Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H\_B^2(X,\mathbb{Z})$ has non-trivial torsion.


Q2. How do we think about these torsion elements? What is the geometrical content behind it? 



Torsion in the Betti cohomology of complex surfaces
The torsion of $H\_B^2(X,\mathbb{Z})$ is that of $H\_1(X,\mathbb{Z})=\pi\_1(X)^{ab}$ (universal coefficient theorem for cohomology), so the simplest case should be a simply connected complex surface quotiented by a fixed point free holomorphic involution (or a prime order automorphism).


I would propose an [Enriques surface](http://en.wikipedia.org/wiki/Enriques_surface), but I'm not at all convinced it is "simplest".


