Submission date: 20 June 2006

Abstract:

I offer a language appropriate to describe the notion of (the homotopy class of) a path on a complex algebraic variety, or equivalenty the universal covering space of a complex algebraic variety, and prove partial results towards categoricity in infinitary logic Lw1w. The proof relates categoricity and stability of the considered structures to conjectures in complex analytic geometry (Shafarevich conjecture on the holomorphic convexity of a universal covering space of a complex algebraic variety) and Galois representations, and an explicit question about Galois action on abelian group extensions associated to elliptic curves.

The thesis generalises the paper of Zilber on group covers.

Mathematics Subject Classification: 03C45 03C98 03C75 32E99 14K99

Keywords and phrases: categoricity, fundamental groupoid functor, Shafarevich conjecture on holomorphic convexity, Galois action on torsion points of elliptic curves

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