2012/10/10 by Rahim Moosa, Anand Pillay, Moosa, Rahim +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1210.2793
openalex publication_date 2012/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.