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Does anyone know a reference that rigorously formulates the notion of a "stack of $D$-modules over a fixed smooth variety $X$"?

Also, naturally, one can ask the question of whether or not holonomic $D$-modules form an stack and that they form in particular an algebraic stack. It is known that $D$-modules themselves cannot form an algebraic stack.

Searching around, it seems the best I can find is Ben-Zvi's comments about gluing $D$-modules, but he does not go all the way to the point of saying that $D$-modules form a stack. He does mention how to define $D$-modules on stacks though which is nice, but not quite what I am seeking.

Edit: I was told by someone irl that this is known but never written down. The statement is that one can define the stack of $D$-modules over schemes on the 'etale site using Kashiwara's Theorem together with the fact that pullbacks and pushforwards of D-modules along 'etale morphisms behave nicely. So to me, the more interesting question would be when it is algebraic.

user1515097
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  • For a complex projective manifold $X$ there is "an equivalence of categories" between the category of finite dimensional complex representations of the topological fundamental group of $X$ and the "category of finite rank vector bundles" $(E, \nabla)$ with a flat algebraic connection. There is a parameter space of finite dimensional complex representations of $X$ of dimension $n$. On this space there is an algebraic group $G$ acting - is this what you are looking for? – hm2020 Apr 25 '25 at 09:21
  • Unfortunately no. The situation there would only handle $O$-coherent $D$-modules and I think this is studied in depth by Simpson. – user1515097 Apr 25 '25 at 13:58
  • If $k$ is the complex numbers and $M(X,n)$ is the parameter space of all $n$-dimensional representations $k^n$ of $\pi_1(X)$, it follows you get a parameter space $M(X)$ "containing" $M(X,n)$ for all $n \geq 1$ - and using this parameter space it may be you could "study" holonomic $D$-modules that are not $\mathcal{O}_X$-coherent. But I believe this parameter space is very large - possibly infinite dimensional. You may be aware of the study of $D$-modues on flag varieties and $D$-affinity. Here I believe they construct such holonomic modules that are not $\mathcal{O}_X$-coherent. – hm2020 Apr 26 '25 at 08:48
  • https://en.wikipedia.org/wiki/Beilinson%E2%80%93Bernstein_localization - When studying D-modules on flag varieties they study certain modules on the enveloping algebra (Verma modules) - There is a book "Enveloping algebras" by Dixmier that gives a Lie theoretic introduction to this topic. – hm2020 Apr 26 '25 at 09:00

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