Anabelian Intersection Theory I: the Conjecture of Bogomolov-pop and Applications

A. Grothendieck first coined the term “anabelian geometry” in a letter to G. Faltings [Gro97a] as a response to Faltings’ proof of the Mordell conjecture and in his celebrated Esquisse d’un Programme [Gro97b]. The “yoga” of Grothendieck’s anabelian geometry is that if the étale fundamental group πét 1 pX,xq of a variety X at a geometric point x is rich enough, then it should encode much of the information about X as a variety; such varietiesX are called anabelian in the sense of Grothendieck, and have the property that two anabelian varieties have isomorphic étale fundamental groups if and only if they are isomorphic; and that the isomorphisms between their étale fundamental groups are precisely the isomorphisms between the varieties. Grothendieck did not specify how much extra information should be encoded, and there is currently not a consensus on the answer. An anabelian theorem (or conjecture) is a theorem (or conjecture) which asserts that a class of varieties are anabelian. Grothendieck wrote in [Gro97a] about a number of anabelian conjectures, one regarding the moduli of curves, defined over global fields (which is still open); one regarding hyperbolic curves, defined over global fields; and a birational anabelian conjecture, which asserts that Spec of finitely-generated, infinite fields are anabelian (in this case, we say the fields themselves are anabelian). The anabelian conjecture for hyperbolic curves was proved in the 1990’s by A. Tamagawa and S. Mochizuki ([Tam97], [Moc99]). The birational anabelian conjecture for finitely-generated, infinite fields is a vast generalization of the pioneering NeukirchIkeda-Uchida theorem for global fields ([Neu69], [Uch77], [Ike77], [Neu77]), and is now a theorem due to F. Pop [Pop94]. Grothendieck remarked that “the reason for [anabelian phenomena] seems. . . to lie in the extraordinary rigidity of the full fundamental group, which in turn springs from the fact that the (outer) action of the ‘arithmetic’ part of this group. . . is extraordinarily strong” [Gro97a]. F. Bogomolov had the surprising insight [Bog91] that as long as the dimension of a variety is ě 2, anabelian phenomena can be exhibited — at least birationally — even over an algebraically closed field, even in the complete absence of the “arithmetic” part of the group Grothendieck referenced. Given a field K, we let GK denote the absolute Galois group of K, the profinite group of field automorphisms of its algebraic closure K (see [NSW08] for more details). Given two fields F1 and F2, we let IsompF1, F2q denote the set of isomorphisms between the pure inseparable closures of F1 and F2, up to Frobenius twists. Given two profinite groups Γ1 and Γ2, we let IsompΓ1,Γ2q denote the set of equivalence classes of continuous isomorphisms from Γ1 to Γ2, modulo conjugation by elements of Γ2. There is a canonical map φF1,F2 : Isom pF1, F2q Ñ IsompGF2 , GF1q (1)