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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.