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Baker-Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators. (Russian. English summary) Zbl 1286.14060
Summary: We give a review of several results about commutative subrings of partial differential operators. We show that \(n\)-dimensional commutative ring of partial differential operators with scalar (not matrix) coefficients (with certain mild conditions) corresponds to a Baker-Akhiezer module on the spectral algebraic variety. We also show that there is a family of coherent torsion free sheaves of special type. The existence of such sheaves gives a stpong restriction on the structure of the spectral variety, in particular, it is possible to find the selfintersection index of a divisor at infinity.

14K25 Theta functions and abelian varieties