Positivity properties of toric vector bundles
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 607-640.

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 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.

Nous prouvons qu’un fibré vectoriel équivariant 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 et un point xX, il existe une section de non-nulle en x ; on déduit de cela que 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 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.

DOI: 10.5802/aif.2534
Classification: 14M25, 14F05
Keywords: Toric variety, toric vector bundle
Mot clés : variété torique, fibré vectoriel torique
Hering, Milena 1; Mustaţă, Mircea 2; Payne, Sam 3

1 University of Connecticut Department of Mathematics 196 Auditorium Road U-3009 Storrs CT 06269-3009 (USA)
2 University of Michigan Department of Mathematics East Hall Ann Arbor, MI 48109 (USA)
3 Stanford University Department of Mathematics Bldg 380 Stanford, CA 94305 (USA)
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     title = {Positivity properties of toric vector bundles},
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Hering, Milena; Mustaţă, Mircea; Payne, Sam. Positivity properties of toric vector bundles. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 607-640. doi : 10.5802/aif.2534. http://www.numdam.org/articles/10.5802/aif.2534/

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