$ℵ_{0}$-categorical strongly minimal compact complex manifolds

Abstract

Essential א0-categoricity; i.e., א0-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential א0-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 Kahler manifold with trivial geometry but which is not א0-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.