Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

2018 ◽  
pp. 471-502
Author(s):  
Brian Collier
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Surface Group ◽  
Closed Surface ◽  
Connected Components ◽  
Character Variety ◽  
Higgs Bundles ◽  
Group Representations ◽  
Symmetric Powers ◽  
Higher Rank

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

scholarly journals Non-injective representations of a closed surface group into PSL(2, )

2007 ◽  
Vol 142 (2) ◽  
pp. 289-304 ◽  
Author(s):  
LOUIS FUNAR ◽  
MAXIME WOLFF
Keyword(s):  
Fundamental Group ◽  
Euler Class ◽  
Discrete Subgroup ◽  
Surface Group ◽  
Closed Surface ◽  
The Other ◽  
Connected Components ◽  
Discrete Representation ◽  
Discrete Representations

AbstractLet e denote the Euler class on the space ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ of representations of the fundamental group Γg of the closed surface Σg of genus g. Goldman showed that the connected components of ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ are precisely the inverse images e−1(k), for 2−2g≤ k≤ 2g−2, and that the components of Euler class 2−2g and 2g−2 consist of the injective representations whose image is a discrete subgroup of ${PSL(2,{\mathbb R})}$. We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.

scholarly journals Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

scholarly journals Surface Group Representations and U(p, q)-Higgs Bundles

2003 ◽  
Vol 64 (1) ◽  
pp. 111-170 ◽  
Author(s):  
Steven B. Bradlow ◽  
Oscar García-Prada ◽  
Peter B. Gothen
Keyword(s):  
Surface Group ◽  
Higgs Bundles ◽  
Group Representations ◽  
Surface Group Representations

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

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