I've been reading about D-modules this summer in preparation for a learning seminar on intersection cohomology. Unfortunately, many of the ideas are not sticking while I learn about the theory. What are examples of D-modules, which demonstrate
- monodromy
- D-modules on nontrivial spaces, such as projective curves or surfaces
- D-modules with support on a singular space (so I can apply Kashiwara's theorem to see what D-modules can say about singularities)
- D-modules on families of varieties over a fixed base
Also, what are some classical systems of differential equations I should be looking at while doing computations? I'm not very familiar with the classical theory.
Best Answer
(I'm not an expert, but you don't seem to have gotten any answers so far.) Let's start from the beginning. Let $A$ be an $n\times n$ matrix of regular functions on a Zariski open subset $X\subset \mathbb{P}^1$ over $\mathbb{C}$. Then consider the system of differential equations $$\frac{df}{dx} = A f$$ Basic theory tells us that the space of holomorphic solutions near $x_0\in X$ is $n$ dimensional. Choose a basis, and analytically continue one of these solutions along a closed path based at $x_0$. It will return to different solution. This is monodromy, which is measuring the "multivaluedness" of the solutions. To connect this to D-modules, let $M=O_X^n$ (sheaf of regular or holomorphic functions). This is clearly an $O_X$-module, it becomes a left $D_X$-module by letting $\partial$ act via the rule $f\mapsto f'-Af$. If you look up the literature on hypergeometric differential equations, you will see a lot of explicit examples worked out in detail.
You can jazz up the last example, by replacing $X$ by any complex manifold/smooth variety and $M$ by a vector bundle with an integrable connection $\nabla$. This gives a rule for letting vector fields act on $M$, integrability ensures that it extends to a $D$-module structure. The kernel $\nabla$ forms a locally constant sheaf, and conversely any locally constant sheaf of $\mathbb{C}$-vector spaces arises this way. So you get plenty of examples of this type on say a curve of genus at least 2. These $D$-modules are quite special in that they are both $O_X$-coherent and holonomic. For an example which is neither, take $D_X$ itself.