundary of an affine patch of the Liouville torus.