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Holomorphic Connections and Extension of Complex Vector Bundles
Author(s) -
Buchdahl N. P.,
Harris Adam
Publication year - 1999
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19992040103
Subject(s) - holomorphic function , mathematics , complex manifold , connection (principal bundle) , identity theorem , extension (predicate logic) , analyticity of holomorphic functions , vector bundle , pure mathematics , complex dimension , surjective function , compact riemann surface , riemann surface , mathematical analysis , geometry , computer science , programming language
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).