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 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.
Keywords
Cite
@article{arxiv.1210.2793,
title = {Some model theory of fibrations and algebraic reductions},
author = {Rahim Moosa and Anand Pillay},
journal= {arXiv preprint arXiv:1210.2793},
year = {2013}
}
Comments
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