Relative internality and definable fibrations

TL;DR

This paper explores the theory of relative internality in stable theories, connecting it to various properties and applying it to differential and complex geometry, providing new insights and examples in model theory.

Contribution

It advances the understanding of relative internality, corrects previous results on the strong canonical base property, and studies definable fibrations in differential fields and complex manifolds.

Findings

$ ext{DCF}_0$ lacks the strong canonical base property

$ ext{CCM}$ does not have the strong CBP

New examples of higher rank types orthogonal to constants

Abstract

We first elaborate on the theory of relative internality in stable theories, focusing on the notion of uniform relative internality (called collapse of the groupoid in an earlier work of the second author), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA. We prove that $\mathrm{DCF}_0$ does not have the strong canonical base property, correcting an earlier proof. We also prove that the theory $\mathrm{CCM}$ of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in $\mathrm{DCF}_0$, where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants.