scholarly journals HODGE POLYNOMIALS OF THE MODULI SPACES OF PAIRS

2007 ◽  
Vol 18 (06) ◽  
pp. 695-721 ◽  
Author(s):  
VICENTE MUÑOZ ◽  
DANIEL ORTEGA ◽  
MARIA-JESÚS VÁZQUEZ-GALLO
Keyword(s):  
Moduli Spaces ◽  
Holomorphic Section ◽  
Complex Numbers ◽  
Projective Curve ◽  
Hodge Structures ◽  
Smooth Projective Curve ◽  
Holomorphic Bundle ◽  
Rank 2 ◽  
Hodge Polynomials ◽  
Mixed Hodge Structures

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

scholarly journals Involutions of Rank 2 Higgs Bundle Moduli Spaces

2018 ◽  
pp. 535-550 ◽  
Author(s):  
Oscar García-Prada ◽  
S. Ramanan
Keyword(s):  
Vector Bundle ◽  
Fundamental Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Higgs Field ◽  
Higgs Bundle ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Genus 2 ◽  
Rank 2

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector bundle with an element α‎ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ±1. It describes the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α‎, and gives an interpretation in terms of the moduli space of representations of the fundamental group.

scholarly journals Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve

2021 ◽  
Vol 30 (1) ◽  
pp. 66-89
Author(s):  
Lie Fu ◽  
◽  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Grothendieck Group ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Chow Motives ◽  
Wall Crossing ◽  
Rank 2 ◽  
Semistable Vector Bundles

<abstract><p>We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $ 3 $ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.</p></abstract>

scholarly journals HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS

2010 ◽  
Vol 21 (11) ◽  
pp. 1505-1529 ◽  
Author(s):  
VICENTE MUÑOZ
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hodge Structure ◽  
Projective Curve ◽  
Hodge Structures ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Algebraic Vector

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

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 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 Moduli spaces of orthogonal and symplectic bundles over an algebraic curve

2008 ◽  
Vol 144 (3) ◽  
pp. 721-733 ◽  
Author(s):  
Olivier Serman
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Vector Bundles ◽  
Explicit Description ◽  
Representation Space ◽  
Classical Groups ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Polynomial Invariants

AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

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 Rank 2 Stable Bundles and Hecke Curves

2016 ◽  
Vol 59 (4) ◽  
pp. 865-877
Author(s):  
Sarbeswar Pal
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Short Note ◽  
Equivalence Classes ◽  
Complex Numbers ◽  
Stable Bundles ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Arbitrary Genus ◽  
Rank 2

AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves

2000 ◽  
Vol 43 (2) ◽  
pp. 129-137 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Stable Rank ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Projective Curves ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

AbstractLet E be a stable rank 2 vector bundle on a smooth projective curve X and V(E) be the set of all rank 1 subbundles of E with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, E, on X with fixed deg(E) and deg(L), L ∈ V(E) and such that .

scholarly journals Counting Higgs bundles and type quiver bundles

2020 ◽  
Vol 156 (4) ◽  
pp. 744-769
Author(s):  
Sergey Mozgovoy ◽  
Olivier Schiffmann
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Moduli Spaces ◽  
Closed Formula ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Closed Expression ◽  
Smooth Projective Curve ◽  
Quiver Bundles ◽  
Fixed Rank

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

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