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. Intuitively, a line bundle is q-ample if it is positive "in all but at most q directions". 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. We also show that a variant of q-ampleness defined by Demailly-Peternell-Schneider is equivalent to the naive notion. As a consequence, q-ampleness defines an open cone (not convex) in the Neron-Severi space N^1(X).