Submission date: 5 July 2010.

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 Kähler manifold with trivial geometry but which is not \aleph_0-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.

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