  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).
