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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.

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  • $\begingroup$ Do you want an honest connection on $E$ or a relative connection with respect to $\pi$? There is a somewhat related paper by Biswas & Singh entitled "On the relative Connections" (2019). It is on arXiv. As far as I understand it they only show the existence of a relative connection under the assumption of connections along the fibers and this connection may not agree with the connections along the fibers. But it might still be a place to start. I do not know that paper in depth. $\endgroup$ Jun 4, 2023 at 17:50
  • $\begingroup$ I think this is certainly along the lines of what I'm looking for. I'll have to read it in more detail, but I very much appreciate the suggestion $\endgroup$
    – Aidan
    Jun 5, 2023 at 18:15

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