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 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.

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.

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.

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].

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
Kiumars Kaveh
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Projective Variety ◽  
Cohomology Ring ◽  
Holomorphic Vector Field ◽  
Zero Set ◽  
Graded Algebras ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

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 Higher-dimensional Clifford–Severi equalities

2019 ◽  
Vol 22 (08) ◽  
pp. 1950079 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Lower Bound ◽  
Line Bundle ◽  
Abelian Variety ◽  
Projective Variety ◽  
Line Bundles ◽  
Smooth Complex ◽  
Higher Dimensional ◽  
Complex Projective Variety ◽  
Irregular Varieties ◽  
Complex Projective

Let [Formula: see text] be a smooth complex projective variety, [Formula: see text] a morphism to an abelian variety such that [Formula: see text] injects into [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text]; denote by [Formula: see text] the minimum of [Formula: see text] for [Formula: see text]. The so-called Clifford–Severi inequalities have been proven in [M. A. Barja, Generalized Clifford–Severi inequality and the volume of irregular varieties, Duke Math. J. 164(3) (2015) 541–568; M. A. Barja, R. Pardini and L. Stoppino, Linear systems on irregular varieties, J. Inst. Math. Jussieu (2019) 1–39; doi:10.1017/S1474748019000069]; in particular, for any [Formula: see text] there is a lower bound for the volume given by: [Formula: see text] and, if [Formula: see text] is pseudoeffective, [Formula: see text] In this paper, we characterize varieties and line bundles for which the above Clifford–Severi inequalities are equalities.

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