Line bundles with partially vanishing cohomology
Abstract
Define a line bundle L on a projective variety to be qample, for a natural number q, if tensoring with high powers of L kills coherent sheaf cohomology above dimension q. Thus 0ampleness is the usual notion of ampleness. Intuitively, a line bundle is qample if it is positive "in all but at most q directions". We show that qampleness 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 qampleness is a Zariski open condition, which is not clear from the definition. We also show that a variant of qampleness defined by DemaillyPeternellSchneider is equivalent to the naive notion. As a consequence, qampleness defines an open cone (not convex) in the NeronSeveri space N^1(X).
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 DOI:
 10.48550/arXiv.1007.3955
 arXiv:
 arXiv:1007.3955
 Bibcode:
 2010arXiv1007.3955T
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 14F17;
 14F05;
 32L20
 EPrint:
 24 pages, 3 figures