# Uniformization theorem in higher dimensions

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.

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

Given any smooth complex surface, you can blow up any point on the surface:

http://en.wikipedia.org/wiki/Blowing_up

This gives you another smooth, complex surface with isomorphic fundamental group but with the second Betti number increased by 1. (There is, in some sense, essentially one more curve on the new surface than there was on the old one, called the "exceptional divisor".)

Starting with any one simply connected smooth complex surface (e.g. the projective plane) and repeatedly blowing up shows that your question has a negative answer.

As a complement to Georges Elencwajg's answer, there are lots of different simply connected hypersurfaces in projective space. The second volume of Shafarevich's introduction to algebraic geometry has a discussion of just this question of uniformization of complex manifolds.