study of cup-product maps, is the fact that non-degenerate line bundles on an abelian variety $X$, with nonzero index, are *partially positive*.

In §\[section:vanishing:theorem\], we establish

\[vanishing:thm\] Let $Y$ be an abelian variety defined over an algebraically closed field. Let $L$ and $F$ denote, respectively, a line bundle and a coherent sheaf on $Y$. If $\chi(L) \not = 0$, then there exists a positive integer $n_0$ such that, for all isogenies $f\colon X\rightarrow Y$, we have ${\mathrm{H}}^j(X,f^*(F\otimes L^n) \otimes \alpha) = 0$ for all $j > {\operatorname{i}}(L)$, for all $n {\geqslant}n_0$, and all $\alpha \in {\operatorname{Pic}}^0(X)$.

We state additional immediate consequences of Theorem \[vanishing:thm\] in §\[section:vanishing:theorem\].

Asymptotic nature of the pair index condition
---------------------------------------------

To prove our main result concerning cup-product maps, we first determine the asymptotic nature of the pair index condition.

\[Step2\] Let $X$ be an abelian variety and let $(L,M)$ be a pair of line bundles on $X$ which satisfies the pair index condition. If $E$ and $F$ are vector bundles on $X$, then there exists a positive integer $n_0$ such that, for all $n {\geqslant}n_0$, the pair $(L^n \otimes E, M^n \otimes F)$ satisfies the pair index condition and $${\operatorname{i}}(T^*_x(L^n \otimes E)) = {\operatorname{i}}(L) \text{,} {\operatorname{i}}(M^n 