Define a line bundle on a projective variety to be -ample, for a natural number , if tensoring with high powers of kills coherent sheaf cohomology above dimension . Thus 0-ampleness is the usual notion of ampleness. We show that -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 -ampleness is a Zariski open condition, which is not clear from the definition.
Cite this article
Burt Totaro, Line bundles with partially vanishing cohomology. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 731–754DOI 10.4171/JEMS/374