Let $\Omega\subset\mathbb{C}^n$ be a domain of holomorphy, $m<n$. Is $\Omega\cap\mathbb{C}^m$ necessarily a domain of holomorphy?
If true, it seems that it will be useful in inductive arguments. However, I could neither prove nor disprove this. It is true in some nice cases, e.g., if $\Omega$ is Reinhardt, but I have no idea in general.
Thanks in advance!
It need not be a domain to begin with, that is, the intersection could be disconnected. Furthermore, even if the original has smooth boundary, the intersection may not. But other than that, using the Hartogs-pseudoconvex definition (there exists a continuous plurisubharmonic exhaustion function), the intersection is Hartogs-pseudoconvex if the original was. (That a restriction of a plurisubharmonic function to a subspace is plurisubharmonic is obvious from the definition).