[EDIT: A previous version mistakenly argued that the fundamental group of *X* was responsible for torsion in the Picard group. I hope that this is correct now!  Btw, there is probably a more direct way of arguing, but I cannot find one at the moment.]

The Picard group of *X* is torsion free if and only if the group ${\rm H_1}(X,\mathbf{Z})$ vanishes.

By the exponential sequence, the torsion in the Picard group of *X* comes from the torsion in ${\rm H^2}(X,\mathbf{Z})$ and from ${\rm H^1}(X,\mathbf{C})/{\rm H^1}(X,\mathbf{Z})$. Thus the vanishing of ${\rm H_1}(X,\mathbf{Z})$ is equivalent (by the Universal Coefficient Theorem) to the torsion-freeness of the Picard group of *X*.