Positivity properties of toric vector bundles

Abstract

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.

Equivariant vector bundles on complete symmetric varieties of minimal rank

“…In [6], a similar result is proved for vector bundles on toric varieties. Before stating the next result, we recall that for the conjugation action of G on itself, Steinberg proved that for a maximal torus T of G, the restriction homomorphism…”

Mori dream spaces

“…Conjecture 4.11 (Hering, Mustaţȃ, Payne, [12]). Let M be a smooth projective toric variety, let E be a toric vector bundle on M and let X be the associated projective bundle.…”

Quantum field theory overF1

“…The vector bundle E is determined, as an equivariant bundle, by its restrictions E| Uσ and by the equivariant gluings over the intersections U σi ∩ U σj . Moreover, as observed in [29], the restrictions E| Uσ decompose as sums of line bundles…”

“…A result of [31] shows that E is equivariant if an only if the bundles γ * E are all isomorphic to E, for all elements γ in the torus T . For further characterizations and results on toric bundles see [29] and [14]. Vector bundles that are sums of line bundles admit an equivariant structure, and so do tangent and cotangent bundle.…”