Jacobian variety and Integrable system -- after Mumford, Beauville and Vanhaecke

TL;DR

This paper explores advanced integrable systems related to Jacobian varieties, extending Mumford's and Beauville's frameworks by constructing variants that involve complements of theta divisors and their intersections.

Contribution

It introduces a new integrable system variant with level sets as complements of intersections of translated theta divisors, generalizing previous Mumford and Vanhaecke systems.

Findings

Constructed a system with level sets as complements of intersections of translated theta divisors.

Connected the new system to generalizations of Mumford and Vanhaecke's systems.

Provided geometric interpretations of the integrable systems in terms of Jacobian varieties.

Abstract

Beauville introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the `intersection' of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke.