\otimes F) = {\operatorname{i}}(M) \text{, and} {\operatorname{i}}(T^*_x(L^n \otimes E) \otimes M^n \otimes F) = {\operatorname{i}}(L \otimes M)\text{,}$$ for all $x \in X$.

Statement of main result
------------------------

First, recall that $p_i$ denotes the projection of $X \times X$ onto the first and second factors respectively (for $i = 1,2$), and $m := p_1 + p_2$. We prove, in Proposition \[mainprop1\], that the image of the cup-product map , for $x \in X$, coincides with that of the fiberwise evaluation map $${\mathrm{H}}^0(X, {\mathrm{R}}^{{\operatorname{i}}(E \otimes F)}_{p_1*}(m^*E \otimes p_2^* F)) \otimes \kappa(x) \rightarrow {\mathrm{R}}^{{\operatorname{i}}(E \otimes F)}_{p_1*}(m^*E \otimes p_2^* F){\mid_x}$$ of the vector bundle ${\mathrm{R}}^{{\operatorname{i}}(E \otimes F)}_{p_1*}(m^*E \otimes p_2^* F)$ on $X$. Using this relationship, Theorem \[vanishing:thm\], Theorem \[Step2\], and results of Pareschi-Popa [@Par], [@Par:Popa] and [@Par:Popa:II], we give two proofs of our main result. (See §\[proof:main:theorem\] and §\[main:theorem:proof2\].)

\[MainTheorem\] Let $X$ be an abelian variety, let $E$ and $F$ be vector bundles on $X$, and let $(L,M)$ be a pair of line bundles on $X$ which satisfies the pair index condition. There exists a positive integer $n_0$ such that the cup-product maps $$\label{asym:cp1} {\mathrm{H}}^{{\operatorname{i}}(L)}(X, T^*_x(L^n \otimes E)) \otimes {\mathrm{H}}^{{\operatorname{i}}(M)}(X , M^n \otimes F) \xrightarrow{\cup} {\mathrm{H}}^{{\operatorname{i}}(L\otimes M)}(X, T^*_x(L^n \otimes E)\otimes M^n \otimes F)$$ are 