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arXiv · 2206.03279

Non-elementary categoricity and projective locally o-minimal classes

Boris Zilber
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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.

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Boris Zilber. 2022-06-07. Non-elementary categoricity and projective locally o-minimal classes. https://doi.org/10.2140/mt.2024.3.101

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