$H_2$, and $H_1 + H_2$ are non-degenerate; and]{}

-   [the Hermitian forms $H_1$, $H_2$, and $H_1 + H_2$ have, respectively, exactly $p$, $q$, and $p + q$ negative eigenvalues?]{}

Summary of results and organization of paper
--------------------------------------------

We prove several theorems all of which are related to Problem \[cp-prod-intro-prob\] and generalizations thereof.

Theorems \[existence:prop\] and \[Pair:index:theorem\] address Problem \[cp-prod-intro-prob\] *(b)* and its relation to Problem \[complex:pic\]. For example, a consequence of Theorem \[existence:prop\], or the more general Theorem \[Pair:index:theorem\], is that there exists, for $g$ fixed, complex simple abelian varieties of dimension $g$ and having real multiplication by a totally real number field of degree $g$ over $\QQ$, for which every possible instance of condition can actually occur. Both Theorems \[existence:prop\] and \[Pair:index:theorem\] rely on work of S. Shimura [@Shimura], while Theorem \[Pair:index:theorem\] also relies on work of Y. Matsushima [@Matsushima76].

Theorem \[vanishing:thm\] concerns the positivity of non-degenerate line bundles. One implication of this theorem can be phrased in the language of naive $q$-ampleness as defined by B. Totaro [@Tot p. 731]. In particular, Corollary \[q-ample\] asserts that if $L$ is a line bundle on $X$ with nonzero Euler characteristic and ${\operatorname{i}}(L) {\leqslant}q$, then $L$ is naively $q$-ample. Notions of 