# Intersection of a domain of holomorphy with a linear subspace

Let $$\Omega\subset\mathbb{C}^n$$ be a domain of holomorphy, $$m. Is $$\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.