scholarly journals PARABOLIC k-AMPLE BUNDLES

2011 ◽  
Vol 22 (11) ◽  
pp. 1647-1660 ◽  
Author(s):  
INDRANIL BISWAS ◽  
FATIMA LAYTIMI
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Parabolic Bundles ◽  
Tautological Line Bundle

We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E* is defined to be k-ample if the tautological line bundle [Formula: see text] is k-ample. We establish some properties of parabolic k-ample bundles.

Double Covers and Metastable Immersions of Spheres

1974 ◽  
Vol 26 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Robert Wells
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Projective Space ◽  
Unit Sphere ◽  
Real Line ◽  
Double Cover ◽  
Sphere Bundle ◽  
Canonical Line Bundle ◽  
Projective Space Bundle ◽  
Double Covers

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

scholarly journals Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
Massimo Pippi
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Local Ring ◽  
Global Section ◽  
Regular Local Ring ◽  
Regular Scheme ◽  
Zero Locus

AbstractWe study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

scholarly journals Stable Higgs bundles over positive principal elliptic fibrations

2018 ◽  
Vol 5 (1) ◽  
pp. 195-201
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Misha Verbitsky
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Complex Manifold ◽  
Higgs Bundle ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Higgs Bundles ◽  
Stable Vector Bundle ◽  
Hermitian Metric ◽  
The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE

1993 ◽  
Vol 04 (03) ◽  
pp. 467-501 ◽  
Author(s):  
JONATHAN A. PORITZ
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Sobolev Space ◽  
Moduli Space ◽  
Vector Bundles ◽  
Weighted Sobolev Space ◽  
Conjugacy Classes ◽  
Stable Bundle ◽  
Yang Mills ◽  
Parabolic Bundles

We study a certain moduli space of irreducible Hermitian-Yang-Mills connections on a unitary vector bundle over a punctured Riemann surface. The connections used have non-trivial holonomy around the punctures lying in fixed conjugacy classes of U (n) and differ from each other by elements of a weighted Sobolev space; these connections give rise to parabolic bundles in the sense of Mehta and Seshadri. We show in fact that the moduli space of stable parabolic bundles can be identified with our moduli space of HYM connections, by proving that every stable bundle admits a unique unitary gauge orbit of Hermitian-Yang-Mills connections.

Sums of stably trivial vector bundles

1980 ◽  
Vol 87 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Jacques Allard
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Real Vector

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

scholarly journals ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

2015 ◽  
Vol 16 (2) ◽  
pp. 223-349 ◽  
Author(s):  
Jean-Michel Bismut ◽  
Xiaonan Ma ◽  
Weiping Zhang
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Differential Form ◽  
Toeplitz Operators ◽  
Vector Bundles ◽  
Direct Image ◽  
Analytic Torsion ◽  
Leading Term ◽  
Positive Line Bundle ◽  
Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

scholarly journals Ulrich Bundles on Three-Dimensional Scrolls

2019 ◽  
Author(s):  
Maria Lucia Fania ◽  
Margherita Lelli-Chiesa ◽  
Joan Pons-Llopis
Keyword(s):  
Line Bundle ◽  
Three Dimensional ◽  
Low Rank ◽  
Tautological Line Bundle

Abstract In this paper we construct Ulrich bundles of low rank on three-dimensional scrolls (with respect to the tautological line bundle). We pay special attention to the four types of three-fold scrolls in ${\mathbb{P}}^5$, which were classified in [31].

scholarly journals THE VERLINDE FORMULA FOR PARABOLIC BUNDLES

2001 ◽  
Vol 63 (3) ◽  
pp. 754-768 ◽  
Author(s):  
LISA C. JEFFREY
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Conjugacy Class ◽  
Fundamental Group ◽  
Moduli Space ◽  
Boundary Component ◽  
Compact Riemann Surface ◽  
Central Element ◽  
Parabolic Bundles ◽  
Verlinde Formula

Let Σg be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e2πid/n, …, e2πid/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space [Mscr ]g, 1(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of cexp(Λ) (for Λ ∈ t+), and also for the moduli space [Mscr ]g, b(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with s + 1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ(j)) for Λ(j) ∈ t+. The proof is valid for Λ(j) in suitable neighbourhoods of 0.

scholarly journals Stable vector bundles on algebraic surfaces II

1973 ◽  
Vol 52 ◽  
pp. 173-195 ◽  
Author(s):  
Fumio Takemoto
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
Algebraic Surfaces ◽  
Complex Numbers ◽  
Stable Vector Bundle ◽  
Projective Varieties ◽  
Stable Vector Bundles ◽  
Direct Sum

This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

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