Is there some kind of classification of (connected) smooth complex varieties such that every homotopy group of the manifold of complex points is torsion-free? Any reference on this topic will be most welcome.



Complex varieties with non-torsion homotopy groups
Following Mark Grant's comment, referencing David Chataur's answer [here](https://mathoverflow.net/questions/207448/simply-connected-cw-complex-with-only-finitely-many-nontrivial-homotopy-and-homo): 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.


