# 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

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.

• I don't recall the details (I am not an analyst), but the motivation comes primarily from distribution theory; the characteristic variety of a holonomic D-module (which as you know is cyclic, generated by a distribution) is related to the singular spectrum of the distribution. I would go have a look at the original work of Kashiwara and Saito, it might be enlightening. Sep 8, 2016 at 8:15
• You may also think of it as an invariant of the PDE, for example the classical distinction between elliptic parabolic or hyperbolic PDE can be read from the characteristic variety. Sep 8, 2016 at 18:12

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.
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 $$M$$.