Are there holonomic $\mathcal{D}$-modules besides flat connections?

Question

Question: are there interesting holonomic $\mathcal{D}$-modules on smooth variety $X$ except those coming from flat meromorphic connections?

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Since

• $M$ is holonomic iff $\dim \text{ch}(M)=\dim X$

• $M$ comes from a holomorphic flat connection iff $\text{ch}(M)=X\subset T^*X$

one can show that holonomic $M$ has $\text{ch}(M)\supset X$, so $$M\vert_U $$ comes from a flat holonomorphic connection for some open $U\subset X$.

So it feels like holonomic $\mathcal{D}$-modules ''are'' extensions of a flat holomorphic connection $\nabla$ from $U$ to $X$, on which the connection aquires meromorphic (maybe essential, if that even makes sense?) singularities.

Thus either meromorphic connections are everything, they are almost everything (the only other examples being ''bad'' singularities), or this heuristic argument is completely off. The question asks what actually happens.

Answer 1

By the definition of characteristic variety, any exact sequence of $D_X$-modules $$0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow 0$$ satisfies $$\text{Ch}(M)=\text{Ch}(L)\cup \text{Ch}(N)$$ Thus, if $M$ is holonomic, and since $D_X$-coherence is built into the definition of a holonomic module, it has a sequence $$0\subseteq M_1\subseteq \cdots \subseteq M_n\subseteq M$$ where each quotient is holonomic. So we get an OK idea of what they look like if we know what the simple holonmics look like.

$$\text{}$$ Theorem 3.4.2 in ''D-Modules, Perverse Sheaves, and Representation Theory'' implies

Theorem: If $X$ is smooth, all simple holonomic $D_X$-modules are of the form $$\text{the unique simple module inside }\int_\iota^0M$$ where $Y$ is a locally closed connected smooth subvariety, $\iota:Y\hookrightarrow X$ is affine, $M$ is a integrable connection on $Y$ and $\int_\iota^0$ is the direct image (which is exact here). Any two such are isomorphic iff $$\text{closure}(Y) \ = \ \text{closure} (Y')$$ and there is an open dense set $U\subseteq Y \cap Y'$ on which $$M\vert_U \ = \ M'\vert_U$$ (Since $M,M'$ are local systems of rank $r$ on $U$, it's enough to check that the maps $\pi_1(U)\to \text{GL}(\text{C}^r)$ they correspond to are the same).

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For instance, if $X=\mathbb{C}$, there are two possibilities:

• $Y$ is a point, and $M=\mathbb{C}$. Then $$\int_\iota^0M\ \simeq \ \mathbb{C}[\partial]$$ where the action of $\mathbb{C}[x,\partial]$ is: $\partial$ acts by multiplication and $x$ acts my multiplication by $a\in \mathbb{C}$ corresponding to $Y$. So there are $\mathbb{C}$-many of these.
• $Y$ is $\mathbb{C}$ minus finitely many points. There are lots of these; I think they biject with $$\{(p_1,...,p_n),\text{ a local system on }\mathbb{C}\setminus \{p_1,...,p_n\} \text{ nontrivial monodromy around all }p_i\}$$ When $n=0,1$ and the rank is $1$ these are $\mathbb{C}[x]$, $\mathbb{C}[x,x^{-1}]x^\lambda$ for $\lambda \in \mathbb{C}\setminus \mathbb{Z}$, I think.