On canonical bases and internality criteria

Abstract

A criterion is given for a strong type in a finite rank stable theory $T$ to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for a compact complex analytic space to be “algebraic” (namely Moishezon). The canonical base property for a stable theory states that the type of the canonical base of a stationary type over a realisation is almost internal to the minimal types of the theory. It is conjectured that every finite rank stable theory has the canonical base property. It is shown here, that in a theory with the canonical base property, if $p$ is a stationary type for which there exists a family of types $q_b$, each internal to a nonlocally modular minimal type $r$, and such that any pair of independent realisations of $p$ are “connected” by the $q_b$’s, then $p$ is almost internal to $r$.