The O-minimal Zilber Conjecture in Higher Dimensions

Abstract

We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal{R}$ be an o-minimal expansion of a real closed field, let $M$ be an interpretable set in $\mathcal{R}$, and let $\mathcal{M}=(M,...)$ be a reduct of the induced structure on $M$. If $\mathcal{M}$ is strongly minimal and not locally modular, then ${\dim }_{\mathcal{R}}(M)=2$. As an application, we prove the Zilber trichotomy for all strongly minimal structures interpreted in the theory of compact complex manifolds.