Let $E$ be a compact set in $\mathbb{C^n}$. Let $D=\mathbb{C^n}-E$.
a. Find an example for an $E$ so that $D$ is not a domain of holomorphy.
b. Are there examples for $E$ so that $D$ is a domain of holomorphy?
So, if $E$ is a compact set so that $D$ is connected, then by Hartogs Lemma, $D$ is not a domain of holomorphy. So an example for $E$ would be closed unit ball.
For part (b) if there are such examples then $D$ should be disconnected. I assume being disconnected is not suffices to decide that it is not a domain of holomorphy even though we use the term "domain". I tried few examples like several variable annulus etc. but everytime I got a component of $D$ which is not a domain of holomorphy. So far I was unable to prove that no such examples too.
Any help is appreciated.
If $E$ is a nonempty compact set and $f \in \mathscr{O}(D \backslash E)$, then there exists a holomorphic function $\widetilde{f}$ such that $f(z) = \widetilde{f}(z)$ for all $z \in D \backslash E$ and $\widetilde{f}$ is holomorphic for all $z \in D$. That is, compact sets are removable. This forms the content of Hartog's extension theorem, see page 172 of Shabat's Introduction to Complex Analysis -- Part II.
So unless $E$ is empty, $D$ cannot be a domain of holomorphy.
Unless you would like me to clarify anything, could you please mark my answer as the accepted one?
Thanks.