scholarly journals On $$\mathsf {G} $$-isoshtukas over function fields

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Paul Hamacher ◽  
Wansu Kim
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Rational Functions ◽  
Conjugacy Classes ◽  
Projective Curve ◽  
Smooth Projective Curve

AbstractIn this paper we classify isogeny classes of global $$\mathsf {G} $$ G -shtukas over a smooth projective curve $$C/{\mathbb {F}}_q$$ C / F q (or equivalently $$\sigma $$ σ -conjugacy classes in $$\mathsf {G} (\mathsf {F} \otimes _{{\mathbb {F}}_q} \overline{{\mathbb {F}}_q})$$ G ( F ⊗ F q F q ¯ ) where $$\mathsf {F} $$ F is the field of rational functions of C) by two invariants $${\bar{\kappa }},{\bar{\nu }}$$ κ ¯ , ν ¯ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $$\mathsf {G} $$ G -shtukas and thus is helpful to calculate their cohomology.

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].

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.

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 -algebras associated with curves and rational functions on . I

2014 ◽  
Vol 150 (4) ◽  
pp. 621-667 ◽  
Author(s):  
Robert Fisette ◽  
Alexander Polishchuk
Keyword(s):  
Homotopy Class ◽  
Rational Functions ◽  
Coherent Sheaf ◽  
Linear Series ◽  
Projective Curve ◽  
Canonical Embedding ◽  
Rational Map ◽  
Projective Embedding ◽  
Smooth Projective Curve ◽  
Weierstrass Points

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

scholarly journals Comparison of Poisson structures on moduli spaces

2021 ◽  
Author(s):  
Indranil Biswas ◽  
Francesco Bottacin ◽  
Tomás L. Gómez
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Poisson Structures ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraic Vector ◽  
Pure Dimension ◽  
Stable Sheaves ◽  
Fixed Rank ◽  
Dimension One

AbstractLet X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.

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.

Integral orbits over function fields

2014 ◽  
Vol 10 (08) ◽  
pp. 2187-2204
Author(s):  
Hsiu-Lien Huang ◽  
Chia-Liang Sun ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Forward Orbit ◽  
Siegel Theorem

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

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>

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