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 is an element of A such that the zero-cycle {x}-{0A} is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that dimV(k)(A)<= k-1 and dimV(k)(A) is countable for a very general abelian variety of dimension at least 2k-1. We study in particular the locus V-g,V-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 Y subset of V-g,V-2, g >= 3, such that for a very general y is an element of Y there is a curve in V-2(Ay) generating A satisfies dim Y <= 2g - 1. The hyperelliptic locus shows that this bound is sharp.