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PreprintCC BY 4.0arXiv

Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces

  • Sheng Tan
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Abstract

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.

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Cite

BibTeXRIS
BibTeX
@misc{tan2026finiteness,
  title = {{Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces}},
  author = {Sheng Tan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17116},
  eprint = {2610.17116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17116},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
AU  - Sheng Tan
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17116
UR  - https://arxiv.org/abs/2610.17116
AB  - We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.
ER  -