Holomorphic connections and extension of complex vector bundles

Let X (pi) under right arrow;Y be a regular, surjective holomorphic map bet ween complex manifolds such that for all t is an element of Y, n(-1)(t) is a connected, simply connected Riemann surface. Let K subset of X be compact , and E --> X \ K a holomorphic vector bundle, equipped with a holomorphic relative connection along the fibres of pi. The main result of this note es tablishes unique existence of a holomorphic vector bundle extension (E) ove r cap --> X under the added assumptions that pi(K) is a proper subset of Y, and a-l(t) boolean AND (X \ K) is always non-empty and connected. As a cor ollary of the main theorem, it follows that if X is an arbitrary complex ma nifold, and A subset of X is an analytic subset of codimension at least two , then E --> X \ A admits a unique extension if there exists a holomorphic connection del:O-X(E) --> Omega(X)(1)(E).