# 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

## 1 Answer

Really just a small remark: in analogy to the simplest way to get a non-separable real manifold---gluing uncountably many copies of the unit interval to get the long line, one can extend the standard construcion of the universal covering surface of the punctured plane, i.e., the Riemann surface of the logarithm. The latter glues together countably many copies of the ruptured plane obtained by cutting open the punctured plane along the positive real axis. One can do the same with uncountably many to obtain a non-separable, non paracompact complex manifold.

Edited for the sceptics. If all you want is a non-separable complex manifold, then you need only take the square of the long line. You could also just take the product of the open half-line with the long line which is essentially the example I gave.

• Is there no difficulty handling limit steps of such a construction? – Eric Wofsey Jul 22 '15 at 13:47
• If I understand this answer correctly, this would give a Riemann surface, but every Riemann surface is paracompact, so I do not think this construction would work. – Joseph Van Name Jul 23 '15 at 17:51
• if you glue uncountably many then the resulting space will not be separable. It will also contain copies of the long line as closed subsets. – priel Jul 24 '15 at 4:22
• 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
• Completely mysified by the down-vote and your objection. The simplest example I can think of---real line times long line satisfies all condtions---tschüß und baba. – priel Jul 24 '15 at 14:58