scholarly journals Complete Integrability of the Parahoric Hitchin System

2018 ◽  
Vol 2019 (21) ◽  
pp. 6499-6528
Author(s):  
David Baraglia ◽  
Masoud Kamgarpour ◽  
Rohith Varma
Keyword(s):  
Integrable System ◽  
Abelian Varieties ◽  
Group Scheme ◽  
Complete Integrability ◽  
Projective Curve ◽  
Nilpotent Cone ◽  
Hitchin System ◽  
Moduli Stack ◽  
Complex Projective ◽  
Complex Projective Curve

Abstract Let $\mathcal {G}$ be a parahoric group scheme over a complex projective curve X of genus greater than one. Let $\mathrm {Bun}_{\mathcal {G}}$ denote the moduli stack of $\mathcal {G}$-torsors on X. We prove several results concerning the Hitchin map on $T^{\ast }\!\mathrm {Bun}_{\mathcal {G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm {Bun}_{\mathcal {G}}$ is “very good” in the sense of Beilinson–Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.

On the Image of Certain Extension Maps. I

2007 ◽  
Vol 50 (3) ◽  
pp. 427-433
Author(s):  
Israel Moreno Mejía
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Vector Bundles ◽  
Projective Curve ◽  
Rational Map ◽  
Stable Vector Bundles ◽  
Linear Subspaces ◽  
Smooth Complex ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be a smooth complex projective curve of genus g ≥ 1. Let ξ ∈ J1(X) be a line bundle on X of degree 1. LetW = Ext1(ξn, ξ–1) be the space of extensions of ξn by ξ–1. There is a rational map Dξ : G(n,W) → SUX(n + 1), where G(n,W) is the Grassmannian variety of n-linear subspaces of W and SUX(n + 1) is the moduli space of rank n + 1 semi-stable vector bundles on X with trivial determinant. We prove that if n = 2, then Dξ is everywhere defined and is injective.

CONNECTION ON PARABOLIC VECTOR BUNDLES OVER CURVES

2011 ◽  
Vol 22 (04) ◽  
pp. 593-602 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MARINA LOGARES
Keyword(s):  
Vector Bundle ◽  
Direct Summand ◽  
Vector Bundles ◽  
Degree Zero ◽  
Projective Curve ◽  
Smooth Complex ◽  
Semistable Vector Bundles ◽  
Complex Projective ◽  
Complex Projective Curve

Let E* be a parabolic vector bundle over a smooth complex projective curve. We prove that E* admits an algebraic connection if and only if the parabolic degree of every parabolic vector bundle which is a direct summand of E* is zero. In particular, all parabolic semistable vector bundles of parabolic degree zero admit an algebraic connection.

scholarly journals Torelli theorem for the moduli space of framed bundles

2009 ◽  
Vol 148 (3) ◽  
pp. 409-423 ◽  
Author(s):  
I. BISWAS ◽  
T. GÓMEZ ◽  
V. MUÑOZ
Keyword(s):  
Moduli Space ◽  
Isomorphism Class ◽  
Coherent Sheaf ◽  
Projective Curve ◽  
Smooth Complex ◽  
Torelli Theorem ◽  
The One ◽  
Pointed Curve ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Ex → r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

Families of vector bundles and linear systems of theta divisors

2017 ◽  
Vol 28 (06) ◽  
pp. 1750039 ◽  
Author(s):  
Sonia Brivio
Keyword(s):  
Linear Systems ◽  
Vector Bundles ◽  
Stable Bundle ◽  
Projective Curve ◽  
Determinant Formula ◽  
Smooth Complex ◽  
Genus Formula ◽  
Semistable Vector Bundles ◽  
Complex Projective ◽  
Complex Projective Curve

Let [Formula: see text] be a smooth complex projective curve of genus [Formula: see text] and let [Formula: see text] be a point. From Hecke correspondence, any stable bundle on [Formula: see text] of rank [Formula: see text] and determinant [Formula: see text] defines a rational family of semistable vector bundles on [Formula: see text] of rank [Formula: see text] and trivial determinant. In this paper, we study linear systems of theta divisors associated to these families.

Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Indranil Biswas ◽  
Amit Hogadi ◽  
Yogish Holla
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Brauer Group ◽  
Projective Curve ◽  
Connected Component ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

scholarly journals Hecke cycles associated to rank 2 twisted Higgs bundles on a curve

2019 ◽  
Vol 30 (12) ◽  
pp. 1950062
Author(s):  
Sang-Bum Yoo
Keyword(s):  
Moduli Space ◽  
Rational Maps ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Complex ◽  
Rank 2 ◽  
Genus Formula ◽  
Hecke Modifications ◽  
Complex Projective ◽  
Complex Projective Curve

Let [Formula: see text] be a smooth complex projective curve of genus [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the moduli space of semistable rank 2 [Formula: see text]-twisted Higgs bundles with trivial determinant on [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank 2 [Formula: see text]-twisted Higgs bundles with determinant [Formula: see text] for some [Formula: see text] on [Formula: see text]. We construct a cycle in the product of a stack of rational maps from nonsingular curves to [Formula: see text] and [Formula: see text] by using Hecke modifications of a stable [Formula: see text]-twisted Higgs bundle in [Formula: see text].

scholarly journals AUTOMORPHISMS OF MODULI SPACES OF SYMPLECTIC BUNDLES

2012 ◽  
Vol 23 (05) ◽  
pp. 1250052 ◽  
Author(s):  
INDRANIL BISWAS ◽  
TOMAS L. GÓMEZ ◽  
VICENTE MUÑOZ
Keyword(s):  
Automorphism Group ◽  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Projective Curve ◽  
Smooth Complex ◽  
Complex Projective ◽  
Complex Projective Curve

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where [Formula: see text], together with the automorphisms induced by automorphisms of X.

scholarly journals Segre invariant and a stratification of the moduli space of coherent systems

2020 ◽  
pp. 2050117
Author(s):  
L. Roa-Leguizamón
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Projective Curve ◽  
Coherent Systems ◽  
Genus Formula ◽  
Complex Projective ◽  
Complex Projective Curve

The aim of this paper is to generalize the [Formula: see text]-Segre invariant for vector bundles to coherent systems. Let [Formula: see text] be a non-singular irreducible complex projective curve of genus [Formula: see text] and [Formula: see text] be the moduli space of [Formula: see text]-stable coherent systems of type [Formula: see text] on [Formula: see text]. For any pair of integers [Formula: see text] with [Formula: see text], [Formula: see text] we define the [Formula: see text]-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the [Formula: see text]-Segre invariant induces a stratification of the moduli space [Formula: see text] into locally closed subvarieties [Formula: see text] according to the value [Formula: see text] of the function. We determine an above bound for the [Formula: see text]-Segre invariant and compute a bound for the dimension of the different strata [Formula: see text]. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype [Formula: see text].

(t, ℓ)-STABILITY AND COHERENT SYSTEMS

2019 ◽  
Vol 62 (3) ◽  
pp. 661-672
Author(s):  
L. BRAMBILA-PAZ ◽  
O. MATA-GUTIÉRREZ
Keyword(s):  
Moduli Spaces ◽  
Finite Family ◽  
Real Parameter ◽  
Projective Curve ◽  
Coherent Systems ◽  
Positive Real ◽  
Positive Real Parameter ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

Generated coherent systems and a conjecture of D. C. Butler

2019 ◽  
Vol 30 (05) ◽  
pp. 1950024 ◽  
Author(s):  
L. Brambila-Paz ◽  
O. Mata-Gutiérrez ◽  
P. E. Newstead ◽  
Angela Ortega
Keyword(s):  
Coherent System ◽  
Projective Curve ◽  
Coherent Systems ◽  
Type Formula ◽  
Complex Projective ◽  
Unknown Case ◽  
Complex Projective Curve

Let [Formula: see text] be a general generated coherent system of type [Formula: see text] on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of [Formula: see text] to the semistability of the kernel of the evaluation map [Formula: see text]. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type [Formula: see text], which is the first previously unknown case.

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