  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.
