The geometry of maximal representations of surface groups into SO0(2,n)

Collier, Brian; Tholozan, Nicolas; Toulisse, Jérémy

doi:10.1215/00127094-2019-0052

Cited by 49 publications

(71 citation statements)

References 67 publications

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“…Another construction of geometric structures on higher-dimensional manifolds using Higgs bundles was done by Collier, Tholozan and Toulisse [14]. They constructed photon structures whose holonomy factors through maximal representations in O(2, n).…”

Section: Projective Structures With Hitchin or Quasi-hitchin Holonomiesmentioning

confidence: 99%

“…The main purpose of this survey paper is to explain these constructions. The first ones were presented by Baraglia in his Ph.D. Thesis [9], more recent ones are in Alessandrini-Li [2,3,5], and Collier-Tholozan-Toulisse [14]. The main idea behind these constructions is that a geometric structure corresponds to a section of a flat bundle which is transverse to the parallel foliation.…”

Section: Introductionmentioning

confidence: 99%

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Higgs Bundles and Geometric Structures on Manifolds

Alessandrini¹

2019

SIGMA

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Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.

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Section: Projective Structures With Hitchin or Quasi-hitchin Holonomiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higgs Bundles and Geometric Structures on Manifolds

Alessandrini¹

2019

SIGMA

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“…Since we have not said much about this situation, we will only discuss the non-Hermitian case, that is, for 2 < p ≤ q. For the case of SO(2, q) we refer the reader to [2,10].…”

Section: So(p Q)-higgs Bundlesmentioning

confidence: 99%

Studying Deformations of Fuchsian Representations with Higgs Bundles

Collier¹

2019

SIGMA

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This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747].

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“…Cyclic Higgs bundles generalize the elements of the Hitchin section in the moduli space of Higgs bundles, are closely related to the affine Toda equations [4], and have been crucial to establishing some of the known cases [23] of Labourie's conjecture [22,24,9]. They also occupy a special position within nonabelian Hodge theory because they admit diagonal harmonic metrics solving Hitchin's equations.…”

Section: Introductionmentioning

confidence: 99%

Twisted cyclic quiver varieties on curves

Rayan

Sundbo

2019

European Journal of Mathematics

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We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the number of points at which the cyclic moduli space intersects a Hitchin fibre; and we describe explicitly certain C × -flows into the nilpotent cone. We also extend this description to moduli of certain twisted cyclic quivers whose rank vector has components larger than 1. We show that, for certain choices of underlying bundle, such moduli spaces decompose as a product of cyclic quiver varieties in which each node is a line bundle.Definition 2.1. An L-twisted Higgs bundle on X is a pair (E, Φ), where E is a holomorphic vector bundle on X and Φ is a vector bundle morphism Φ :for all proper subbundles F of E which satisfy Φ(F ) ⊆ F ⊗ L (this condition is known as Φ-invariance). We will sometimes denote µ(E) by µ tot . The data of Φ is commonly referred to as a Higgs field for E.Definition 2.2. We say that two (twisted) Higgs bundles (E, Φ) and (E ′ , Φ ′ ) on X are equivalent if there exists a vector bundle isomrophism Ψ : E ∼ = → E ′ for which Φ = ΨΦ ′ Ψ −1 .

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The geometry of maximal representations of surface groups into SO0(2,n)

Cited by 49 publications

References 67 publications

Higgs Bundles and Geometric Structures on Manifolds

Higgs Bundles and Geometric Structures on Manifolds

Studying Deformations of Fuchsian Representations with Higgs Bundles

Twisted cyclic quiver varieties on curves

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