Let $\mathbb{B}^n \subset \mathbb{C}^n$ be the unit ball and $V \subset \mathbb{B}^n$ be a closed complex submanifold of codimension one. How do we show that $\mathbb{B}^n \setminus V$ is a domain of holomorphy?
Showing that $\mathbb{B}^n \setminus V$ is a domain of holomorphy is equivalent to showing that there is a holomorphic function on the domain that is singular at every boundary point.