Moduli spaces of bundles on an abelian variety

2017 ◽  
Vol 14 (02) ◽  
pp. 1750030
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Abelian Variety ◽  
Moduli Spaces ◽  
Character Variety ◽  
Higgs Bundles ◽  
Connected Component ◽  
Trivial Bundle

Let [Formula: see text] be a complex abelian variety and [Formula: see text] a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal [Formula: see text]-bundles and [Formula: see text]-Higgs bundles on [Formula: see text]. We also describe the moduli spaces of [Formula: see text]-connections and the [Formula: see text]-character variety for [Formula: see text].

scholarly journals Brauer Group of Moduli of Higgs Bundles and Connections

2018 ◽  
pp. 387-398
Author(s):  
David Baraglia ◽  
Indranil Biswas ◽  
Laura P. Schaposnik
Keyword(s):  
Riemann Surface ◽  
Algebraic Group ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Brauer Group ◽  
Higgs Bundles ◽  
Smooth Locus

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.

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 On completeness of holomorphic principal bundles

1975 ◽  
Vol 56 ◽  
pp. 121-138 ◽  
Author(s):  
Shigeru Takeuchi
Keyword(s):  
Algebraic Group ◽  
Lie Groups ◽  
Abelian Variety ◽  
Lie Group ◽  
Basic Structure ◽  
Factor Group ◽  
The Other ◽  
Algebraic Subgroup ◽  
Points Of View ◽  
Lie Subgroup

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
Author(s):  
Tanmay Deshpande
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Groups ◽  
Unipotent Group ◽  
Connected Component ◽  
Unipotent Groups ◽  
Character Sheaves ◽  
Treat All ◽  
Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

Involutions of Higgs moduli spaces over elliptic curves and pseudo-real Higgs bundles

2019 ◽  
Vol 142 ◽  
pp. 47-65 ◽  
Author(s):  
Indranil Biswas ◽  
Luis Angel Calvo ◽  
Emilio Franco ◽  
Oscar García-Prada
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Higgs Bundles

A Torelli theorem for moduli spaces of principal bundles on curves defined over ℝ

2017 ◽  
Vol 28 (06) ◽  
pp. 1750049
Author(s):  
Indranil Biswas ◽  
Olivier Serman
Keyword(s):  
Algebraic Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Isomorphism Class ◽  
Projective Curve ◽  
Real Numbers ◽  
Principal Bundles ◽  
Smooth Projective Curve ◽  
Simple Factor ◽  
Torelli Theorem

Let [Formula: see text] be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let [Formula: see text] be a connected reductive affine algebraic group, defined over [Formula: see text], such that [Formula: see text] is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal [Formula: see text]-bundles on [Formula: see text] determine uniquely the isomorphism class of [Formula: see text].

Moduli of wild Higgs bundles on with -actions

2021 ◽  
pp. 1-34
Author(s):  
LAURA FREDRICKSON ◽  
ANDREW NEITZKE
Keyword(s):  
Fixed Points ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Central Charge ◽  
Higgs Field ◽  
Vertex Algebra ◽  
Higgs Bundles ◽  
Irregular Singularity ◽  
Isomorphism Classes ◽  
Type Decomposition

Abstract We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$ , such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$ , where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$ -action analogous to the famous $\mathbb{C}^\times$ -action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$ -action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$ . We classify the fixed points of this $\mathbb{C}^\times$ -action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L 2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

Sign in / Sign up

Export Citation Format

Share Document