Geometric Classification of Commutative Algebras of Ordinary

1. The purpose of this paper is to give a geometric classification of all commutative algebras consisting of linear ordinary differential operators whose coefficients are scalar-valued functions. The problem of determining commuting ordinary differential operators has a long history in mathematics – it started in 1879. Since then it has been studied by many people in various contexts and from different motivations. The so-called rank one case (see Section 2 for definition) was essentially worked out in the 1920’s and the most general theorem in the rank one case was obtained by Krichever and Mumford in 1970’s. We will present here a complete solution of this problem which is valid for all ranks and generalizes naturally the theorem of Krichever in terms of the geometry of vector bundles on algebraic curves. Let B be a commutative algebra of linear ordinary differential operators with scalar-valued functions as coefficients. We say two operators P and Q commute with one another if the operator product P ·Q coincides with Q ·P . We impose the following conditions on B. (B-1) B is a C-algebra with the identity operator 1 ∈ B.