Covers of Abelian varieties as analytic Zariski structures

Abstract

We use tools of mathematical logic to analyse the notion of a path on a complex algebraic variety, and are led to formulate a “rigidity” property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for ℵ1-categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space.Technically, for a variety A defined over a finite field extension of the field Q of rational numbers, we introduce a countable language L(A) describing the universal covering space of A(C), or, equivalently, homotopy classes of paths in π1(A(C)). Under some assumptions on A we show that the universal covering space of A(C) is an analytic Zariski structure (2010) [26]), and present an Lω1ω(L(A))-sentence axiomatising the class containing the structure and that is stable and homogeneous over elementary submodels. The “rigidity” condition on fundamental groups says that projection of the fundamental group of a variety is the fundamental group of the projection, up to finite index and under some irreducibility assumptions, and is used to prove that the projection of an irreducible closed set is closed in the analytic Zariski structure.In particular, we define an analytic Zariski structure on the universal covering space of an Abelian variety defined over a finite extension of the field Q of rational numbers.