We study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { 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 $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

中文翻译：

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阿贝尔簇模空间中voisin集的维数

我们研究了复杂阿贝尔变体 A 的子集 $$V_k(A)$V k ( A ) 包含在点集合 $$x\in A$$ x ∈ A 中，使得零循环 $$\{ x\}-\{0_A\}$$ { x } - { 0 A } 对于 Chow 组中的 Pontryagin 乘积是 k 幂零的。这些集合最近由 Voisin 引入，她证明了 $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 和 $$\dim V_k(A)$$ dim V k (A) 对于维数至少为 $$2k-1$$2 k-1 的非常一般的阿贝尔变体是可数的。我们特别研究了具有固定极化的 g 维阿贝尔变体的模空间中的轨迹 $${\mathcal {V}}_{g,2}$$ V g , 2，其中 $$V_2(A)$ $ V 2 ( A ) 是正维数。我们证明了一个不可约子变体 $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , 这样对于一个非常一般的 $$y \in {\mathcal {Y}}$$ y ∈ Y 有一条曲线 $$V_2(A_y)$$ V 2 ( A y ) 生成 A 满足 $$\dim { \mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 。超椭圆轨迹表明这个界限是尖锐的。