Baker–Akhiezer Modules on the Intersections of Shifted Theta Divisors

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 rings of partial differential operators with matrix coefficients and their non-trivial evolution equations are obtained.