Definable Galois theory for bimeromorphic geometry
Abstract
The outlines of a "Galois theory" for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal meromorphic bundles with algebraic structure group, and admitting no horizontal subvarieties, is deduced. Examples of algebraic groups arising as binding groups are provided, as is a characterisation of when they are linear. Using binding groups in CCM it is shown that, in contrast to the situation in differentially closed fields, there are many algebraic groups which admit nontrivial definable torsors over acl-closed sets in the theory DCCM of existentially closed differential CCM-structures. A self-contained exposition of the binding group theorem in totally transcendental theories, that emphasises the bitorsorial nature of the construction, is also included.
Keywords
Cite
@article{arxiv.2508.19524,
title = {Definable Galois theory for bimeromorphic geometry},
author = {Rahim Moosa and Anand Pillay},
journal= {arXiv preprint arXiv:2508.19524},
year = {2025}
}
Comments
An earlier version claimed that all projective general linear groups arise as binding groups in CCM, but, as Remi Jaoui pointed out to us, the argument given was incorrect and only produces a family of linear algebraic groups of arbitrarily high dimension. The changes only affect Section 5.3