Abstract
We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and relate a conjecture of Shafarevich in complex analytic geometry via a model-theoretic approach of logically perfect structures developed by Zilber.
We show how this question can be formulated, and arises from, as a question of $L_{\omega_1\omega}$-categoricity naturally arising in the approach of analytic Zariski structures.
In the presented paper we answer the question partially by proving $\omega$-homogeneity and $\omega$-stability \emph{over models} of a related $L_{\omega_1\omega}$-class.