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 $$\dim V_k(A) \le k-1$$ and $$\dim V_k(A)$$ 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.