Complex manifold with no divisors

Question

I read in Griffith Harris P132 that a complex manifold of dimension greater than one can have no divisors on it at all. I want to find examples. Is there an example? Does the Hopf manifolds $S^1\times S^{2n-1}$ have such properties?

Answer 1

The example you want, as mentioned in the comments, is that of an Inoue surface. Since they have no curves, they cannot have any divisors. On the other hand, consider the Hopf surface $S$ given by $(\mathbb{C}^2-{0})/\sim$ where $(x,y) \sim (2^nx,2^ny)$ for all $n\in\mathbb{Z}$, and the morphism $\varphi:S\to\mathbb{P}^{1}$ given by $[(x,y)] \mapsto [x:y]$. The pull back of a point $y\in\mathbb{P}^1$ will be a divisor on $S$. In fact, since $\varphi$ induces an elliptic fibration, the divisors are just elliptic curves. To finish, one can show that these are all the divisors on $S$, which establishes $\text{Cl}(S) \cong \mathbb{Z}$.

Answer 2

For $n=2$ there is a nontrivial picard group. There are a few decent references about this listed on the wikipedia page for Hopf surfaces.