The affine geometry of meromorphic connections with irregular singularities

TL;DR

This work develops a comprehensive local-to-global framework for meromorphic connections on Riemann surfaces, translating them into complex affine structures with irregular singularities. It introduces a complete invariant system, notably the asymptotic values invariant, to classify irregular poles up to local isomorphism, and constructs canonical local models that realize every invariant. Building on these local models, the authors extend the Delaunay decomposition to finite-type affine surfaces, classify maximal immersions into several types, and derive sharp bounds on the combinatorial complexity of the core and exterior components. The results connect flat affine geometry to dilation/translation dynamics, moduli of flat structures, and holomorphic dynamics, offering a robust toolkit for understanding geodesic flow, holonomy, and degenerations in affine surface theory. Overall, the paper provides a rigorous framework for analyzing irregular singularities and their global geometry, with potential implications for moduli theory and related dynamical systems.

Abstract

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.

Figures (50)

• Figure 1: In gray: a face in a graph complement. It is bounded by 10 edges, two of which bound it on both sides, so its side count is 12.
• Figure 2: Illustration of the proof of \ref{['lem:GB']}
• Figure 3: A translation cylinder of finite height and a dilation cylinder of finite angle, obtained by quotienting the gray fundamental domain by a translation and a dilation.
• Figure 4: A skew cone of angle close to $\pi/4$ and dilation factor close to $1.4$. It is the quotient of $\tilde{\mathbb{C}}^*$ by a subgroup $\Lambda$ of $\operatorname{Aut}(\mathbb{C}^*)$ generated by some linear map $\lambda$. A portion of width $2\pi$ of $\tilde{\mathbb{C}}^*$ is shown as $\mathbb{C} \setminus (-\infty,0]$. The orbit of a geodesic by $\Lambda$ is shown in red. Its intercept with the fundamental sector trace a single geodesic in the skew cone.
• Figure 5: Finite angle bounded sector. We have a sector of angle 30° and a gluing of factor 3 between its sides. We restrict to the subset of the sector that ends at the dotted (straight) line, whose shape is not so important. Right: a Riemann chart of the flat model is given by a branch of the map $z\mapsto z^\beta$ with $\beta = 2\pi i/(\log(3) + 2\pi i\frac{30}{360})$.

Theorems & Definitions (236)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10