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Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be). Let $\mathcal{E}$ be a locally free sheaf on $X$, and $$\nabla:\mathcal{E} \to \Omega_{X/S}\otimes\mathcal{E}$$ an $\mathcal{O}_S$-linear connection.

It would also be very useful to me if we allow $\nabla$ to be a logarithmic connection, i.e. taking values in $\Omega_{X/S}(D)\otimes\mathcal{E}$ for some divisor $D \subset X$.

We have the sheaf of $\nabla$-constants, a sub-$\mathcal{O}_S$ module of $\mathcal{E}$: $$\mathcal{E}^\nabla := \mathrm{ker}\left[\mathcal{E} \xrightarrow{\nabla} \Omega_{X/S}\otimes\mathcal{E}\right]$$

Question: What is the sheaf cohomology $H^1(X,\mathcal{E}^\nabla$)? (or rather the higher direct image $R^1\pi_*(\mathcal{E}^\nabla))$

If $\mathcal{E}^\nabla$ is a constant sheaf, then it is flasque and hence acyclic. However I haven't been able to figure out how to show that $\mathcal{E}^\nabla$ is a constant sheaf. What can we say here?

I would even be content to understand the simplified case of: $\mathcal{E}$ a line bundle and $\pi$ relative dimension 1 with normal crossing singularities.

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    $\begingroup$ I would guess that, at least in characteristic 0, $\mathcal{E}^{\nabla}$ is a vector bundle on $S$ which is trivialized by a finite etale covering $S' \rightarrow S$ (i.e. a finite vector bundle in the sense of Nori). $\endgroup$
    – Jesse Silliman
    Commented Mar 3, 2022 at 1:31
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    $\begingroup$ Probably you can check that this sheaf satisfies a relative form of flasqueness, where for $U \subset V$ open sets, if the image of $U$ and $V$ in $S$ is the same, then the restriction map is surjective. The point is to consider a section on $U$, which has a pole on some component of $V \setminus U$, and check that the derivative must have a pole of higher order to get a contradiction. Then that claim should suffice for vanishing of the higher direct image. $\endgroup$
    – Will Sawin
    Commented Mar 3, 2022 at 3:27
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    $\begingroup$ @WillSawin I think I should have said that I'm guessing that $\mathcal{E}^{\nabla}$ is a finite rank $\pi^*\mathcal{O}_S$-module on $X$ which becomes a "constant", i.e. pulled back from on $S$, after a finite etale covering $X' \rightarrow X$. This is not a Nori finite vector bundle anymore, though. $\endgroup$
    – Jesse Silliman
    Commented Mar 3, 2022 at 5:31
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    $\begingroup$ I'm not sure what is needed exactly but normal sounds right. $\endgroup$
    – Will Sawin
    Commented Mar 3, 2022 at 15:55
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    $\begingroup$ @PrimeRibeyeDeal Yes, I meant $\pi^{-1} \mathcal{O}_S$, and you are right that some sort of hypothesis, such as normal, is necessary. If $S = \mathrm{Spec}(\mathbb{C})$, this type of object is a sheaf whose analytification is a $\mathbb{C}$-local system on $X^{\mathrm{an}}$ with finite monodromy. $\endgroup$
    – Jesse Silliman
    Commented Mar 3, 2022 at 16:23

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