A Universal Family of Deformations for the Uniformising Higgs bundle

2011/11/28 by Peter Dalakov, Dalakov, Peter
Mathematics · #14C30 #14D15 #14D20 #53C07 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.DG #msc:14C30 #msc:14D15 #msc:14D20 #msc:53C07

paper · pdf · doi:10.48550/arxiv.1111.6457

Preliminary version. Comments welcome!

arxiv created 2011/11/28 · openalex publication_date 2011/11/28 · arxiv updated 2011/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of deformations which provides holomorphic Darboux coordinates in a neighbourhood of the section. This is a special case of a more general deformation-theoretic construction in the spirit of Kuranishi theory. As a toy example of the latter we consider the tautological family of centralisers over the Kostant slice.

Citations

• The Self-Duality Equations on a Riemann Surface
• Langlands duality for Hitchin systems

Related

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• A universal Higgs bundle moduli space
• Construction of the Moduli Space of Higgs Bundles using Analytic Methods
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• Real Higgs pairs and Non-abelian Hodge correspondence on a Klein surface