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Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve.

Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex $n$-dimensional vector bundle map on $X-\{P_1, \ldots, P_m\}$ since in $P_1, \ldots, P_m$ the local triviality condition is not satisfied.

I am looking for a general notion of this phenomena. In the spirit of, "locally free sheaves correspond to vector bundles". I am not sure if coherent sheaves are the right objects because I do not want the fibre dimension to vary but only to have no local triviality at a finite number of points.

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Do you want the fibers over the $P_i$'s to have a linear structure? Do you want $E$ to be an analytic space? Do you require the map $E\to X$ to be continuous/holomorphic? – Qfwfq Apr 14 '11 at 12:15
may be parabolic bundles are what you are looking for ? – Niels Apr 15 '11 at 7:14

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