Abstract:We formulate hyperbolic uniformization as a Tannakian reconstruction theorem. For a hyperbolic log-orbi curve C, we construct an intrinsic canonical maximal parahoric PSL2-Higgs object. A tensor-functorial parahoric non-abelian Hodge-Riemann-Hilbert correspondence identifies its Betti realization with a representation of the orbifold fundamental group whose image is the uniformizing cofinite Fuchsian lattice. This construction quasi-inverts the compactified quotient functor from cofinite Fuchsian lattices in PSL2(R) to hyperbolic log-orbi curves. We also prove a Galois enhancement, identifying finite etale covers of C with finite continuous sets for the profinite completion of the reconstructed lattice. As an application, for a one-variable complex function field F, we recover its absolute Galois group as the inverse limit of etale fundamental groups of orbifold models and identify the hyperbolic sector with a compatible Fuchsian-lattice-valued pro-system.
| Comments: | 48 pages |
| Subjects: | Algebraic Geometry (math.AG); Geometric Topology (math.GT) |
| MSC classes: | 14A20, 14A21, 14C30, 14D07, 14F30, 14H30, 14H57 |
| Cite as: | arXiv:2605.25468 [math.AG] |
| (or arXiv:2605.25468v1 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25468 arXiv-issued DOI via DataCite (pending registration) |
From: Xiaojin Lin [view email]
[v1]
Mon, 25 May 2026 06:19:22 UTC (57 KB)
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