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.