Remarks on Seshadri constants of vector bundles

Hacon, Christopher D.

doi:10.5802/aif.1772

Cited by 16 publications

(11 citation statements)

References 9 publications

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“…Note that the curves in C ρ,x are precisely the irreducible curves C on Y for which mult x ρ * C > 0. See also [Hac00] for the case of bundles.…”

Section: Definition and Properties Of Seshadri Constantsmentioning

confidence: 99%

“…where π is the blow-up of X at x with exceptional divisor E. Seshadri constants have attracted much attention as interesting invariants that capture subtle geometric properties of both X and L; see [Laz04a,Chapter 5] and [BDRH + 09]. In higher rank, a version of Seshadri constants for ample vector bundles (of arbitrary rank) appears implicitly in work of Beltrametti-Schneider-Sommese [BSS93,BSS96], and has been further studied by Hacon [Hac00].…”

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Seshadri constants for vector bundles

Fulger

Murayama

2019

Preprint

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We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci.We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose tangent bundle has positive Seshadri constant at a point is isomorphic to a projective space. We conjecture that the Fano condition can be removed. Among other results in this direction, we prove the conjecture for surfaces. Second, we restate a classical conjecture on the nef cone of self-products of curves in terms of semistability of higher conormal sheaves, which we use to identify new nef classes on self-products of curves. Third, we prove that our Seshadri constants can be used to control separation of jets for direct images of pluricanonical bundles, in the spirit of a relative Fujita-type conjecture of Popa and Schnell.Contents 28 8. Direct images of pluricanonical sheaves 31 References 36

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“…Note that the curves in C ρ,x are precisely the irreducible curves C on Y for which mult x ρ * C > 0. See also [Hac00] for the case of bundles.…”

Section: Definition and Properties Of Seshadri Constantsmentioning

confidence: 99%

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confidence: 99%

Seshadri constants for vector bundles

Fulger

Murayama

2019

Preprint

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“…The global Seshadri constant ε(E) is defined as inf x∈X ε(E, x). See [19] for background and further details about Seshadri constants of vector bundles.…”

Section: Q-twisted Bundles and Seshadri Constantsmentioning

confidence: 99%

Positivity properties of toric vector bundles

Hering¹,

Mustaţă²,

Payne³

2010

Ann. inst. Fourier

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We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M L that arise as the kernel of the evaluation map H 0 (X, L) ⊗ O X → L, for ample line bundles L. We give examples of twists of such bundles that are ample but not globally generated.Résumé. Nous prouvons qu'un fibré vectoriel equivariant sur une variété torique complète est nef ou ample si et seulement si sa restriction à chaque courbe invariante est nef ou ample, respectivement. Nous montrons également qu'étant donne un fibré vectoriel torique nef E et un point x ∈ X, il existe une section de E non-nulle en x; on déduit de cela que E est trivial si et seulement si sa restriction à chaque courbe invariante est triviale. Nous appliquons ces résultats et méthodes pour étudier en particulier les fibrés vectoriels M L , définis en tant que noyau des applications d'évaluation H 0 (X, L) ⊗ O X → L, ou L est un fibré en droites ample. Finalement, nous donnons des exemples des fibrés vectoriels toriques qui sont amples mais non engendrés par leur sections globales.

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“…where the infimum is taken over all curves C ⊂ P(E) that intersect p −1 (x), but are not contained in p −1 (x). For more details on Seshadri constants of vector bundles, see [14,13].…”

Section: Now We Will Prove (3) Since T Acts On the Exceptional Divisormentioning

confidence: 99%

“…While most of the works on Seshadri constants have considered the case of line bundles, there has been recent interest in Seshadri constants for vector bundles of arbitrary rank. An explicit definition of these was first given by Hacon [14]; see Definition 3.4 below. For a comprehensive survey and a generalization to the relative setting, see [13].…”

Section: Introductionmentioning

confidence: 99%

On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

Biswas¹,

Hanumanthu²,

Kannan³

2022

Preprint

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We study torus-equivariant vector bundles E on a complex projective variety X which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that E is nef (respectively, ample) if and only if its restriction to every torus-invariant curve in X is nef (respectively, ample). We also compute the Seshadri constants ε(E, x), where x ∈ X is any point fixed by the action of a maximal torus.

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Remarks on Seshadri constants of vector bundles

Cited by 16 publications

References 9 publications

Seshadri constants for vector bundles

Seshadri constants for vector bundles

Positivity properties of toric vector bundles

On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

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