On the dimension of Voisin sets in the moduli space of abelian varieties

We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting in the collection of points $x\in A$ such that the zero-cycle $\{x\}-\{0_A\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $\dim V_k(A) \leq k-1$ and $V_k(A)$ is countable for a very general abelian variety of dimension at least $2k-1$. We study in particular the locus $\mathcal V_{g,2}$ in the moduli space of abelian varieties of dimension $g$ with a fixed polarization, where $V_2(A)$ is positive dimensional. We prove that an irreducible subvariety $\mathcal Y \subset \mathcal V_{g,2}$, $g\ge 3$, such that for a very general $y \in \mathcal Y $ there is a curve in $V_2(A_y)$ generating $A$ satisfies $\dim \mathcal Y \le 2g - 1.$ The hyperelliptic locus shows that this bound is sharp.