Galois sections for abelianized fundamental groups

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the \'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\'el\`ene and Sansuc.