What is the motivation behind the characteristic variety of a D-module and what does it's geometry tell me about the D-module?

Question

Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as $$ \text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M})}\right) \subset T^*X $$ These varieties have a number of nice properties

1. Their dimension is equal to the dimension of the underlying $D_X$-module
2. Their dimension is greater than or equal to the dimension of $X$
3. Behaves well with restriction to open subsets of $X$
4. They behave well with respect to exact sequences of coherent $D_X$-modules
5. They are coisotropic subvarieties of $T^*X$
6. They are lagrangian iff the underlying D-module is holonomic

Unfortunately, it's not clear why these varieties are useful and what their motivation for construction is.

Answer 1

Here’s one way to think about them: they tell you how far a $D$-module is from being an integrable connection (i.e. finitely generated over $\mathcal O$). Here’s what I mean: let $M$ be a $D$-module on $X$. Then $M$ is an integrable connection in a neighborhood of a point $x\in X$ if and only if $\operatorname{Char}(M)\cap T^*_x X$ is zero (i.e. is contained in the zero section).

I also want to correct your point number 1. The dimension of a $D$-module is by definition the dimension of its characteristic variety.

Answer 2

Another way to think about them: they are just support of a coherent $\mathcal{O}_{T^*X}$ module $\widetilde{gr^FM}$.It is a generalization of the notion of a singular support of a module, and that is a refinement of the usual notion of "support" appear in geometry.

For module $M$ over a ring, we define the support to be $\{p|M_p \neq 0\}$, and note that $V(\sqrt{Ann(M)} ) = supp(M)$. This is still true when you replace $M$ with $gr^F_M$, and $supp(gr^F_M)$ is denoted as the singular support of $M$.

Here is a paper on generalizing this notion to 2-category. https://arxiv.org/pdf/1201.6343.pdf (talk version here: https://pirsa.org/16040075)