Partially positive line bundles
Seminar Room 1, Newton Institute
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 prove some of the basic properties of q-ample line bundles. Related ideas have been used by Ottem to define what an "ample subvariety" of any codimension should mean.