Abstract

Given a cover $\mathbb{𝕌}$ of a family of smooth complex algebraic varieties, we associate with it a class $\mathfrak{𝔘}$, containing $\mathbb{𝕌}$, of structures locally definable in an o-minimal expansion of the real numbers. We prove that the class is ${\aleph }_0$-homogenous over submodels and stable. It follows that $\mathfrak{𝔘}$ is categorical in cardinality ${\aleph }_1$. In the case when the algebraic varieties are curves we prove that a slight modification of $\mathfrak{𝔘}$ is an abstract elementary class categorical in all uncountable cardinals.

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