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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Ketil Tveiten Sep 8 '16 at 8:15
  • 4
    $\begingroup$ 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. $\endgroup$ – 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.


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