Analytic Zariski structures and non-elementary categoricity

TL;DR

This paper explores the model-theoretic properties of analytic Zariski structures, establishing their stability and homogeneity, and connecting geometric notions like Hrushovski's predimension to non-elementary categoricity.

Contribution

It introduces a method to associate abstract elementary classes with analytic Zariski structures and analyzes their stability, quasi-minimality, and homogeneity, linking geometric and model-theoretic concepts.

Findings

The class associated with a one-dimensional analytic Zariski structure is stable.

The class is quasi-minimal and homogeneous over models.

Hrushovski's predimension naturally arises in this context as a geometric tool.

Abstract

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski's predimension arises in this general context as a natural geometric notion and use it as one of our main tools.