Compact Subvarieties of the Moduli Space of Complex Abelian Varieties

We determine the maximal dimension of compact subvarieties of ${\mathcal{A}}_g$, the moduli space of complex principally polarized abelian varieties of dimension $g$, and the maximal dimension of a compact subvariety through a very general point of ${\mathcal{A}}_g$. This also allows us to draw some conclusions for compact subvarieties of the moduli space of complex curves of compact type.