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


*

*Their dimension is equal to the dimension of the underlying $D_X$-module

*Their dimension is greater than or equal to the dimension of $X$

*Behaves well with restriction to open subsets of $X$

*They behave well with respect to exact sequences of coherent $D_X$-modules

*They are coisotropic subvarieties of $T^*X$

*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.
 A: 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. 
A: 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)
