On canonical bases and internality criteria

TL;DR

This paper establishes a criterion for types in finite rank stable theories to be almost internal to nonmodular minimal types, inspired by complex geometry results, under the assumption of the canonical base property (CBP).

Contribution

It provides a model-theoretic analogue of Campana's geometric criteria, extending understanding of internality in stable theories with CBP.

Findings

Criterion for internality to nonmodular minimal types

Connection between model theory and complex geometry

Results assuming the canonical base property (CBP)

Abstract

A criterion is given for a type in a finite rank stable theory to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for compact complex analytic spaces to be algebraic (namely Moishezon), in terms of the existence of "generating" families of algebraic subvarieties. A model-theoretic anologue/generalisation of Campana's results is given under the hypothesis that the theory has the "canonical base property" (CBP), a property that is conjectured to hold in all stable finite rank theories and which states that the type of the canonical base over a realisation is almost internal to the minimal types of the theory.