Isolated types of finite rank: an abstract Dixmier-Moeglin equivalence

Abstract

Suppose $T$ is totally transcendental and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type $p=tp(a/A)$ is isolated if and only if $a$ is independent from $q(\mathcal U)$ over $Ab$ for every $b\in \operatorname{acl}(Aa)$ and $q\in S(Ab)$ nonisolated and minimal. This applies to the theory of differentially closed fields -- where it is motivated by the differential Dixmier-Moeglin equivalence problem -- and the theory of compact complex manifolds.