Non-compact complex manifolds which are not Stein

Question

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.

Answer 1

Any non-pseudoconvex open subset of $\mathbb C^n$ will do. For example, $\mathbb C^2$ minus a point is not holomorphically convex, because Hartogs' extension theorem shows that any holomorphic function defined on a punctured neighborhood of a point extends holomorphically across the missing point. Therefore, it's not Stein.

Answer 2

Since no compact holomorphic manifold of positive dimension embeds in a complex Euclidean space, any non-compact holomorphic manifold containing compact holomorphic submanifolds of positive dimension fails to embed. Since the complement of a point in $\mathbf{CP}^{2}$ contains families of embedded curves, it is indeed a non-compact manifold that isn't Stein.

Alternatively, as Jack Lee notes, if $X$ is compact and $Y \subset X$ is compact and has complex codimension at least two, then $X \setminus Y$ is non-compact but not Stein (because every holomorphic function on $X \setminus Y$ extends to $X$ by Hartogs' theorem. Particularly, the complement of a point $Y$ in $X = \mathbf{CP}^{2}$ is not Stein.

Answer 3

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.