Line bundles with partially vanishing cohomology

Abstract

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.