arXiv 2004.069072020-04math.AG

关于阿贝尔簇模空间中Voisin集的维数

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

E. Colombo, G. P. Pirola, J. C. Naranjo

备注：: Final version to appear in Mathematische Annalen

摘要：

我们研究了复阿贝尔簇$A$的子集$V_k(A)$，它包含在点$x\in A$的集合中，使得零圈$\{x\}-\{0_A\}$对于Chow群中的Pontryagin积是$k$ -幂零的.这些集介绍了最近的Voisin和她表明，$\dim V_k(A) \leq k-1$和$V_k(A)$是可数的一个非常普遍的阿贝尔各种尺寸至少$2k-1$。我们研究特别是轨迹$\mathcal V_{g,2}$的模空间中的阿贝尔簇的尺寸$g$与固定的极化，其中$V_2(A)$是积极的维。我们证明了一个不可约的子簇$\mathcal Y \subset \mathcal V_{g,2}$，$g\ge 3$，使得对于一个非常一般的$y \in \mathcal Y $，有一条曲线在$V_2(A_y)$生成$A$满足$\dim \mathcal Y \le 2g - 1.$的超椭圆轨迹表明，这个界限是尖锐的。

英文摘要：

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 $\dim V_k(A) \leq k-1$ 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.

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摘要：

英文摘要：

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