Krichever Correspondence for Algebraic Surfaces

Abstract

In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. The construction was successfully used in the theory of integrable systems, particularly, for the KP and KdV equations. There were also found some applications to the moduli of algebraic curves. But there remained a hard restriction by the case of curves, so by dimension 1. Recently, it was pointed out by the author that there are some connections between the theory of the KP-equations and the theory of n-dimensional local fields. From this point of view it becomes clear that the Krichever construction should have a generalization to the case of higher dimensions. This generalization is suggested in the paper for the case of algebraic surfaces.