# Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.

Is there a non-separable complex manifold? Can a non-separable complex manifold be normal (or non-normal)? Also, does there exist a separable normal complex manifold which is not paracompact? For these questions, what is the minimum dimension for such a counterexample?

This paper gives examples of separable complex manifolds which are not normal, but I have not been able to find any reference to other kinds of non-paracompact complex manifolds. I would also be interested in examples of non-paracompact complex manifold besides the ones that Calabi and Rosenlicht have constructed that have different general topological properties than the properties that are satisfied by the manifolds of Calabi and Rosenlicht. Other references for such manifolds will also be helpful. Furthermore, every Riemann surface is paracompact, so any of these strange manifolds must have complex dimension at least 2 (and hence real dimension at least 4).

The motivation behind this question is mainly curiosity. This question may depend on set theoretic hypotheses, so these sorts of questions could form a rare connection between the fields of set theory and complex variables.

• David Gauld recently published a book whose title is "Non-metrisable Manifolds". While I think it just brushes complex manifolds (as it focuses on topological manifolds), you might find it interesting. – Mathieu Baillif Jul 22 '15 at 10:02

• A complex manifold must have a set of charts with holomorphic transition functions, not just be locally homeomorphic to $\mathbb{C}$. – Eric Wofsey Jul 24 '15 at 12:39