Complex varieties with non-torsion homotopy groups

Following Mark Grant's comment, referencing David Chataur's answer here: McGibbon and Neisendorfer proved that a finite-dimensional 1-connected space with any nonzero reduced homology has infinitely many homotopy groups with torsion. In particular, this applies to the universal cover $\tilde{X}$ of the space $X$ we care about. So $\tilde{X}$ must be acyclic (and is simply-connected), and is thus contractible by Hurewicz's theorem. Therefore $X$ is aspherical.

The answer is likely to be a bit messy, since every projective curve is a $K(\pi, 1),$ and so are cartesian products and (iterated) fibrations of such (classifying which of these are algebraic is, presumably, still open, even for surface bundles over surfaces). I assume, in my ignorance, that complements of hyperplane arrangements are not of this form, which makes me think that the question is hopeless.