scholarly journals Hodge polynomials of the SL(2, ℂ)-character variety of an elliptic curve with two marked points

2014 ◽  
Vol 25 (14) ◽  
pp. 1450125 ◽  
Author(s):  
Marina Logares ◽  
Vicente Muñoz
Keyword(s):  
Elliptic Curve ◽  
Moduli Space ◽  
Betti Numbers ◽  
Conjugacy Classes ◽  
Character Variety ◽  
Higgs Bundles ◽  
Hodge Numbers ◽  
Hodge Polynomials ◽  
Marked Points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

Model Higgs bundles in exceptional components of the Sp(4, ℝ)-character variety

2021 ◽  
pp. 2150067
Author(s):  
Georgios Kydonakis
Keyword(s):  
Elliptic Operator ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Higgs Bundle ◽  
Character Variety ◽  
Topological Invariants ◽  
Higgs Bundles ◽  
Connected Sum ◽  
Anosov Representations

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the [Formula: see text]-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the [Formula: see text] exceptional components of the maximal [Formula: see text]-Higgs bundle moduli space, which correspond to components solely consisting 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 Guichard and Wienhard.

MIXED HODGE STRUCTURES OF THE MODULI SPACES OF PARABOLIC CONNECTIONS

2016 ◽  
Vol 225 ◽  
pp. 185-206
Author(s):  
ARATA KOMYO
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Higgs Bundles ◽  
Hodge Structures ◽  
Character Varieties ◽  
Generic Eigenvalues ◽  
Hodge Polynomials ◽  
Mixed Hodge Structures

In this paper, we investigate the mixed Hodge structures of the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable parabolic Higgs bundles and the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable regular singular parabolic connections. We show that the mixed Hodge polynomials are independent of the choice of generic eigenvalues and the mixed Hodge structures of these moduli spaces are pure. Moreover, by the Riemann–Hilbert correspondence, the Poincaré polynomials of character varieties are independent of the choice of generic eigenvalues.

scholarly journals Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

2007 ◽  
Vol 187 (879) ◽  
pp. 0-0 ◽  
Author(s):  
O. García-Prada ◽  
P. B. Gothen ◽  
V. Muñoz
Keyword(s):  
Moduli Space ◽  
Betti Numbers ◽  
Higgs Bundles

scholarly journals Twisted spectral correspondence and torus knots

2020 ◽  
Vol 29 (06) ◽  
pp. 2050040
Author(s):  
Wu-Yen Chuang ◽  
Duiliu-Emanuel Diaconescu ◽  
Ron Donagi ◽  
Satoshi Nawata ◽  
Tony Pantev
Keyword(s):  
Conjugacy Classes ◽  
Present Approach ◽  
Higgs Bundles ◽  
Torus Knots ◽  
Cohomological Invariants ◽  
Character Varieties ◽  
Spectral Correspondence ◽  
Marked Points ◽  
Chern Simons

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to [Formula: see text] torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

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

2018 ◽  
Vol 14 (2) ◽  
pp. 7744-7786 ◽  
Author(s):  
Georgios Kydonakis
Keyword(s):  
Elliptic Operator ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Higgs Bundle ◽  
Character Variety ◽  
Topological Invariants ◽  
Higgs Bundles ◽  
Connected Sum ◽  
Anosov Representations

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 alsoallows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.

scholarly journals INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS

2021 ◽  
pp. 1-40
Author(s):  
Mirko Mauri
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Character Variety ◽  
Degree Zero ◽  
Surface Groups ◽  
Higgs Bundles ◽  
Character Varieties ◽  
Rank 2

Abstract For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350010 ◽  
Author(s):  
GILBERTO BINI ◽  
FILIPPO F. FAVALE ◽  
JORGE NEVES ◽  
ROBERTO PIGNATELLI
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
General Type ◽  
Picard Number ◽  
Surfaces Of General Type ◽  
Geometric Genus ◽  
Genus Zero ◽  
New Family ◽  
Hodge Numbers ◽  
Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

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