Vector bundles satisfying the point property

2020 ◽  
pp. 2250069
Author(s):  
Punam Gupta ◽  
Sanjay Kumar Singh
Keyword(s):  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Surface ◽  
Point Property

In this note, we prove that vector bundles which satisfy the point property over a very general principally polarized Jacobian, Prym and abelian variety are indecomposable. We also compare two known constructions of vector bundles satisfying the point property over a very general principally polarized abelian surface.

scholarly journals Continuous CM-regularity of semihomogeneous vector bundles

2020 ◽  
Vol 20 (3) ◽  
pp. 401-412
Author(s):  
Alex Küronya ◽  
Yusuf Mustopa
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Upper Bound ◽  
Vector Bundles ◽  
Projective Variety ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Piecewise Constant ◽  
Generic Vanishing

AbstractWe ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

scholarly journals A theorem of Matsushima

1974 ◽  
Vol 54 ◽  
pp. 123-134 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Complex Torus ◽  
Frobenius Endomorphism

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

scholarly journals Index conditions and cup-product maps on Abelian varieties

2014 ◽  
Vol 25 (04) ◽  
pp. 1450036 ◽  
Author(s):  
Nathan Grieve
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
General Setting ◽  
Index Theorem ◽  
Line Bundles ◽  
One Dimensional ◽  
Cup Product ◽  
Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

scholarly journals Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

2021 ◽  
Vol 20 (1) ◽  
pp. 95-119
Author(s):  
Nathan Grieve
Keyword(s):  
New York ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Index Theorem ◽  
Polynomial Functions ◽  
Algebraic Varieties ◽  
Wedderburn Decomposition ◽  
Generic Vanishing ◽  
Roch Theorem

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

scholarly journals On the S-fundamental group scheme. II

2012 ◽  
Vol 11 (4) ◽  
pp. 835-854 ◽  
Author(s):  
Adrian Langer
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Group Scheme ◽  
Line Bundles ◽  
Fundamental Groups ◽  
Determinant Line Bundles

AbstractThe S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial étale fundamental group.

scholarly journals Ample vector bundles on a rational surface

1975 ◽  
Vol 59 ◽  
pp. 135-148 ◽  
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).

scholarly journals Stable Vector Bundles on Algebraic Surfaces

1972 ◽  
Vol 47 ◽  
pp. 29-48 ◽  
Author(s):  
Fumio Takemoto
Keyword(s):  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Abelian Surface ◽  
Closed Field ◽  
Stable Bundles ◽  
Stable Vector Bundles ◽  
Protective Plane ◽  
Numerical Equivalence ◽  
Fixed Rank ◽  
Definition Of

AbstractLet k be an algebraically closed field, and X a nonsingular irreducible protective algebraic variety over k. These assumptions will remain fixed throughout this paper. We will consider a family of vector bundles on X of fixed rank r and fixed Chern classes (modulo numerical equivalence). Under what condition is this family a bounded family? When X is a curve, Atiyah [1] showed that it is so if all elements of this family are indecomposable. But when I is a surface, he showed also that this condition is not sufficient. We give the definition of an H-stable vector bundle on a variety X. This definition is a generalization of Mumford’s definition on a curve. Under the condition that all elements of a family are H-stable of rank two on a surface X, we prove that the family is bounded. And we study H-stable bundles, when X is an abelian surface, the protective plane or a geometrically ruled surface.

scholarly journals Positivity of vector bundles on homogeneous varieties

2020 ◽  
Vol 31 (12) ◽  
pp. 2050097
Author(s):  
Indranil Biswas ◽  
Krishna Hanumanthu ◽  
D. S. Nagaraj
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Algebraic Group ◽  
Abelian Variety ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Projective Variety ◽  
Closed Curve ◽  
Seshadri Constant ◽  
Higher Rank

We study the following question: Given a vector bundle on a projective variety [Formula: see text] such that the restriction of [Formula: see text] to every closed curve [Formula: see text] is ample, under what conditions [Formula: see text] is ample? We first consider the case of an abelian variety [Formula: see text]. If [Formula: see text] is a line bundle on [Formula: see text], then we answer the question in the affirmative. When [Formula: see text] is of higher rank, we show that the answer is affirmative under some conditions on [Formula: see text]. We then study the case of [Formula: see text], where [Formula: see text] is a reductive complex affine algebraic group, and [Formula: see text] is a parabolic subgroup of [Formula: see text]. In this case, we show that the answer to our question is affirmative if [Formula: see text] is [Formula: see text]-equivariant, where [Formula: see text] is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on [Formula: see text].

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.

scholarly journals On a certain type of vector bundles over an abelian variety

1976 ◽  
Vol 64 ◽  
pp. 31-45 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Abelian Variety ◽  
Vector Bundles ◽  
Theta Functions ◽  
Higher Rank

It seems to the author that the theory of vector bundles of rank r over an abelian variety of dimension g is not sufficiently developed except for the two cases r = 1 and g = 1. The case r = 1 is the theory of theta functions and is one of the richest branches of mathematics. It is quite natural to try to explore vector bundles in the higher rank case as we did in the case r = 1.

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