TITLE: Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
QUESTION [7 upvotes]: Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What can we say about $X$?
I am mainly interested in the case where $\dim X = 2$, and $\Omega^1_X$ satisfies some positivity properties such as ampleness.
What is "known" about surfaces $X$  with infinite abelian fundamental group, finite $\pi_2$ and ample $\Omega^1_X$?

REPLY [7 votes]: With regard to surfaces: it is a simple theorem that $\pi_2$ of any closed $4$-manifold is torsion free; so one doesn't need any complex geometry to see that there are no such projective surfaces.  The argument I will give was observed by the late Jerry Levine and me in 2004 after some correspondence with Gurjar following his paper referred to in ulrich's answer.  The idea is to apply the argument that follows to the universal cover of a closed 4-manifold. It seems that the more interesting question, addressed by Gurjar for complex surfaces, is whether $\pi_2$ is actually free; our attempts to resolve this ran into some well-known problems related to group cohomology and ends of manifolds.
Let X be a simply connected $4$-manifold with empty boundary. Then $H_2 (X)$ is isomorphic to 
$H^2_c (X)$, cohomology with compact supports. This is the direct 
limit of $H^2 (X,X-C)$, where $C$ ranges over compact subsets of X.  we 
have an exact sequence
$$
0 \to H^1 (X-C) \to H^2 (X,X-C) \to H^2 (X)
$$
Now $H^1 (X-C)$ is torsion-free because $H^1$ always is.  $H^2 
(X)=Hom(H_2 (X),Z)$ is also torsion-free since $X$ is simply connected, and so $H^2 (X,X-C)$ is 
torsion-free.
But now it is easy to see that the direct limit of a family of 
torsion-free groups is torsion-free.