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Let $X$ be a scheme and $\mathcal{F}$ an $\mathcal{O}_X$-module with a connection $\nabla$. If $f:X\to Y$ is a morphism of schemes, then we have a pushforward of $(\mathcal{F},\nabla)$ as a $\mathcal{D}$-module, namely a connection on $f_*\left(\mathcal{F}^{\nabla}\right)$. Its of course the right definition in as far as it is the ajoint functor to the pullback functor for $\mathcal{D}$-modules, but my question is if there exists a connection on $f_*\mathcal{F}$. In the special case that $f$ is étale, the pushforward as a $\mathcal{D}$-module gives a connection on the $\mathcal{O}$-module pushforward. Can get this for other classes of $f$, say smooth, proper, or is there a good obstruction for this? The only obstruction I know for the existence of a connection is for line bundles, where the vanishing of the first chern class is a necessary and sufficient condition (see here ), but I don't think this works here, unless this chern obstruction generalizes to arbitrary $\mathcal{O}_X$-modules.

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    $\begingroup$ "I don't think this works here": yes, it does. Take for instance for $f$ a double covering of smooth projective curves, branched along $2d$ points of $Y$. Then $f_*\mathscr{O}_X=\mathscr{O}_X\oplus L$, with $\deg L=-d$, thus $f_*\mathscr{O}_X$ doesn't admit a connection. $\endgroup$
    – abx
    Jun 17 at 12:00

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