Non-elementary categoricity and projective locally o-minimal classes

TL;DR

This paper constructs a class of structures from complex algebraic varieties within an o-minimal framework, proving categoricity and stability properties, with implications for model theory of such classes.

Contribution

It introduces a new class of structures associated with algebraic varieties, demonstrating categoricity and stability in uncountable cardinals, extending the understanding of non-elementary classes.

Findings

The class is $eth_0$-homogeneous over submodels.

The class is stable.

Categorical in $eth_1$ and all uncountable cardinals in one-dimensional case.

Abstract

Given a cover $\mathbb{U}$ of a family of smooth complex algebraic varieties, we associate with it a class $\mathcal{U},$ containing $\mathbb{U}$, of structures locally definable in an o-minimal expansion of the reals. We prove that the class is $\aleph_0$-homogenous over submodels and stable. It follows that $\mathcal{U}$ is categorical in cardinality $\aleph_1.$ In the one-dimensional case we prove that a slight modification of $\mathcal{U}$ is an abstract elementary class categorical in all uncountable cardinals.