Positivity for toric vector bundles

Details Positivity for toric vector bundles Positivity for toric vector bundles

2008-05-27

arxiv.org/abs/0805.4035v2

Ann. Inst. Fourier 60 (2010), no. 2, 607--640

Mathematics

Algebraic Geometry

28 pages; revised version, to appear in Ann. Inst. Fourier (Grenoble)

Scientific paper

We show that an 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 on sections of L, when L is an ample line bundle. We give examples of twists of such bundles that are ample but not globally generated.

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