A Counterexample to Hartogs’ Type Extension of Holomorphic Line Bundles

Consider a domain Ω in Cⁿ 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.