# Existence of connection on the pushforward of a sheaf with connection

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.

• "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.
– abx
Jun 17 at 12:00