Index conditions and cup-product maps on abelian varieties

TL;DR

This paper investigates cup-product maps on abelian varieties, demonstrating their positivity properties, asymptotic behaviors, and applications to vector bundles, with examples and conditions for their occurrence.

Contribution

It extends Mumford's index theorem to analyze positivity and asymptotics of cup-product maps on abelian varieties, providing new insights and examples.

Findings

Non-degenerate line bundles exhibit positivity similar to ample bundles.

Families of cup-product maps are asymptotically globally generated.

Constructed examples show zero maps on one-dimensional loci.

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.