Positivity properties of toric vector bundles

Hering, Milena; Mustaţă, Mircea; Payne, Sam

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 607-640.

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 $ℰ$ et un point $x\in X$, 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 $H^0(X,L)\otimes {𝒪}_X\to L$, 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.

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 $H^0(X,L)\otimes {𝒪}_X\to L$, for ample line bundles $L$. We give examples of twists of such bundles that are ample but not globally generated.

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