A counter example to Hartogs' type extension of holomorphic line bundles (Zhangchi Chen - Université Paris-Sud)

The story begins with the well known Hartogs' extension theorem of holomorphic
functions: Let <nobr>Ω⊂Cn</nobr> with <nobr>n⩾2</nobr> be a domain, <nobr>K⊂⊂Ω</nobr> be a compact subset such that <nobr>Ω∖K</nobr> is
connected. Then every holomorphic function over <nobr>Ω∖K</nobr> can be
uniquely extended as a holomorphic function over <nobr>Ω</nobr>.

For holomorphic line bundles we can ask the same question: with the same
geometric assumptions above, could every holomorphic line bundle defined over
<nobr>Ω∖K</nobr> be extended as a holomorphic line bundle over <nobr>Ω</nobr>?
Such extension uniquely exists under some extra geometrical assumptions.

However, we cannot extend in general. In any dimension <nobr>n⩾2</nobr> we can
construct <nobr>Ω</nobr> and <nobr>K</nobr> such that there exists a non-extendable holomorphic
line bundle over <nobr>Ω∖K</nobr>. 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.