The Picard group is torsion free if and only if *X* is simply connected.  Indeed, if *L* is a line bundle on *X* representing a torsion element in the Picard group, then there is an associated connected \'etale cover of *X* of degree the order of *L* in the Picard group. (See e.g. Barth-Hulek-Peters-van de Ven, Compact Complex Surfaces, p. 54.) Conversely, if *X* is simply connected, then the fact that the Picard group is torsion free follows from the exponential sequence.