Positivity properties of toric vector bundles

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.