$\aleph_0$-categorical strongly minimal compact complex manifolds

TL;DR

The paper investigates the model-theoretic property of essential -categoricity in strongly minimal compact complex manifolds, providing characterizations and counterexamples that challenge existing conjectures.

Contribution

It introduces the notion of essential -categoricity for these manifolds, characterizes triviality and categoricity via automorphisms and correspondences, and presents a counterexample to a prior conjecture.

Findings

Essential -categoricity is a robust property for strongly minimal compact complex manifolds.

Characterizations are given in terms of automorphisms and correspondences.

A counterexample shows a manifold with trivial geometry that is not -categorical.

Abstract

Essential $\aleph_0$-categoricity; i.e., $\aleph_0$-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential $\aleph_0$-categoricity are given in terms of complex-analytic automorphisms, in the simply connected case, and correspondences in general. As a consequence it is pointed out that an example of McMullen yields a strongly minimal compact K\"ahler manifold with trivial geometry but which is not $\aleph_0$-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.