Resumen de Some model theory of fibrations and algebraic reductions
Let p=tp(a/A) be a stationary finite rank type in an arbitrary stable theory and PP an AA -invariant family of partial types. The following property is introduced and characterised: whenever cc is definable over (A,a) and aa is not algebraic over (A,c), then tp(c/A) is almost internal to PP . 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 hyperkähler 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) hyperkähler 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.
