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Lelong classes on toric manifolds and a theorem of Siciak

Maritza M. Branker, Małgorzata Stawiska (2012)

Annales Polonici Mathematici

We generalize a theorem of Siciak on the polynomial approximation of the Lelong class to the setting of toric manifolds with an ample line bundle. We also characterize Lelong classes by means of a growth condition on toric manifolds with an ample line bundle and construct an example of a nonample line bundle for which Siciak's theorem does not hold.

Line bundles with partially vanishing cohomology

Burt Totaro (2013)

Journal of the European Mathematical Society

Define a line bundle L on a projective variety to be q -ample, for a natural number q , if tensoring with high powers of L kills coherent sheaf cohomology above dimension q . Thus 0-ampleness is the usual notion of ampleness. We show that q -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that q -ampleness is a Zariski open condition, which is not clear from the definition.

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