Baker-Akhiezer Modules on the Intersections of Shifted Theta Divisors

TL;DR

This paper investigates how Baker-Akhiezer modules restrict to intersections of shifted theta divisors on abelian varieties, revealing their structure as free modules and deriving new integrable systems and differential operators.

Contribution

It demonstrates that these restricted BA-modules are free over differential operators and introduces new examples of commutative rings of PDEs with matrix coefficients.

Findings

Restricted BA-modules are free modules over differential operators.

Derived evolution equations for BA-module generators.

Constructed new commutative rings of PDEs with matrix coefficients.

Abstract

The restriction, on the spectral variables, of the Baker-Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of differential operators in $k$ variables. The remaining g-k derivations define evolution equations for generators of the BA-module. As a corollary new examples of commutative ring of partial differential operators with matrix coefficients and their non-trivial evolution equations are obtained.