Covers of Abelian varieties as analytic Zariski structures

We use tools of mathematical logic to analyse the notion of a path on an 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 uncountable 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. We also show that, with this topology, the universal covering space of the variety is an analytic Zariski structure. Technically, we present a countable $L_{omega_1\omega}$-sentence axiomatising a class of analytic Zariski structures containing the universal covering space of an algebraic variety over a number field, under some assumptions on the variety.