Restricted trichotomy in higher dimensions

TL;DR

This paper proves that algebraic varieties of dimension greater than one, when equipped with certain strongly minimal structures, are necessarily locally modular, answering a question posed by Zilber.

Contribution

It establishes a new restriction on the structure of strongly minimal sets on higher-dimensional algebraic varieties, specifically proving local modularity.

Findings

Algebraic varieties of dimension > 1 with atomic Zariski-definable relations are locally modular.

Answers Zilber's question on the structure of such varieties.

Provides a new understanding of the geometry of strongly minimal structures in higher dimensions.

Abstract

I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic relations definable in the Zariski language on $M$, then $M$ is locally modular.