Density of non-simple abelian varieties over the real numbers

Let $k \leq g$ be nonnegative integers. We consider a family of polarised abelian varieties of dimension $g$ over $\mathbb R$ and give a criterion for the density in the parameter space of those abelian varieties over $\mathbb R$ containing a $k$-dimensional abelian subvariety over $\mathbb R$. As applications, we prove density of such a set in the moduli space of polarised real abelian varieties of dimension $g$, and (for small $k$) density of real algebraic curves mapping non-trivially to real $k$-dimensional abelian varieties in the moduli space of real algebraic curves of genus $g$ as well as in the moduli space of real plane curves. This extends to the real setting results by Colombo and Pirola as outlined in their paper Some density results for curves with non-simple jacobians, $\textit{Math. Ann.}$ 288.1 (1990): 161-178.