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

10.1017/s0027763000014367 ◽

1971 ◽

Vol 43 ◽

pp. 41-72 ◽

Cited By ~ 32

Author(s):

Tadao Oda

Keyword(s):

Vector Bundle ◽

Elliptic Curve ◽

Abelian Variety ◽

Vector Bundles ◽

Closed Field ◽

List Type ◽

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.

Torus-equivariant vector bundles on projective spaces

10.1017/s0027763000000982 ◽

1988 ◽

Vol 111 ◽

pp. 25-40 ◽

Cited By ~ 5

Author(s):

Tamafumi Kaneyama

Keyword(s):

Vector Bundle ◽

Toric Variety ◽

Vector Bundles ◽

Rational Point ◽

Projective Spaces ◽

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718001314 ◽

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 $r\mid d$.

Stable vector bundles on an algebraic surface

10.1017/s0027763000016688 ◽

1975 ◽

Vol 58 ◽

pp. 25-68 ◽

Cited By ~ 38

Author(s):

Masaki Maruyama

Keyword(s):

Vector Bundle ◽

Chern Class ◽

Algebraic Curve ◽

Algebraic Surface ◽

Vector Bundles ◽

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.

On Equivariant Vector Bundles on an Almost Homogeneous Variety

10.1017/s0027763000016561 ◽

1975 ◽

Vol 57 ◽

pp. 65-86 ◽

Cited By ~ 10

Author(s):

Tamafumi Kaneyama

Keyword(s):

Vector Bundles ◽

Multiplicative Group ◽

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.

Homology of SL2 over function fields I: Parabolic subcomplexes

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0047 ◽

2018 ◽

Vol 2018 (739) ◽

pp. 159-205

Author(s):

Matthias Wendt

Keyword(s):

Vector Bundles ◽

Function Fields ◽

Equivalence Classes ◽

Isotropy Group ◽

Closed Field ◽

Explicit Formulas ◽

Group Homology ◽

Equivariant Homology ◽

Smooth Affine

Abstract The present paper studies the group homology of the groups {\operatorname{SL}_{2}(k[C])} and {\operatorname{PGL}_{2}(k[C])} , where {C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}} is a smooth affine curve over an algebraically closed field k. It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve {\overline{C}} . There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of {\operatorname{SL}_{2}(k[C])} above degree s, generalizing a result of Suslin in the case {s=1} .

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

Advances in Geometry ◽

10.1515/advgeom-2018-0039 ◽

2020 ◽

Vol 20 (1) ◽

pp. 109-116

Author(s):

Masahiro Ohno

Keyword(s):

Chern Class ◽

Characteristic Zero ◽

Vector Bundles ◽

Closed Field ◽

Quadric Surface ◽

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.

Ample vector bundles on a rational surface

10.1017/s0027763000016846 ◽

1975 ◽

Vol 59 ◽

pp. 135-148 ◽

Cited By ~ 4

Author(s):

Toshio Hosoh

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Rational Surface ◽

Ruled Surface ◽

Necessary Condition ◽

Abelian Surface ◽

Closed Field ◽

Stable Vector Bundle ◽

Singular Curve ◽

Rank 2

On a complete non-singular curve defined over the complex number field C, a stable vector bundle is ample if and only if its degree is positive [3]. On a surface, the notion of the H-stability was introduced by F. Takemoto [8] (see § 1). We have a simple numerical sufficient condition for an H-stable vector bundle on a surface S defined over C to be ample; let E be an H-stable vector bundle of rank 2 on S with Δ(E) = c1(E)2 - 4c2(E) ≧ 0, then E is ample if and only if c1(E) > 0 and c2(E) > 0, provided S is an abelian surface, a ruled surface or a hyper-elliptic surface [9]. But the assumption above concerning Δ(E) evidently seems too strong. In this paper, we restrict ourselves to the projective plane P2 and a rational ruled surface Σn defined over an algebraically closed field k of arbitrary characteristic. We shall prove a finer assertion than that of [9] for an H-stable vector bundle of rank 2 to be ample (Theorem 1 and Theorem 3). Examples show that our result is best possible though it is not a necessary condition (see Remark (1) §2).

Normal bundles in flag bundles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100058874 ◽

1981 ◽

Vol 90 (3) ◽

pp. 395-402

Author(s):

Samuel A. Ilori

Keyword(s):

Vector Bundle ◽

Normal Bundle ◽

Closed Field ◽

Projective Varieties ◽

Algebraic Vector Bundle ◽

Algebraic Vector ◽

Normal Bundles ◽

Tangent Bundles

AbstractLet i: Y ↪ X be an inclusion map of non-singular irreducible algebraic quasi-projective varieties defined over an algebraically closed field. Let E be an algebraic vector bundle over X and H be a sub-bundle of the induced bundle, i*E. If j:F(H) ↪ F(E) is the corresponding inclusion map of (incomplete) flag bundles, then we derive the normal bundle N(F(H), F(E)) in terms of the bundles H and E, the tangent bundles of Y and X as well as the tautological bundles over F(H).

Formal orbifolds and orbifold bundles in positive characteristic

International Journal of Mathematics ◽

10.1142/s0129167x19500678 ◽

2019 ◽

Vol 30 (12) ◽

pp. 1950067

Author(s):

Manish Kumar ◽

A. J. Parameswaran

Keyword(s):

Fundamental Group ◽

Characteristic Zero ◽

Vector Bundles ◽

Positive Characteristic ◽

Closed Field ◽

Fundamental Groups ◽

Projective Curve ◽

Smooth Projective Curve ◽

Arbitrary Characteristic

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental groups approximate the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statements about vector bundles which are true in characteristic zero are also proved. Some of these are positive characteristic avatars of notions which appear in the second author’s work [A. J. Parmeswaran, Parabolic coverings I: Case of curves, J. Ramanujam Math. Soc. 25(3) (2010) 233–251.] in characteristic zero.