Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, we can say that $U$ is a simply connected domain?


Any hints would be appreciated. 



Domains of holomorphy and simply connected domains
As stated by PVAL, the answer is no. Any domain in $\mathbb{C}$ is a domain of holomorphy, so in particular, for example $\mathbb{D}^\*$, the punctured disk is one. (This particular domain is also easily seen directly to be a domain of holomorphy). Thus, one obtains a counterexample in $\mathbb{C}$, and taking the product with $B$, the unit ball in $\mathbb{C}^{n-1}$, one obtains the same in $\mathbb{C}^n$. 


