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

TL;DR

This paper establishes an abstract criterion for the isolation of finite rank types in totally transcendental theories, linking model-theoretic properties to the differential Dixmier-Moeglin equivalence and complex geometry.

Contribution

It provides a new characterization of isolated types in totally transcendental theories, extending the differential Dixmier-Moeglin equivalence to broader contexts.

Findings

Characterization of isolated types via independence conditions

Application to differential fields and complex manifolds

Connection between model theory and geometric structures

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.