Holomorphic Connections and Extension of Complex Vector Bundles

Letbe a regular, surjective holomorphic map between complex manifolds such that for all t ∈ Y, π −1 (t) is a connected, simply connected Riemann surface. Let K C X be compact, and E ⊂ X \ K a holomorphic vector bundle, equipped with a holomorphic relative connection along the fibres of π. The main result of this note establishes unique existence of a holomorphic vector bundle extension Ê→ X under the added assumptions that π (K) is a proper subset of Y, and π −1 (t) ∪ (X \ K) is always non‐empty and connected. As a corollary of the main theorem, it follows that if X is an arbitrary complex manifold, and A C X is an analytic subset of co dimension at least two, then E → X \ A admits a unique extension if there exists a holomorphic connection ▽:O x (E) → Ω   1 X (E).