Abelian birational sections in characteristic 0

For a smooth and geometrically irreducible variety X over a field k of characteristic 0, the quotient of the absolute Galois group Gk(X) by the commutator subgroup of Gk¯(X) projects onto Gk. We investigate the sections of this projection. We show that such sections correspond to "infinite divisions" of the elementary obstruction of Colliot-Thélène and Sansuc. If k is a number field and the Tate-Shafarevich group of the Picard variety of X is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree 1. Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.