ON RINGS OF COMMUTING PARTIAL DIFFERENTIAL OPERATORS

A natural generalization is given for the classification of commutative rings of ordinary differential operators, as presented by Krichever, Mumford, Mulase. The commutative rings of operators in a completed ring of partial differential operators in two variables (satisfying certain mild conditions) are classified in terms of Parshin's generalized geometric data. This classification involves a generalization of M. Sato's theory and is constructible both ways.