Differential isomorphism and equivalence of algebraic varieties
Yuri Yu. Berest, George V. Wilson · Cambridge University Press eBooks · 2004
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 ).