\eta}$ on $\CC^g$ whose imaginary part is rational valued on $J_Z(\mathcal{M})$.

Matsushima has shown that the Schr[ö]{}dinger representation of the group ${\operatorname{G}}_{H_{Z, \eta}}(J_Z(\mathcal{M}))$ determines a non-degenerate simple semi-homogeneous vector bundle $F_{Z, \eta}$ on $X$ with slope $H_{Z, \eta}$ and index equal to $m {\operatorname{i}}(\eta)$, where ${\operatorname{i}}(\eta)$ denotes the number of negative conjugates of $\eta$, [@Matsushima76 Corollary 8.3, p. 184, Theorem 8.4, p. 188, Theorem 9.3, p. 195].

On the other hand since the image of $K^{\oplus 2}$ in $\RR^{\oplus 2 e}$, under the map $$(\alpha, \beta) \mapsto (\sigma_1(\alpha), \dots, \sigma_e(\alpha), \sigma_1(\beta), \dots, \sigma_e(\beta)) \text{,}$$ is dense, there exists $ \eta , \gamma \in K$ such that ${\operatorname{i}}(\eta) = \ell$, ${\operatorname{i}}(\gamma) = k$, and ${\operatorname{i}}(\eta + \gamma) = \ell + k$.

Using [@Matsushima76 Theorem 9.2, p. 191], we deduce that the vector bundle $F_{Z, \eta} \otimes F_{Z, \gamma}$ is a non-degenerate semi-homogeneous vector bundle (it need not be simple in general although it is a direct sum of vector bundles analytically equivalent to the vector bundle $F_{H_{Z, \eta + \gamma}}$), satisfies the index condition, and has index $m {\operatorname{i}}(\eta + \gamma)$.

Statement of results and outline of their proof {#statement-cup-prod-results}
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[$q$]{}-ampleness and non-degenerate line bundles {#ample:results}
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Important, in our 