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

Author(s):  
Mirko Mauri

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.

2011 ◽  
Vol 151 (3) ◽  
pp. 441-457 ◽  
Author(s):  
INDRANIL BISWAS ◽  
PETER B. GOTHEN ◽  
MARINA LOGARES

AbstractLetXbe a compact Riemann surfaceXof genus at–least two. Fix a holomorphic line bundleLoverX. Letbe the moduli space of Hitchin pairs (E, φ ∈H0(End0(E) ⊗L)) overXof rankrand fixed determinant of degreed. The following conditions are imposed:(i)deg(L) ≥ 2g−2,r≥ 2, andL⊗rKX⊗r;(ii)(r, d) = 1; and(iii)ifg= 2 thenr≥ 6, and ifg= 3 thenr≥ 4.We prove that that the isomorphism class of the varietyuniquely determines the isomorphism class of the Riemann surfaceX. Moreover, our analysis shows thatis irreducible (this result holds without the additional hypothesis on the rank for low genus).


Author(s):  
Bonaventure Loo

AbstractIn this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.


Author(s):  
David Baraglia ◽  
Indranil Biswas ◽  
Laura P. Schaposnik

Given a compact Riemann surface X and a semi-simple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the Brauer group of the smooth locus of these varieties.


2016 ◽  
Vol 225 ◽  
pp. 185-206
Author(s):  
ARATA KOMYO

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.


2005 ◽  
Vol 16 (04) ◽  
pp. 365-386
Author(s):  
MRIDUL MEHTA

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.


2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur

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.


2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.


Author(s):  
Mirko Mauri

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”.


2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].


2003 ◽  
Vol 14 (10) ◽  
pp. 1097-1120 ◽  
Author(s):  
WEI-PING LI ◽  
ZHENBO QIN

In this paper, we apply the technique of chamber structures of stability polarizations to construct the full moduli space of rank-2 stable sheaves with certain Chern classes on Calabi–Yau manifolds which are anti-canonical divisor of ℙ1×ℙn or a double cover of ℙ1×ℙn. These moduli spaces are isomorphic to projective spaces. As an application, we compute the holomorphic Casson invariants defined by Donaldson and Thomas.


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