The affine geometry of meromorphic connections with irregular singularities
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918155618615296 |
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| author | Buff, Xavier Chéritat, Arnaud Tahar, Guillaume |
| author_facet | Buff, Xavier Chéritat, Arnaud Tahar, Guillaume |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The affine geometry of meromorphic connections with irregular singularities Buff, Xavier Chéritat, Arnaud Tahar, Guillaume Complex Variables Geometric Topology 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. |
| title | The affine geometry of meromorphic connections with irregular singularities |
| topic | Complex Variables Geometric Topology |
| url | https://arxiv.org/abs/2510.05677 |