---
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.'
address:
- 'Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada'
- ' Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada'
author:
- Nathan Grieve
title: 'Index conditions and cup-product maps on abelian varieties'
---

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Introduction {#intro}
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Let $X$ be an abelian variety, defined over an algebraically closed field, and let $L$ and $M$ be 