A counterexample to Hartogs' type extension of holomorphic line bundles

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