Colombo, E.Naranjo del Val, Juan CarlosPirola, Gian Pietro2022-11-042022-11-042021-01-120025-5831https://hdl.handle.net/2445/190458We 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 $\{x\}-\left\{0_A\right\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $\operatorname{dim} V_k(A) \leq k-1$ and $\operatorname{dim} V_k(A)$ is countable for a very general abelian variety of dimension at least $2 k-1$. 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 $\mathcal{Y} \subset \mathcal{V}_{g, 2}$, $g \geq 3$, such that for a very general $y \in \mathcal{Y}$ there is a curve in $V_2\left(A_y\right)$ generating $A$ satisfies $\operatorname{dim} \mathcal{Y} \leq 2 g-1$. The hyperelliptic locus shows that this bound is sharp.14 p.application/pdfengcc by (c) Colombo, E. et al., 2021http://creativecommons.org/licenses/by/3.0/es/Varietats abelianesGeometria algebraicaCicles algebraicsAbelian varietiesAlgebraic geometryAlgebraic cyclesinfo:eu-repo/semantics/article7053842022-11-04info:eu-repo/semantics/openAccess