Injectivity in the Section Conjecture

Abstract

We recall the well known injectivity of the profinite Kummer map in the arithmetically relevant cases. There are at least two approaches towards injectivity. The abelian approach relies on the determination of the Kummer map for abelian varieties and their arithmetic, see Corollary 71, and also on the computation of the maximal abelian quotient extension ${\pi }_{{}_{1}}^{\mathrm{ab}}(X/k)$, see Proposition 69, which for later use in Sect. 13.5 we carefully revise also for smooth projective varieties of arbitrary dimension. The second approach is intrinsically anabelian and due to Mochizuki, see Theorem 76.In general, for a geometrically connected variety, the injectivity of the Kummer map (after arbitrary finite scalar extension) implies that the fundamental group must be large in the sense of Kollár, see Proposition 77, which imposes strong geometric constraints on possible higher dimensional anabelian varieties.