$q$-ampleness and partial positivity are subjects of independent interest — the survey article [@Greb:Kur] provides a summary of some other results in this area.

Theorems \[Step2\] and \[MainTheorem\] are applications of Theorem \[vanishing:thm\]. These theorems show how the concept of naive $q$-ampleness, for $q > 0$, can be used to study higher cup-product maps on abelian varieties. They also show that this concept is related to global generation of vector bundles on abelian varieties. Let us now describe some aspects of Theorems \[Step2\] and \[MainTheorem\] in more detail.

Theorem \[Step2\] determines the *asymptotic nature* of the *pair index condition*. We define this condition in §\[pair:index\]; condition is a special case. Theorem \[Step2\] implies that if $E$ and $F$ are vector bundles on $X$ and if $(L,M)$ is a pair of line bundles which satisfies condition , then the cup-product maps $$\label{Intro:CP}
{\mathrm{H}}^{{\operatorname{i}}(L)}(X, T^*_x(L^n \otimes E)) \otimes {\mathrm{H}}^{{\operatorname{i}}(M)}(X, M^n \otimes F) \xrightarrow{\cup} {\mathrm{H}}^{{\operatorname{i}}(L \otimes M)}(X, T^*_x(L^n \otimes E) \otimes M^n \otimes F)$$ have nonzero source and target space for all $x \in X$ and all $n \gg 0$.

Theorem \[MainTheorem\] is our main result and addresses the behaviour of the cup-product map . It implies that the map is nonzero and 