Calogero-Moser Spaces over Algebraic Curves

TL;DR

This paper surveys the classification of ideals in differential operator rings over complex curves, extending classical results to a broader geometric setting using Calogero-Moser spaces.

Contribution

It generalizes the classification of ideals from the Weyl algebra to differential operators on algebraic curves, introducing Calogero-Moser spaces for this purpose.

Findings

Classifies ideals of D(X) via Picard group of X.

Defines Calogero-Moser spaces as representation varieties.

Shows these spaces are smooth irreducible varieties of dimension 2n.

Abstract

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.