Let $\pi:E\to B$ be a holomorphic fibration, and let $\mathcal{F}$ be a sufficiently nice sheaf (coherent for example) of $\mathcal{O}_E$-modules on $E$ that is flat over $B$ i.e. $\mathcal{F}$ is flat as a $\pi^{-1}\mathcal{O}_B$-module. This defines a family of sheaves on each fiber that locally preserve properties such as rank and degree. Suppose now that we have a connection $$\nabla_b:\mathcal{F}|_{\pi^{-1}(b)}\to \mathcal{F}|_{\pi^{-1}(b)}\otimes \Omega^1_{\pi^{-1}(b)}$$ on each fiber such that flat sections locally depend on $b$ holomorphically.
Is it possible to glue these connections together in order to obtain a connection on $E$. What sort of data is needed to do so, and what are the possible obstructions? It seems unlikely that it would be possible to do this globally over $B$, but if we assume $B$ is sufficiently small, like a polydisc of radius $\epsilon>0$, I imagine there should be some possibility. Any references where this problem or a similar problem are discussed would be very much appreciated.
