Restricted trichotomy in characteristic zero

Abstract

We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if M \mathcal M is any non-locally modular strongly minimal structure interpreted in an algebraically closed field K K of characteristic zero, then M \mathcal M itself interprets K K ; in particular, any non-1-based structure interpreted in K K is mutually interpretable with K K . Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.