Holomorphic connections and extension of complex vector bundles

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:disp-formula> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="urn:x-wiley:0025584X:media:MANA19992040103:nueq001"><jats:alt-text>equation image</jats:alt-text></jats:graphic> </jats:disp-formula> be a regular, surjective holomorphic map between complex manifolds such that for all t ∈ Y, π<jats:sup>−1</jats:sup>(t) is a connected, simply connected Riemann surface. Let <jats:italic>K</jats:italic> C <jats:italic>X</jats:italic> 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<jats:sup>−1</jats:sup> (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<jats:sub>x</jats:sub> (E) → Ω<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tex2gif-stack-1.gif" xlink:title="urn:x-wiley:0025584X:media:MANA19992040103:tex2gif-stack-1" />(E).</jats:p>