Krichever Correspondence for Algebraic Surfaces

In 70’s there was discovered a construction how to attach to some algebraicgeometric 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 [10, 19]. There were also found some applications to the moduli of algebraic curves [2, 3]. Now it is known as the Krichever correspondence or the Krichever map [2, 11, 1, 17, 4]. The original work by I. M. Krichever has also included commutative rings of differential operators as a third part of the correspondence. The map we want to study here was first described in an explicit way by G. Segal and G. Wilson [19]. They have used an analytical version of the infinite dimensional Grassmanian introduced by M. Sato [18, 16]. In the sequel we consider a purely algebraic approach as developed in [11]. Let us just note that the core of the construction is an embedding of the affine coordinate ring on an algebraic curve into the field k((z)) corresponding to the power decompositions in a point at infinity (the details see below in section 2). In number theory this corresponds to an embedding of the ring of algebraic integers to the fields C or R. The latter one is well known starting from the XIX-th century. The idea introduced by Krichever was to insert the local parameter z. This trick looking so simple enormously extends the area of the correspondence. It allows to consider all algebraic curves simultaneously. But there still remained a hard restriction by the case of curves, so by dimension 1. Recently, it was pointed out by the author [15] that there are some connections between the theory of the KP-equations and the theory of n-dimensional local fields [13], [6]. From this point of view it becomes clear that the Krichever construction should have a generalization to the case of higher dimensions. This