scholarly journals Character varieties of virtually nilpotent Kähler groups and G–Higgs bundles

2015 ◽  
Vol 65 (6) ◽  
pp. 2601-2612
Author(s):  
Indranil Biswas ◽  
Carlos Florentino
Keyword(s):  
Higgs Bundles ◽  
Character Varieties ◽  
Kähler Groups

scholarly journals Compact connected components in relative character varieties of punctured spheres

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Nicolas Tholozan ◽  
Jérémy Toulisse
Keyword(s):  
Lie Groups ◽  
Invariant Theory ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Connected Components ◽  
Higgs Bundles ◽  
Geometric Invariant ◽  
Projective Varieties ◽  
Elliptic Element ◽  
Character Varieties

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

scholarly journals THE SL2(ℂ) CASSON INVARIANT FOR SEIFERT FIBERED HOMOLOGY SPHERES AND SURGERIES ON TWIST KNOTS

2006 ◽  
Vol 15 (07) ◽  
pp. 813-837 ◽  
Author(s):  
HANS U. BODEN ◽  
CYNTHIA L. CURTIS
Keyword(s):  
Moduli Spaces ◽  
Closed Formula ◽  
Higgs Bundles ◽  
Character Varieties ◽  
Surgery Formula

We derive a simple closed formula for the SL2(ℂ) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL2(ℂ) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL2(ℂ) Casson invariant.

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 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 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 Kähler groups and subdirect products of surface groups

2020 ◽  
Vol 24 (2) ◽  
pp. 971-1017
Author(s):  
Claudio Llosa Isenrich
Keyword(s):  
Surface Groups ◽  
Kähler Groups ◽  
Subdirect Products

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 Topological mirror symmetry for rank two character varieties of surface groups

2021 ◽  
Author(s):  
Mirko Mauri
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Surface Groups ◽  
Character Varieties ◽  
Hitchin System ◽  
Global Topology ◽  
Hodge Numbers

AbstractThe moduli spaces of flat $${\text{SL}}_2$$ SL 2 - and $${\text{PGL}}_2$$ PGL 2 -connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.

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