Moduli spaces of Higgs bundles on degenerating Riemann surfaces

Swoboda, Jan

doi:10.1016/j.aim.2017.10.028

Cited by 10 publications

(35 citation statements)

References 27 publications

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“…The description of these models is obtained by studying the behavior of the harmonic map between a surface X with a given complex structure and the surface X with the corresponding Riemannian metric of constant curvature -4, under degeneration of the domain Riemann surface X to a noded surface; cf. [36], [39] for further details.…”

Section: 2mentioning

confidence: 99%

“…Choosing a local holomorphic trivialization on E and assuming that with respect to it the auxiliary hermitian metric h 0 is the standard hermitian metric on C 2 , the corresponding hermitian metric for this solution on the bundle E = L ⊕ L −1 is globally well-defined with respect to the holomorphic splitting of E into line bundles. Calculations worked out in [36] imply that in particular…”

Section: 2mentioning

confidence: 99%

“…A crucial step in this argument is to show that the linearization of the G-Hitchin operator at our approximate solution A app R , Φ app R is invertible; this is obtained by showing that an appropriate self-adjoint Dirac-type operator has no small eigenvalues. This method was also used by J. Swoboda in [36] to produce a family of smooth solutions of the SL(2,C)-Hitchin equations, which may be viewed as desingularizing a solution with logarithmic singularities over a noded Riemann surface. The analytic techniques from [36], are extended to provide the main theorem from that article for solutions of the Sp(4,R)-Hitchin equations as well, and moreover to obtain our main result: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…This method was also used by J. Swoboda in [36] to produce a family of smooth solutions of the SL(2,C)-Hitchin equations, which may be viewed as desingularizing a solution with logarithmic singularities over a noded Riemann surface. The analytic techniques from [36], are extended to provide the main theorem from that article for solutions of the Sp(4,R)-Hitchin equations as well, and moreover to obtain our main result: Theorem 1.1. Let X 1 be a closed Riemann surface of genus g 1 and D 1 = {p 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…A standard strategy , due largely to C. Taubes [37], for correcting an approximate solution to an exact solution of gauge-theoretic equations involves studying the linearization of a relevant elliptic operator. In the Higgs bundle setting, the linearization of the Hitchin operator was described in [25] and furthermore in [36] for solutions to the SL(2,C)-self-duality equations over a noded surface. We are going to use this analytic machinery to correct our approximate solution to an exact solution over the complex connected sum of Riemann surfaces.…”

mentioning

confidence: 99%

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Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

Kydonakis

2018

JAM

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We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the Sp(4,R)-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the 2g − 3 exceptional components of the maximal Sp(4,R)-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.for s-many possible pairs of points (p i , q j ). Then there is a polystable Sp(4,R)-Higgs bundle (E # , Φ # ) → X # , constructed over the connected sum of Riemann surfaces X # = X 1 #X 2 of genus g 1 + g 2 + s − 1, which agrees with the initial data over X # \X 1 and X # \X 2 .By analogy with the terminology introduced by O. Guichard and A. Wienhard in their construction of hybrid representations, we call the polystable Higgs bundles corresponding to such exact solutions hybrid. The construction can have a wider applicability for identifying smooth points in moduli of Higgs bundles. As one application, we build Higgs bundles corresponding to Zariski dense representations into Sp(4,R). For this purpose, we look at how the Higgs bundle topological invariants behave under the complex connected sum operation. We first show the following: Proposition 1.2. Let X # = X 1 #X 2 be the complex connected sum of two closed Riemann surfaces X 1 and X 2 with divisors D 1 and D 2 of s-many distinct points on each surface, and let V 1 , V 2 be parabolic principal H C -bundles over X 1 and X 2 respectively. For a parabolic subgroup P ⊂ H C , a holomorphic reduction σ of the structure group of E from H C to P and an antidominant character χ of P , the following identity holds:Note that an analogous additivity property for the Toledo invariant was established by M. Burger, A. Iozzi and A. Wienhard in [8] from the point of view of fundamental group representations. It implies in particular that the connected sum of maximal parabolic G-Higgs bundles is again a maximal (non-parabolic) G-Higgs bundle.We find model Higgs bundles in all exceptional components of the maximal Sp(4,R)-Higgs bundle moduli space; these models are described by hybrid Higgs bundles. In the case when G = Sp(4,R), considering all possible decompositions of a surface Σ along a simple, closed, separating geodesic is sufficient in order to obtain representations in the desired components of M max , which are fully distinguished by the calculation of the degree of a line bundle. This degree equals the Euler class for a hybrid representation as defined by O. Guichard and A. Wienhard, although these invariants live naturally in different cohomology groups.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

Kydonakis

2018

JAM

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From Hyperbolic Dehn Filling to Surgeries in Representation Varieties

Kydonakis

2022

In the Tradition of Thurston II

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The Hitchin fibration under degenerations to noded Riemann surfaces

Swoboda

2016

Abh. Math. Semin. Univ. Hambg.

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Abstract. In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces ΣR converging for R 0 to a surface Σ0 with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration M d → ΣR gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in [12]. Hitchin fibration and self-duality equations and noded Riemann surface

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Moduli spaces of Higgs bundles on degenerating Riemann surfaces

Cited by 10 publications

References 27 publications

Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

From Hyperbolic Dehn Filling to Surgeries in Representation Varieties

The Hitchin fibration under degenerations to noded Riemann surfaces

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