I am stuck at the following question :
Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function on $\mathbb{C}^n$.
I think that I have to somehow use the Riemann extension theorem here. But I am not being able to carry out the details.
Can anybody please help ?? Any suggestion or comment is very much appreciated.
Best Answer
Riemann's second extension theorem says that if $M$ is a holomorphic manifold and $S\subset M$ is an analytic subset of codimension at least two, the restriction map $\mathcal O(M) \to \mathcal O(M\setminus S) $ is bijective.
Your question is solved by applying this result to $M=\mathbb C^n$ and $S=\{(z_1, z_2, \ldots, z_n) \in \mathbb C^n| z_1=z_2=0\}$
The theorem is not very difficult but for some reason it is underappreciated and rather difficult to find in the literature.
I located a slightly weaker but very elementary version in Gunning's Introduction to Holomorphic Functions of Several Variables, volume I, Theorem 2, page 30.
(The weaker version is local and supposes $S$ smooth, but this suffices for your question).
The general version can be found in Grauert-Remmert's masterful Coherent analytic Sheaves, page 132.