Geometrically Abelian Sections on Higher-Dimensional Varieties

We investigate how much arithmetic information about a variety can be detected by sections of the geometrically abelian fundamental groups of sufficiently small open subvarieties. Over a $p$-adic field, such a section determines a rational point on the Albanese torsor, and, under a conjectural assumption, this point is represented by a $0$-cycle of degree $1$ on the original variety if the variety has index $1$. For curves, the latter statement holds unconditionally. Over finitely generated fields, such sections still force the relative Brauer group to vanish away from the characteristic. The key input is an identification of the Chern class obstruction attached to a section with the obstruction arising from local Tate duality. As a byproduct, we give a new proof of an open-subvariety strengthening of a theorem of Esnault and Wittenberg.