Here in Berkeley, we are having a small learning seminar on D-modules, trying to read Bernstein’s notes. Thursday we had an organizational meeting and on Monday Anton is giving the first talk.
If I had been more organized, I would have given a motivational talk on Thursday trying to explain why D-modules are interesting. Here is what I would have said.
First, just so that we are on the right page, let me explain the basic concepts. The ring is the (sheaf of) ring(s) of algebraic differential operators on a smooth algebraic variety and a D-module is a (sheaf of) module(s) over this ring. For example, we could take to be the affine line over , in which case Here is the differential operator and is multiplication by . In general the ring is a kind of universal enveloping algebra of vector fields over functions.
So here are three reasons why D-modules are interesting.
1. They give an algebraic way of studying differential equations. More specifically, systems of linear partial differential equations. In algebraic geometry, we study systems of polynomial equations and the way we do it is to take the ring of polynomials and quotient by the ideal generated by the equations. Similarly, here if we have some linear PDEs, we can quotient the ring D by these equations, to get a D-module. For example, if we want to study the equation , we would take .
Then a solution of our differential equation is just a D-module homomorphism from to a D-module of functions . Compare this to points in an algebraic variety. Note that while we are working with algebraic differential operators, any sheaf of smooth functions is a D-module. So we can study algebraic, analytic, or smooth solutions to our equations.
Anyway, this is the story you always hear. I wonder if anyone actually studies PDEs in this way. I’d be glad to know if someone has a reference or more information.
2. D-modules are an easy case of non-commutative algebraic geometry. Of course the ring is non-commutative, but more interesting is to observe that it has a filtration (by order of the differential operator) such that its associated graded is commutative. For example, in our case of the affine line, when you take associated graded you get rid of the lower order term “1” in the defining relation, so you are just left with and commuting, and so just a polynomial ring in two variables.
In general, associated graded of is nothing but the ring of functions on the cotangent bundle of . So is a non-commutative version of the cotangent bundle and in studying D-modules you are doing non-commutative geometry. This has some applications, because if you find a filtration of your module which is compatible with the filtration of , then you can take the associated graded of the module to get a coherent sheaf on the cotangent bundle. This leads to the concept of characteristic variety.
3. D-modules are related to perverse sheaves. Perverse sheaves are a somewhat mysterious abelian subcategory of the derived category of constructible sheaves on , viewed with the complex topology. In particular, the category of perverse sheaves is an interesting _topological_ invariant of . The Riemann-Hilbert correspondence gives us an equivalence of categories between the category of regular, holonomic, D-modules and the category of perverse sheaves. A special case of this is the well-known relation between vector bundles with flat connection and representations of the fundamental group. This correspondence gives us a nice way of thinking about perverse sheaves and also shows that the category D-modules is a topological invariant.