# 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. – Ketil Tveiten Sep 8 '16 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. – Michael Bächtold Sep 8 '16 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.