Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres

2005/03/23 by Mike Krebs, Krebs, Mike
Mathematics · #14D20 #57M50 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #math.AG #math.GT #msc:14D20 #msc:57M50

paper · pdf · doi:10.48550/arxiv.math/0503509

39 pages; replacement contains alternate proofs of verticality

openalex publication_date 2005/03/23 · arxiv created 2006/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface over it, called a Dolgachev surface. Using the theory of Higgs bundles on these Dolgachev surfaces, we explicitly compute all values taken on by the orbifold Toledo invariant.

Citations

• The Self-Duality Equations on a Riemann Surface
• Representation spaces of Seifert fibered homology spheres
• The Geometries of 3-Manifolds
• Moduli of representations of the fundamental group of a smooth projective variety II

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• Cyclic Higgs bundles and the Toledo invariant
• Spectral Data for SU(1,2) Higgs Bundles
• Topological invariants of parabolic G-Higgs bundles
• Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group
• Higgs bundles over elliptic curves for real groups