scholarly journals Stable vector bundles on an algebraic surface

1975 ◽  
Vol 58 ◽  
pp. 25-68 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
First Chern Class

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

2020 ◽  
Vol 20 (1) ◽  
pp. 109-116
Author(s):  
Masahiro Ohno
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Closed Field ◽  
Quadric Surface ◽  
Algebraically Closed Field ◽  
First Chern Class ◽  
Smooth Quadric Surface ◽  
Nef Vector Bundles

AbstractWe classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.

scholarly journals Torus-equivariant vector bundles on projective spaces

1988 ◽  
Vol 111 ◽  
pp. 25-40 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundle ◽  
Toric Variety ◽  
Vector Bundles ◽  
Rational Point ◽  
Projective Spaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Algebraic Torus

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

scholarly journals On the positivity of the first Chern class of an Ulrich vector bundle

2021 ◽  
pp. 2150071
Author(s):  
Angelo Felice Lopez
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Vector Bundles ◽  
First Chern Class

We study the positivity of the first Chern class of a rank [Formula: see text] Ulrich vector bundle [Formula: see text] on a smooth [Formula: see text]-dimensional variety [Formula: see text]. We prove that [Formula: see text] is very positive on every subvariety not contained in the union of lines in [Formula: see text]. In particular, if [Formula: see text] is not covered by lines we have that [Formula: see text] is big and [Formula: see text]. Moreover we classify rank [Formula: see text] Ulrich vector bundles [Formula: see text] with [Formula: see text] on surfaces and with [Formula: see text] or [Formula: see text] on threefolds (with some exceptions).

scholarly journals On the Behavior of Extensions of Vector Bundles Under the Frobenius Map

1972 ◽  
Vol 48 ◽  
pp. 73-89 ◽  
Author(s):  
Hiroshi Tango
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Frobenius Map

Let k be an algebraically closed field of characteristic p > 0, and let X be a curve defined over k. The aim of this paper is to study the behavior of the Frobenius map F*: H1(X, E) → H1(X, F*E) for a vector bundle E.

scholarly journals Vector Bundles on an Elliptic Curve

1971 ◽  
Vol 43 ◽  
pp. 41-72 ◽  
Author(s):  
Tadao Oda
Keyword(s):  
Vector Bundle ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Closed Field ◽  
List Type ◽  
Algebraically Closed Field ◽  
Frobenius Map

Let k be an algebraically closed field of characteristic p≧ 0, and let X be an abelian variety over k.The goal of this paper is to answer the following questions, when dim(X) = 1 and p≠0, posed by R. Hartshorne: (1)Is E(P) indecomposable, when E is an indecomposable vector bundle on X?(2)Is the Frobenius map F*: H1 (X, E) → H1 (X, E(p)) injective?We also partly answer the following question posed by D. Mumford:(3)Classify, or at least say anything about, vector bundles on X when dim (X) > 1.

A CRITERION FOR GETTING A BIG COMPONENT OF THE MODULI OF VECTOR BUNDLES BY CHANGING A POLARIZATION

2001 ◽  
Vol 12 (08) ◽  
pp. 927-942
Author(s):  
KIMIKO YAMADA
Keyword(s):  
Vector Bundles ◽  
Line Bundles ◽  
Closed Field ◽  
Projective Surface ◽  
Chern Classes ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
Normal Surfaces ◽  
Smooth Projective Surface ◽  
Moduli Of Vector Bundles

Let MH(c1, c2) be a coarse moduli scheme parameterizing all rank-two H-μ-stable vector bundles with Chern classes (c1, c2) on a smooth projective surface X over an algebraically closed field. For fixed two ample line bundles H and H′, it is known that if c2 is greater than some constant p(X, H, H′) depending on H and H′ then MH(c1, c2) and MH′(c1, c2) are birationally equivalent. In this paper we show that this constant p(X, H, H′) generally does depends on the choice of H and H′. More precisely, we give some example of surface (and c1) on which, for any number K, there exists an integer c2 with c2≥K such that sup H: ample dim MH(c1, c2) = + ∞. This result is available also for normal surfaces.

scholarly journals GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

2009 ◽  
Vol 20 (11) ◽  
pp. 1363-1396 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Chern Class ◽  
Vector Bundles ◽  
Crossed Products ◽  
Tensor Categories ◽  
Dual Object ◽  
Nontrivial Center ◽  
First Chern Class ◽  
The Given

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

scholarly journals On Equivariant Vector Bundles on an Almost Homogeneous Variety

1975 ◽  
Vol 57 ◽  
pp. 65-86 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundles ◽  
Multiplicative Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Algebraic Torus ◽  
Homogeneous Variety

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

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