Some model theory of fibrations and algebraic reductions

TL;DR

This paper develops a model-theoretic framework for analyzing fibrations and algebraic reductions, introducing a new notion of descent, and applies these results to hyperk"ahler manifolds and differential algebraic groups.

Contribution

It introduces a novel model-theoretic property related to internality and descent, with applications to complex geometry and differential algebra.

Findings

Algebraic reduction of nonalgebraic hyperk"ahler manifolds does not descend.

New notion of descent for stationary types is characterized.

Applications to differential algebraic groups demonstrate the framework's versatility.

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 hyperk\"ahler manifolds. Journal of Differential Geometry, 85(3):397--424, 2010] where structural properties of generalised hyperk\"ahler 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) hyperk\"ahler 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.