Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. What if $X$ is a complex analytic manifold?
In his article "Regular connections after Deligne," Malgrange begins to sketch the proof: fix a point $x \in X$ and a minimal system of generators for $\mathscr{F}_x$. Consider a relation in which some coefficient has minimal order at $x$ and differentiate it. How do we finish the proof? I do not see how this calculation implies that the coefficients must vanish (and therefore $\mathscr{F}_x$ is free).