Gluing constructions for Higgs bundles over a complex connected sum
Kydonakis, Georgios A.
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https://hdl.handle.net/2142/100920
Description
Title
Gluing constructions for Higgs bundles over a complex connected sum
Author(s)
Kydonakis, Georgios A.
Issue Date
2018-04-02
Director of Research (if dissertation) or Advisor (if thesis)
Bradlow, Steven B.
Doctoral Committee Chair(s)
Nevins, Thomas
Committee Member(s)
Albin, Pierre
Dunfield, Nathan M.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Higgs bundles, character variety, topological invariants
Abstract
For a compact Riemann surface of genus $g\ge 2$, the components of the moduli space of $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundles, or equivalently the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety, are partially labeled by an integer $d$ known as the Toledo invariant. The subspace for which this integer attains a maximum has been shown to have $3\cdot {{2}^{2g}}+2g-4$ many components. A gluing construction between parabolic Higgs bundles over a connected sum of Riemann surfaces provides model Higgs bundles in a subfamily of particular significance. This construction is formulated in terms of solutions to the Hitchin equations, using the linearization of a relevant elliptic operator.
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