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

Abstract

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 $V_k(A)$ 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.

arXiv commentFinal version to appear in Mathematische Annalen