The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain in the extended complex plane. In particular, then, the universal cover of a domain in the extended complex plane is biholomorphic to a domain in the extended complex plane. This leads to an analogous question in higher dimensions: Is the universal cover of a domain in complex projective space biholomorphic to a domain in complex projective space? More precisely, I am asking for a counterexample. Many results in one complex variable break in several complex variables, and the Uniformization Theorem is fairly delicate, so it seems reasonable to expect it to break. Perhaps there is a counterexample that one can see just by topology?
Consider a tubular neighborhood of three generic lines on P^2. The fundamental group is Z. The universal covering will contain an infinite chain of P^1's, and in particular two disjoint P^1's. Thus it cannot be a domain in P^2.