We study subvarieties of an $n$-dimensional complex principally polarized abelian variety (ppav) $(X,\theta)$ with class $d$ times the integral class $\theta^2/2$. On a general ppav, the question of their existence can be reformulated as: is the integral class $\theta^2/2$ the class of an algebraic cycle? We show that these subvarieties $V$ are singular if $n\ge 8$, and satisfy $dim(Sing(V))\ge n/4$ for $X$ general of dimension $\ge 26$. We also study an analog of Hartshorne's conjecture for codimension $2$ subvarieties of $X$. If $(X,\theta)$ is a general ppav of dimension $n\ge 11$, we prove that any smooth subvariety of $X$ with class $e\theta^2$, $e\le 8$, is (almost) a complete intersection.