Commuting differential operators and higher-dimensional algebraic varieties

Several algebro-geometric properties of commutative rings of partial differential operators (PDOs) as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of PDOs, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker–Akhieser functions. On the other hand, there is a recent generalization of Sato’s theory, which belongs to the third author of this paper. We compare both approaches to the commutative rings of PDOs in two variables. As a by-product, we get several necessary conditions on geometric data describing commutative rings of PDOs.