Calogero-Moser Spaces over Algebraic Curves

In these notes, we give a survey of the main results of [BC] and [BW]. Our aim is to generalize the geometric classification of (one-sided) ideals of the first Weyl algebra $A_1(C)$ (see [BW1, BW2]) to the ring $D(X)$ of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of D(X) up to stable isomorphism, in terms of the Picard group of X; then, we refine this classification by describing each stable isomorphism class as a disjoint union of (certain quotients of) generalized Calogero-Moser spaces C_n(X, I). The latter are defined as representation varieties of deformed preprojective algebras over a one-point extension of the ring of regular functions on X by the line bundle I. As in the classical case, C_n(X, I) turn out to be smooth irreducible varieties of dimension 2n.