Uniform Vector Bundles Research Articles

Uniform (d + 1)-bundle over the Grassmannian G(d,n)

This paper is dedicated to the classification of uniform vector bundles of rank [Formula: see text] over the Grassmannian [Formula: see text] ([Formula: see text]) over an algebraically closed field in characteristic [Formula: see text]. Specifically, we show that all uniform vector bundles with rank [Formula: see text] over [Formula: see text] are homogeneous.

On uniform flag bundles on Fano manifolds

As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper, we develop the necessary background and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.

Splitting conjectures for uniform flag bundles

We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the validity of these conjectures in the case of classical groups.

On jumping lines of vector bundles on $$\mathbb {P}^n$$ P n

We give bounds on the rank of non uniform vector bundles on \(\mathbb {P}^n\) with a finite number of jumping lines.

Spaces of matrices of constant rank and uniform vector bundles

We consider the problem of determining l(r,a), the maximal dimension of a subspace of a×a matrices of rank r. We first review, in the language of vector bundles, the known results. Then using known facts on uniform bundles we prove some new results and make a conjecture. Finally we determine l(r;a) for every r, 1≤r≤a, when a≤10, showing that our conjecture holds true in this range.

Uniform vector bundles on Fano manifolds and applications

In this paper we give a splitting criterion for uniform vector bundles on Fano manifolds covered by lines. As a consequence, we classify low rank uniform vector bundles on Hermitian symmetric spaces and Fano bundles of rank two on Grassmannians.

Uniform vector bundles on P^N as Syzygy bundles

Uniform vector bundles on P^N as Syzygy bundles

A cohomological class of vector bundles

The goal of this paper is to give a cohomological characterization of F n , t {F_{n,t}} , where F n , t := Ker ⁡ ( ( n + t ; n ) O P n ( − t ) → O P n ) {F_{n,t}}: = \operatorname {Ker} ((n + t;n){\mathcal {O}_{{{\mathbf {P}}^n}}}( - t) \to {\mathcal {O}_{{{\mathbf {P}}^n}}}) .

Uniform vector bundles on quadrics

In this paper we prove that a rankr uniform vector bundle on a nonsingular quadricQ with dimQ≥2r+2 is a direct sum of line bundles. We study also rank 2 uniform vector bundles onP 1×P1. We prove that they are not all homogeneous and that any rank 2 homogeneous vector bundles onP 1×P1 is decomposable.

Uniform vector bundles on a projective space

Uniform vector bundles on a projective space

Decomposing algebraic vector bundles on the projective line

V ( R ) \mathcal {V}(R) denotes the category of algebraic vector bundles on P R 1 , R {\mathbf {P}}_R^1,\;R a commutative, noetherian ring. If K K is a field, it is known that any F ∈ V ( K ) \mathcal {F} \in \mathcal {V}(K) is isomorphic to a (unique) direct sum of line bundles. If p ∈ Spec ⁡ R \mathfrak {p} \in \operatorname {Spec} R and K ( p ) K(\mathfrak {p}) is the quotient field of R / p R/\mathfrak {p} , any F ∈ V ( R ) \mathcal {F} \in \mathcal {V}(R) induces a bundle in V ( K ( p ) ) \mathcal {V}(K(\mathfrak {p})) , and so a decomposition into line bundles. If the decomposition is the same for each p , F \mathfrak {p},\;\mathcal {F} is said to be uniform. It is shown that if R R is reduced, uniform vector bundles on P R 1 {\mathbf {P}}_R^1 are sums of tensor products of (pullbacks of) bundles on Spec ⁡ R \operatorname {Spec} R with line bundles on P R 1 {\mathbf {P}}_R^1 .

On uniform vector bundles

On uniform vector bundles