  In this paper, we develop the main step in the global theory for the
mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for
higher-dimensional function fields over algebraically closed fields. More
precisely, we show how to reconstruct a function field $K$ of transcendence
degree $\geq 5$ over an algebraically closed field, up-to inseparable
extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed
with the collection of mod-$\ell$ rational quotients.
