= {\operatorname{i}}(T^*_xE) + {\operatorname{i}}(F) \text{.}$

Each pair $(E,F)$ of vector bundles on $X$ which satisfies the pair index condition determines cup-product maps $$\label{basic-cup-prod}
 \cup(T^*_x E, F) : {\mathrm{H}}^{{\operatorname{i}}(T_x^*E)}(X,T^*_x E) \otimes {\mathrm{H}}^{{\operatorname{i}}(F)}(X,F) \xrightarrow{\cup} {\mathrm{H}}^{{\operatorname{i}}(T^*_x(E) \otimes F)}(X, T^*_x(E) \otimes F) \text{,}$$ for all $x \in X$, with nonzero source and target space.

[**Remark.**]{} Using upper-semicontinuity, see [@Mum part (a) of the corollary on p. 47] for instance, we deduce

-   [a pair $(E,F)$ of vector bundles on $X$ satisfies the pair index condition if and only if the vector bundles $\text{$T^*_xE$, $F$, and $T^*_x(E) \otimes F$}$ satisfy the index condition, for all $x \in X$, and ${\operatorname{i}}(E) + {\operatorname{i}}(F) = {\operatorname{i}}(E \otimes F) \text{;}$]{}

-   [a pair $(L,M)$ of line bundles on $X$ satisfies the pair index condition if and only if $L$, $M$, and $L \otimes M$ are non-degenerate and ${\operatorname{i}}(L) + {\operatorname{i}}(M) = {\operatorname{i}}(L \otimes M)$.]{}

Real multiplication and the pair index condition {#real:multi:vect}
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In [@Shimura] Shimura constructs families of abelian varieties with real multiplication. On the other hand, in [@Matsushima76], Matsushima showed how the Appell-Humbert theorem can be generalized to construct simple semi-homogeneous vector bundles of higher rank. Here, we use these results to address Problem \[cp-prod-intro-prob\] *(b)*. 