Zilber's Trichotomy in Hausdorff Geometric Structures

TL;DR

This paper develops a new axiomatic framework for Zilber's trichotomy within Hausdorff geometric structures, applying it to classify relics of algebraically closed fields and related structures, advancing understanding of their model-theoretic properties.

Contribution

It introduces Hausdorff geometric structures and completes the proof of Zilber's trichotomy for relics of algebraically closed fields, including real and valued fields.

Findings

Proves the trichotomy for strongly minimal relics on real sorts.

Shows non-locally modular relics interpret one-dimensional groups.

Reduces the trichotomy problem for ACVF relics to a conjectural condition.

Abstract

We give a new axiomatic treatment of the Zilber trichotomy, and use it to complete the proof of the trichotomy for relics of algebraically closed fields, i.e., reducts of the ACF-induced structure on ACF-definable sets. More precisely, we introduce a class of geometric structures equipped with a Hausdorff topology, called \textit{Hausdorff geometric structures}. Natural examples include the complex field; algebraically closed valued fields; o-minimal expansions of real closed fields; and characteristic zero Henselian fields (in particular $p$-adically closed fields). We then study the Zilber trichotomy for relics of Hausdorff geometric structures, showing that under additional assumptions, every non-locally modular strongly minimal relic on a real sort interprets a one-dimensional group. Combined with recent results, this allows us to prove the trichotomy for strongly minimal relics on the real sorts of algebraically closed valued fields. Finally, we make progress on the imaginary sorts, reducing the trichotomy for \textit{all} ACVF relics (in all sorts) to a conjectural technical condition that we prove in characteristic $(0,0)$.