Differential Embedding Problems over Complex Function Fields

TL;DR

This paper develops a new framework using differential torsors and patching techniques to solve differential embedding problems over complex function fields, enhancing understanding of the differential Galois group's structure.

Contribution

It introduces differential torsors and applies patching to differential Galois theory, proving solvability of all embedding problems over complex function fields.

Findings

Patching techniques are valid over complex function fields.

All differential embedding problems over complex function fields are solvable.

Provides new insights into the structure of the absolute differential Galois group.

Abstract

We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category.