Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

Abstract

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V$ if $V \subset \tM_g$ (or $\cA_g$) is complete of codimension $g$.