The O-minimal Zilber Conjecture in Higher Dimensions

TL;DR

This paper proves a higher-dimensional version of the o-minimal Zilber conjecture, establishing that certain strongly minimal structures in o-minimal expansions of real closed fields are two-dimensional, with applications to complex manifolds.

Contribution

It extends the o-minimal Zilber conjecture to higher dimensions and applies the result to strongly minimal structures in complex geometry.

Findings

Proved the higher-dimensional o-minimal Zilber conjecture.

Established the Zilber trichotomy for structures in complex manifolds.

Demonstrated that non-locally modular strongly minimal structures are two-dimensional.

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.