Model Theory of universal covering spaces of complex algebraic varieties

We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and reformulate this question as a question of categoricity of the universal covering space of a complex algebraic variety in a natural countable in nitary language. We provide partial positive results towards the question. Assuming a conjecture of Shafarevich and some assumptions on the fundamental group of the complex algebraic variety, we introduce a Zariski-like topology on the universal covering space of a complex algebraic variety which enjoys properties slightly weaker than those of a Zariski topology: the topology has descending chain condition for irreducible set, the projection of a closed set is closed, and some others, and we prove that a natural countable language is able to de ne rst-order the irreducible closed sets of the topology. Then we axiomatise a class of structures which admit topologies with similar properties; those properties are enough to prove model stability of the class. Following the programme of Zilber of logically perfect structures , the paper aims to provide a new class of examples of such structures.