TITLE: Non-compact complex manifolds which are not Stein
QUESTION [5 upvotes]: I am studying Stein manifolds, and it is clear for me that compact complex manifolds can not be Stein for obviously reasons. On the other hand, there exists some non-compact complex manifolds which are not Stein, otherwise every non-compact complex manifolds is Kähler.
Does anyone know of some explicit examples of non-compact complex surfaces which are not Stein?
In general non-compact non-Kähler manifolds are not Stein; but I do not know any explicit examples.

REPLY [5 votes]: As you pointed out, Stein manifolds are Kähler, so one way to find non-compact complex manifolds which are not Stein is to look for non-compact complex manifolds which are not Kähler. 
Let $S$ be a non-Kähler complex surface (such as a Hopf surface), then for any $k > 0$, $S\times\mathbb{C}^k$ is a non-compact complex manifold which is not Kähler (submanifolds of Kähler manifolds are Kähler). This construction provides examples in dimension at least three. As discussed in this MO question, a Hopf surface with a point removed is an example of a non-compact complex surface which is not Kähler.