CATEGORY OF VECTOR BUNDLES ON ALGEBRAIC CURVES AND INFINITE DIMENSIONAL GRASSMANNIANS

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Abstract
Equivalence between the following categories is established: 1) A category of arbitrary vector bundles on algebraic curves defined over a field of arbitrary characteristic, and 2) a category of infinite dimensional vector spaces corresponding to certain points of Grassmannians together with their stabilizers. Our contravariant functor between these categories gives a full generalization of the well-known Krichever map, which assigns points of Grassmannians to the geometric data consisting of curves and line bundles. As an application, a solution to the classical problem of Wallenberg-Schur of classifying all commutative algebras consisting of ordinary differential operators is obtained. It is also shown that the KP flows produce all generic vector bundles on arbitrary algebraic curves of genus greater than one.

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