Index conditions and cup-product maps on abelian varieties

Abstract

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.