Holonomic = annihilated by some differential operator Let $X$ be a smooth variety over a characteristic zero field $k$. It seems to be well-known that
"A coherent $\mathcal{D}_X$-module is holonomic if and only if 'every element is annihilated by a nonzero differential operator'."
I know that, when $X=\mathbb{A}^1$, this is true in the sense that "a finitely generated $k[x]\langle\partial\rangle$-module $M$ is holonomic if and only if, for every $m\in M$, there exists $P\in k[x]\langle\partial\rangle\setminus 0$ such that $Pm=0$."
Is a similar statement true in more generality? (For all $X$? For affine $X$?) If so, why?
 A: "A coherent $D_X$-module is holonomic if and only if 'every element is annihilated by a nonzero differential operator'."
This is very much not true in dim>1.
For example, let $D=k[x_1,x_2]\langle \partial_1,\partial_2\rangle$ be the ring of differential operators on $\mathbb A^2$, and consider the left $D$-module
$$M= D/D\partial_1.$$
Then every element is annihilated by a suitably high power of $\partial_1$, but $M$ is not holonomic (its characteristic variety is the hyperplane in $T^\ast\mathbb A^2= \operatorname{Spec} k[x_1,x_2,\xi_1,\xi_2]$ cut out by the equation $\xi_1=0$).
EDIT: In fact, we can make a stronger statement. Let $M$ be a $D$-module, and suppose we can find $m\in M$ which is not annihilated by any non-zero differential operator. Then the homomorphism $D\to M$ given by $P \mapsto Pm$ is injective. It follows that $\mathit{SS}(D) \subseteq \mathit{SS}(M)$ (the characteristic variety operation  takes short exact sequences to unions). Thus $\mathit{SS}(M)$ is the entire cotangent bundle in this case.
Put another way, any $D$-module whose characteristic variety is a proper subset of the cotangent bundle will satisfy the property that each element is annihilated by a non-zero differential operator.
In dimension 1, every proper closed coisotropic subset of the cotangent bundle is lagrangian, so every such $D$-module is holonomic. But this is definitely not the case in higher dimensions.
