Some model theory of fibrations and algebraic reductions

Let be a stationary finite rank type in an arbitrary stable theory and an -invariant family of partial types. The following property is introduced and characterised: whenever is definable over and is not algebraic over , then is almost internal to . The characterisation involves among other things an apparently new notion of descent for stationary types. Motivation comes partly from results in Sect. 2 of (Campana et al. in J Differ Geom 85(3):397-424, 2010) where structural properties of generalised hyperkahler manifolds are given. The model-theoretic results obtained here are applied back to the complex-analytic setting to prove that the algebraic reduction of a nonalgebraic (generalised) hyperkahler manifold does not descend. The results are also applied to the theory of differentially closed fields, where examples coming from differential-algebraic groups are given.