Abstract
In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank in , a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold. These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve with and ; 2) a divisor with , , , and . Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations. Bibliography: 45 titles.
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