Let $M$ be a 4-manifold with a complex structure.


Does there exist a finite list of simply connected complex 4-manifolds $M\_1, ... , M\_n$ such that M is the quotient of some $M\_i$ by the action of a group acting discretely on $M$?


This would be an analog of the Poincare-Koebe uniformization theorem in (real) dimension 2. People who I've asked this question to think that it's unlikely that there is such a list, but haven't been able to offer an argument or a reference.



Uniformization theorem in higher dimensions
There exist infinitely many holomorphically non-isomorphic complex structures on the unit ball of R^4 (or more generally R^2n): this is a beautiful theorem of Burns, Shnider, Wells ( Inventiones Math. 46, p. 237-253 ,1978) .
Since these complex manifolds are simply connected (even contractible) they are their own covering spaces and this proves that the set (hum, hum) of universal covers of complex manifolds of dimension 2 is infinite.
As a historical aside, Poincaré was first in noticing that the unit ball and the bidisk (DXD) in C^2 were not holomorphically isomorphic.
Here is the reference to Kang-Tae's nice survey on the subject.


<http://mathnet.kaist.ac.kr/mathnet/kms_tex/100248.pdf>


