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I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection $$ \nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X, $$ then $\mathcal{F}$ is automatically locally free.

Does this result remain valid when one has a connection with logarithmic singularities along some normal crossings divisor?

Any reference, examples, counter-examples will be appreciated.

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I may be misunderstanding something here, but isn't the following a counterexample?

Consider $X=\mathbb A^1 = Spec(\mathbb C[z])$, with the divisor $D= \{0\}$. Let $\mathcal F$ be the 1-dimensional skyscraper sheaf at $0$. Let $\delta$ denote a non-zero section of the stalk at $0$ (so $z\delta = 0$). The sheaf $\mathcal F$ admits a connection with logarithmic singularities defined by:

$$ \nabla (\delta) = - \delta \otimes \frac{dz}{z}.$$

But $\mathcal F$ is certainly not locally free (I am really thinking of $\mathcal F$ as a lattice inside the $D$-module of delta functions at $0$).

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    $\begingroup$ Yes, this is the standard counterexample. $\endgroup$ – S. Carnahan Jun 28 '13 at 2:42

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