Calogero–Moser spaces over algebraic curves

Abstract

In these notes, we give a survey of the main results of [5] and [7]. Our aim is to generalize the geometric classification of (left) ideals of the first Weyl algebra \({A_{1}(\mathbb{C})}\) (see [8, 9]) to the ring \({\mathcal{D}}(X)\) of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of \({\mathcal{D}}(X)\) 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 (quotients of) generalized Calogero–Moser spaces \({\mathcal{C}}_{n}(X,{\mathcal{I}})\). The latter are defined as representation varieties of deformed preprojective algebras over a certain noncommutative extension of the ring of regular functions on X. As in the classical case, \({\mathcal{C}}_{n}(X,{\mathcal{I}})\) turn out to be smooth irreducible varieties of dimension 2n.