The Riemann extension theorem in several complex variables is the following. Let $f$ be a holomorphic function on $D \setminus A$ where $A$ is a proper analytic set in a domain $D \subset \mathbb{C}^n$. If $f$ is bounded it uniquely extends to a holomorphic function on $D$.

What is the conclusion if $A$ is replaced by a smooth real manifold ? If $A$ is a real hyperplane the theorem is false (take $A=\{Re(z_1=0\}$ and $f(z)= 1$ if $Re(z_1)>0$ and $-1$ if $Re(z_1)<0$ which is holomorphic on $C^2 \setminus A$ but can not be continuously extended on $C^2$).

If the real codimension of $A$ is great than $2$, what can I say ? I think the theorem is true but I don't know how to prove that.

Thank you


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