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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/190458
On the dimension of Voisin sets in the moduli space of abelian varieties
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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 $\{x\}-\left\{0_A\right\}$ 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.
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COLOMBO, E., NARANJO DEL VAL, Juan Carlos and PIROLA, Gian Pietro. On the dimension of Voisin sets in the moduli space of abelian varieties. Mathematische Annalen. 2021. Vol. 381, num. 91-104. ISSN 0025-5831. [consulted: 18 of August of 2026]. Available at: https://hdl.handle.net/2445/190458