Zilber’s restricted trichotomy in characteristic zero

Abstract

We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that ifM\mathcal Mis any non-locally modular strongly minimal structure interpreted in an algebraically closed fieldKKof characteristic zero, thenM\mathcal Mitself interpretsKK; in particular, any non-1-based structure interpreted inKKis mutually interpretable withKK. 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.