To any space $X$ you can associate its [de Rham space](https://ncatlab.org/nlab/show/de+Rham+space) $X\_{dR}$. Vector bundles on $X\_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.


Can anything like this be said for **meromorphic** connections?




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For instance, a naive idea is that there might literally be a space $X\_{mdR}$ whose vector bundles are vector bundles on $X$ with a flat meromorphic connection. Then there would be maps $\eta\_{dR}\to X\_{mdR}\to X\_{dR}$, where $\eta$ is the generic point of $X$ (on which all meromorphic bundles are holomorphic), suggesting that maybe $X\_{mdR}$ could be constructed as some sort of intermediate object between $\eta\_{dR}$ and $X\_{dR}$.


I'm aware that Deligne has a structure theorem for regular meromorphic connections, so maybe it's more reasonable to restrict to the regular meromorphic case.



A de Rham space for meromorphic connections?
Here is one way to construct an object $X\_{mdR}$ for $X$ say, a complex variety. Recall that $X\_{dR}$ can be constructed as follows. Let $\hat{X}$ be the formal completion of the diagonal inside $X\times X$. Then $X\_{dR}$ is the quotient of $X$ by the equivalence relation defined by $\hat{X}$.


Let $\eta$ denote the generic point of $X$. We can define an affine scheme $\hat{\eta}'$ as the inverse limit of the $\hat{U},$ as $U$ ranges over all affine opens in $X$. (I believe, though I didn't carefully check, that this is not the formal completion of $\eta$ inside $\eta\times\eta$.) It comes with two natural maps $p\_1,p\_2:\eta'\rightarrow\eta$ making $\hat{\eta}'$ an equivalence relation. Then we can define $X\_{mdR}$ to be the quotient of $\eta$ by $\hat{\eta}'$.


A vector bundle on $X\_{mdR}$ is a vector bundle $V$ on $\eta$ with an isomorphism $p\_1^\*V\cong p\_2^\*V.$ We can find an open $U\_1$ so that $V$ can be defined over $U\_1$, and we can find a smaller open $U\_2\subseteq U\_1$ so that the isomorphism $p\_1^\*V\cong p\_2^\*V$ is defined over $\hat{U\_2}$. (This follows from the general framework of "Noetherian approximation", but can also be seen explicitly.) Thus $V$ comes from a vector bundle with connection on $U\_2$.


That being said, I'm not convinced that this is a good way of thinking about meromorphic connections. My experience is that in non-finite type settings, usually trying to work with $D$-modules via constructions like the de Rham stacks runs into issues fairly quickly. For starters, I'm not sure if this works well in families, i.e., if there is any nice interpretation of vector bundles on $A\times X\_{mdR}$.


