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Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.

A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent crystal on the infinitesimal site of $X$, realising an equivalence between these two types of object.

Deligne's comparison of differential equations on $X$ and $X^{an}$ leads one to restrict one's attention to the subcategory of differential equations on $X$ with regular singularities along the boundary divisor of a good compactification of $X$.

My question: is there a clean way to describe this category as a subcategory of the category of crystals on $Inf(X)$?

My guess (fear?) is the answer might have something to do with log-geometry, but I know very little about such things so would also appreciate a good reference suggestion if indeed it is the required tool.

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The correspondence can be unpacked from Theorem 6.2 in Kato's "Logarithmic structures of Fontaine-Illusie" (pdf scans available by internet search), in particular by identifying $X = Y = D$, and $S = \operatorname{Spec} \mathbb{C}$. As you suggest, using Nagata and Hironaka, we may choose a smooth compactification with boundary ncd $(\bar{X},Z)$. Then the category of locally free crystals on the corresponding log scheme is equivalent to the category of locally free sheaves on $\bar{X}$ with integrable connection on $X$ that extends to the boundary with log poles.

See also the introduction to Ogus's paper "On the Logarithmic Riemann-Hilbert Correspondence".

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