Some model theory of fibrations and algebraic reductions

Abstract

Let p=tp(a/A) be a stationary type in an arbitrary finite rank stable theory, and P an A-invariant family of partial types. The following property is introduced and characterised: whenever c is definable over (A,a) and a is not algebraic over (A,c) then \tp(c/A) is almost internal to P. The characterisation involves among other things an apparently new notion of ``descent" for stationary types. Motivation comes partly from results in Section~2 of [Campana, Oguiso, and Peternell. Non-algebraic hyperkähler manifolds. Journal of Differential Geometry, 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.

Substantially revised and augmented. A new section applying the results to differentially closed fields has been added; title, abstract, and introduction are new, and several new examples are added. 14 pages