Introduction

Let X be an irreducible complex affine algebraic variety, and let D(X) be the ring of (global, linear, algebraic) differential operators on X (we shall review the definition in Section 2). This ring has a natural filtration (by order of operators) in which the elements of order zero are just the ring (X) of regular functions on X. Thus, if we are given D(X) together with its filtration, we can at once recover the variety X. But now suppose we are given D(X) just as an abstract noncommutative -algebra, without filtration; then it is not clear whether or not we can recover X. We shall call two varieties X and Y differentially isomorphic if D(X) and D(Y) are isomorphic.

The first examples of nonisomorphic varieties with isomorphic rings of differential operators were found by Levasseur, Smith and Stafford (see [LSS] and Section 9 below). These varieties arise in the representation theory of simple Lie algebras; they are still the only examples we know in dimension > 1 (if we exclude products of examples in lower dimensions). For curves, on the other hand, there is now a complete classification up to differential isomorphism; the main purpose of this article is to review that case. The result is very strange. It turns out that for curves, D(X) determines X (up to isomorphism) except in the very special case when X is homeomorphic to the affine line (we call such a curve a framed curve).