2025/10/07 by Xavier Buff, Buff, Xavier, Arnaud Chéritat +3
Mathematics · #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2510.05677
A meromorphic connection on the tangent bundle of a Riemann surface induces a complex affine structure on the complement of the poles. Local models for Fuchsian singularities are already known. In this paper, we introduce a complete set of local invariants for a meromorphic connection and provide local models for a complex affine structure in a punctured neighborhood of an irregular singularity. Generalizing a construction attributed to Veech, we introduce the Delaunay decomposition of a compact Riemann surface endowed with a meromorphic connection with irregular singularities. In particular, we give upper bounds on the complexity of the decomposition.