Cohomologies of Affine Jacobi Varieties and Integrable Systems

TL;DR

This paper investigates the structure of the affine ring of affine Jacobi varieties associated with hyper-elliptic curves, using Mumford's matrix construction to explore cohomology groups and their relation to integrable systems.

Contribution

It introduces conjectures on the cohomology of affine hyper-elliptic Jacobi varieties and links the affine ring to classical observables in integrable systems.

Findings

Character of the affine ring calculated using Mumford's matrix construction

Decomposition of the character leads to conjectures on cohomology groups

Affine ring generated by highest cohomology group under invariant vector fields

Abstract

We study the affine ring of the affine Jacobi variety of a hyper-elliptic curve. The matrix construction of the affine hyper-elliptic Jacobi varieties due to Mumford is used to calculate the character of the affine ring. By decomposing the character we make several conjectures on the cohomology groups of the affine hyper-elliptic Jacobi varieties. In the integrable system described by the family of these affine hyper-elliptic Jacobi varieties, the affine ring is closely related to the algebra of functions on the phase space, classical observables. We show that the affine ring is generated by the highest cohomology group over the action of the invariant vector fields on the Jacobi variety.