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We use tools of mathematical logic to analyse the notion of a path on ancomplex algebraic variety, and are led to formulate a "rigidity" property offundamental groups specific to algebraic varieties, as well as to define a bonafide topology closely related to etale topology. These appear as criteria foruncountable categoricity, or rather stability and homogeneity, of the formalcountable language we propose to describe homotopy classes of paths on avariety, or equivalently, its universal covering space. We also show that, withthis topology, the universal covering space of the variety is an analyticZariski structure.  Technically, we present a countable $L_{omega_1\omega}$-sentence axiomatisinga class of analytic Zariski structures containing the universal covering spaceof an algebraic variety over a number field, under some assumptions on thevariety.

Covers of Abelian varieties as analytic Zariski structures