Commuting Partial Differential Operators and Vector Bundles Over Abelian Varieties

Abstract

Since then the case of rings of rank r were studied by several people [14], [25], [34]. Then the complete determination of geometrical object which classifies commutative rings of rank r for r > 2 was accomplished by Mulase [22] in 1989. It is thus natural to ask what happens if we replace ordinary differential operators by partial differential operators. There are a few results on this problem [4], [14], [3]. In these papers, all operators are of scalar coefficients and spectral varieties are singular varieties, curves or their products. In [26], we have constructed commuting partial differential operators from line bundles on abelian varieties. The particular feature of these operators is that they are of matrix coefficients. The main purpose of this paper is to construct commuting partial differential operators with matrix coefficients from a class of vector bundles on abelian varieties. In studying commuting operators, their common eigenfunctions play a central role. Krichever [13] characterized such eigenfunctions as Baker-Akhiezer (BA) functions in the case of ordinary differential operators. The concept of BA-functions are generalized to higher dimensions ([26][7]); BA-functions are constructed from (X, D, ri), where X is a projective variety, D an ample divisor on X and {ri} is a base of differentials of the second kind with poles on D modulo exact forms. In this paper we introduce BA-functions of coherent sheaves on abelian varieties, which are our main objects of study in this paper.