Uniform vector bundles on a projective space

Satô, Eiichi

doi:10.2969/jmsj/02810123

Cited by 47 publications

(31 citation statements)

References 6 publications

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“…Here we recall some of the results on uniform vector bundles on P n needed later, results due to Van de Ven [50], Sato [49], Elencwajg, Hirschowitz, and Schneider [13], Elencwajg [11], [12], Ellia [14], and Ballico [7]. Theorem 2.3.…”

Section: Preliminaries and Proofs Of Propositionsmentioning

confidence: 96%

Classification of generalized polarized manifolds by their nef values

Ohno

2006

Advg

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Let (M, E) be a generalized polarized manifold, i.e., a pair of an n-dimensional smooth projective variety and an ample vector bundle E of rank r on M . Let τ be the nef value of a polarized manifold (M, det E), i.e., the minimum of the set of real numbers t such that K M + t det E is nef; we have τ r ≤ n + 1 by Mori's theory. In this paper we classify the pairs (M, E) in the following two cases: (1) n − 2 ≤ τ r and τ ≥ 1; (2) n + 1 − τ r < τ ≤ 1.Proposition 1.1. We have τ r ≤ n + 1. Now we have the following classification.Proposition 1.2. If τ r = n + 1, then (M, E) is isomorphic to (P n , O(1) ⊕r ).

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Section: Preliminaries and Proofs Of Propositionsmentioning

confidence: 96%

Classification of generalized polarized manifolds by their nef values

Ohno

2006

Advg

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“…which restricts on U to (3.5.1). Hence S and Q are direct sums of line bundles by [25,61] (1) X ≃ P 6 and every fiber of the morphism e( U ) → X has dimension ≤ 1.…”

Section: Comparison Theoremmentioning

confidence: 99%

Classification of Mukai pairs with corank $3$

Kanemitsu¹

2019

Annales De l'Institut Fourier

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We classify pairs (X, E ) where X is a smooth Fano manifold of dimension n ≥ 5 and E is an ample vector bundle of rank n − 2 on X with c 1 (E ) = c 1 (X). Japan. 0.1. Study of generalized polarized pairs gives another motivation to investigate Mukai pairs. A pair (X, E ) is called a generalized polarized pair of dimension n and rank r if X is a smooth projective n-fold and E is an ample vector bundle of rank r. The adjoint divisor K X + c 1 (E ) is attached to a given generalized polarized pair (X, E ), and a fundamental problem in this field is to determine when the adjoint divisor K X + c 1 (E ) satisfy positivity (e.g., ampleness or nefness) or to distinguish generalized polarized pairs whose adjoint divisors lack positivity from general ones. Such a problem is carried out in a number of papers, including [4,7,20,49,65,69,[72][73][74]. In [7], Andreatta and Mella studied the case r = n − 2 and they clarified when the adjoint divisor is not nef. Also, assuming that K X +c 1 (E ) is nef but not ample, they (roughly) described the structure of the contraction defined by the adjoint divisor. Understandably the contraction can be trivial, which implies that (X, E ) is a Mukai pair [7, Theorem 5.1 (2) (i)]. Our result gives a detailed classification in such a case.Also, given a generalized polarized pair (X, E ) of dimension n and rank r, the geometry of the zero locus S of a section s ∈ H 0 (E ) is studied in several context, provided that S has the expected dimension n − r. For example, in [35, Corollary 1.3], it is proved that if S as above is a minimal surface of Kodaira dimension = 0, then S is a K3 surface and (X, E ) is a Mukai pair of corank 3. Thus:Corollary 0.5. Let (X, E ) be a generalized polarized pair of dimension n ≥ 5 and rank n − 2. Suppose that there is a K3 surface S ⊂ X which is a zero locus of a section s ∈ H 0 (E ). Then (X, E ) is one of the pairs as in Theorem 0.3.

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“…If the closed subset is empty, such bundles are called uniform vector bundles. Uniform bundles are widely studied not only on projective spaces [ 3 , 8 – 10 , 26 , 27 , 29 ] but also on special Fano manifolds of Picard number one [ 2 , 6 , 11 , 14 , 17 , 25 ]. Please see Introduction in [ 6 ] for the details.…”

Section: Introductionmentioning

confidence: 99%

Vector bundles on rational homogeneous spaces

Fang

Gao

2021

Annali di Matematica

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We consider a uniform r -bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X , then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian and the rank r is less than or equal to some number that depends only on X , then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces.

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Uniform vector bundles on a projective space

Abstract: The aim of this paper is to generalize the above result to higher dimension. Our main theorem which will be proved in \S 2 is as follows: MAIN THEOREM.

Cited by 47 publications

References 6 publications

Classification of generalized polarized manifolds by their nef values

Classification of generalized polarized manifolds by their nef values

Classification of Mukai pairs with corank $3$

Vector bundles on rational homogeneous spaces

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