surjective for all $x \in X$ and all $n \gg 0$.

A further consequence of Theorem \[MainTheorem\] concerns global generation of a particular class of vector bundles. More specifically, in Proposition \[mainprop1\] we prove that the condition that the map is surjective, for $n$ fixed and all $x \in X$, is equivalent to the condition that the vector bundle $$\label{PP:glob}
{\mathrm{R}}^{{\operatorname{i}}(L \otimes M)}_{p_1 *}(m^*(L^n \otimes E) \otimes p_2^*(M^n \otimes F))$$ is globally generated. Since Theorem \[MainTheorem\] implies that the cup-product maps are surjective for all $n \gg 0$, it also implies that the vector bundle is globally generated for all $n \gg 0$.

We prove Theorems \[vanishing:thm\], \[Step2\], and \[MainTheorem\] in §\[Proofs\]. These theorems apply to line bundles with nonzero index on abelian varieties defined over algebraically closed fields of positive characteristic. As hinted at above, one theme of our results is that non-degenerate line bundles with nonzero index exhibit positivity analogous to that of ample line bundles. This positivity follows from, and can be expressed in terms of, Mumford’s index theorem.

In §\[ab-var-examples\] we consider examples which illustrate Theorems \[vanishing:thm\], \[Step2\], and \[MainTheorem\] as we now explain. When considering cup-product maps such as , arising from a pair of 