A Counterexample to Hartogs’ Type Extension of Holomorphic Line Bundles

Abstract

Consider a domain $$\varOmega $$ in $$\mathbb {C}^n$$ with $$n\geqslant 2$$ and a compact subset $$K\subset \varOmega $$ such that $$\varOmega \backslash K$$ is connected. We address the problem whether a holomorphic line bundle defined on $$\varOmega \backslash K$$ extends to $$\varOmega $$ . In 2013, Fornæss, Sibony, and Wold gave a positive answer in dimension $$n\geqslant 3$$ , when $$\varOmega $$ 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 $$n\geqslant 2$$ . 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.