Cycle Classes in Anabelian Geometry

Abstract

Cycle classes appear in the context of anabelian geometry in different incarnations, see for example Parshin (The Grothendieck Festschrift, vol. 3, 1990), Mochizuki (Invent. Math. 138(2):319–423, 1999; Mathematical Sciences Research Institute Publications, vol. 41, 2003; J. Math. Kyoto Univ. 47(3):451–539, 2007), Esnault and Wittenberg ( Mosc. Math. J. 9(3):451–467, 2009). After recalling and comparing several known constructions we describe yet another construction of the cycle class of a section.Our construction relies on the principle that for a hyperbolic curve, or more generally for an algebraic K(π, 1)-space, see Sect. 6.2 below, cohomological arguments could take place on the finite étale site. This version of the cycle class originates again from the class of the diagonal, which we consider as the universal cycle class for points on the curve. The cycle class of a section arises as the evaluation of this universal family of cycle classes in the section just as if the section were a closed point. The construction by evaluation extends to subvarieties and also to the relative case of cycles in families.As an application we partially present Parshin’s proof of the geometric Mordell Theorem using fundamental groups and hyperbolic geometry (Parshin, The Grothendieck Festschrift, vol. 3, 1990) from an algebraic viewpoint, at least concerning the use of fundamental groups and cycle classes.