Real moduli spaces and density of non-simple real abelian varieties

For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$ over $\mathbb{R}$, we 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 application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension $g$, and density of real algebraic curves mapping non-trivially to real $k$-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the 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", $\textit{Math. Ann.}$ (1990). We then consider the real locus of an algebraic stack over $\mathbb{R}$, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for $\mathcal{M}_g$ and $\mathcal{A}_g$, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross-Harris ("Real algebraic curves", $\textit{Ann. Sci. Ecole. Norm. Sup.}$ (1981)) and Sepp\"al\"a-Silhol ("Moduli Spaces for Real Algebraic Curves and Real Abelian Varieties", $\textit{Math. Z.}$ (1989)).