A Counterexample to Hartogs’ Type Extension of Holomorphic Line Bundles

Consider a domain Ω in C n with n ⩾ 2 and a compact subset K ⊂ Ω such that Ω \ K is connected. We address the problem whether a holomorphic line bundle defined on Ω \ K extends to Ω . In 2013, Fornæss, Sibony, and Wold gave a positive answer in dimension n ⩾ 3 , when Ω is pseudoconvex and K is a sub...

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Bibliographic Details
Published in:The Journal of geometric analysis 2018-07, Vol.28 (3), p.2624-2643
Main Author: Chen, Zhangchi
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Bundles
Bundling
Complex Variables
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Exhaustion
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Gluing
Mathematical analysis
Mathematics
Mathematics and Statistics
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Summary:Consider a domain Ω in C n with n ⩾ 2 and a compact subset K ⊂ Ω such that Ω \ K is connected. We address the problem whether a holomorphic line bundle defined on Ω \ K extends to Ω . In 2013, Fornæss, Sibony, and Wold gave a positive answer in dimension n ⩾ 3 , when Ω is pseudoconvex and K is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for K of general shape, we construct counterexamples in any dimension n ⩾ 2 . The key is a certain gluing lemma by means of which we extend any two holomorphic line bundles which are isomorphic on the intersection of their base spaces.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-017-9923-z