Definable Galois theory for bimeromorphic geometry

TL;DR

This paper develops a Galois theory framework for bimeromorphic geometry using model-theoretic definable groups, leading to structural theorems and examples in complex and differential geometry.

Contribution

It introduces a novel Galois theory approach for bimeromorphic geometry via definable groups, with applications to principal bundles and algebraic group characterizations.

Findings

Structural theorem for principal meromorphic bundles without horizontal subvarieties

Examples of algebraic groups as binding groups and their linearity characterization

Existence of nontrivial definable torsors over acl-closed sets in DCCM

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.