A counter example to Hartogs' type extension of holomorphic line bundles

Zhangchi Chen
Université Paris-Sud
Lundi, 7 Janvier, 2019 - 14:00 - 15:00

The story begins with the well known Hartogs' extension theorem of holomorphic
functions: Let $\Omega\subset\mathbb{C}^n$ with $n\geqslant 2$ be a domain, $K
\subset\subset \Omega$ be a compact subset such that $\Omega\backslash K$ is
connected. Then every holomorphic function over $\Omega\backslash K$ can be
uniquely extended as a holomorphic function over $\Omega$.

For holomorphic line bundles we can ask the same question: with the same
geometric assumptions above, could every holomorphic line bundle defined over
$\Omega\backslash K$ be extended as a holomorphic line bundle over $\Omega$?
Such extension uniquely exists under some extra geometrical assumptions.

However, we cannot extend in general. In any dimension $n\geqslant 2$ we can
construct $\Omega$ and $K$ such that there exists a non-extendable holomorphic
line bundle over $\Omega\backslash K$. 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.