${\mathrm{\aleph }}_0$-categorical strongly minimal compact complex manifolds

Essential ${\mathrm{\aleph }}_0$-categoricity; i.e., ${\mathrm{\aleph }}_0$-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential ${\mathrm{\aleph }}_0$-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 ${\mathrm{\aleph }}_0$-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.