A model-theoretic counterpart to Moishezon morphisms

TL;DR

This paper introduces a model-theoretic concept analogous to Moishezon morphisms in complex geometry, providing criteria for when certain types are Moishezon to nonmodular minimal types, under specific model-theoretic assumptions.

Contribution

It establishes a model-theoretic analogue of Campana's algebraicity criterion using internality and the canonical base property, linking complex geometry concepts to model theory.

Findings

Provides a criterion for Moishezon-ness of types under canonical base property.

Connects complex geometry notions like coreductions to model-theoretic structures.

Extends the understanding of internality and algebraicity in model theory.

Abstract

In this note a natural strengthening of internality motivated by complex geometry, being "Moishezon" to a set of types, is introduced. Under the hypothesis of Pillay's canonical base property, and using results of Chatzidakis, a criterion is given for when a finite U-rank stationary type that is internal to a nonmodular minimal type is in fact Moishezon to the set of all nonmodular minimal types. This result is a model-theoretic analogue of (a special case of) Campana's "first algebraicity criterion". Other related abstractions from complex geometry, including "coreductions" and "generating fibrations" are also discussed.