Compact Subvarieties of the Moduli Space of Complex Abelian Varieties

We determine the maximal dimension of compact subvarieties of Ag, 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 Ag. This also allows us to draw some conclusions for compact subvarieties of the moduli space of complex curves of compact type.