What are examples of D-modules that I should have in mind while learning the theory? 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.
 A: I will edit this question as I learn more, but here's one useful example: Consider the $\mathcal{D}_{\mathbb{A}^1}$-module
$$
\frac{\mathcal{D}_{\mathbb{A}^1}}{\mathcal{D}_{\mathbb{A}^1}(t\partial_t - \beta)}
$$
is a local system on $\mathbb{C}^*$ with monodromy $\text{exp}(2\pi i \beta)$. Apparently this is a useful example for showing the definition of nearby cycles in mixed hodge modules is the right one.
A: (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.
