论文标题
在阿贝利亚品种的模量空间中的Voisin集的尺寸上
On the dimension of Voisin sets in the moduli space of abelian varieties
论文作者
论文摘要
我们研究复杂的Abelian品种的子集$ v_k（a）$ a $ a $ a $组成$ x \的集合，以使零周期$ \ {x \} - \ {0_a \} $是$ k $ -nillpotent ch pontryagin chow groups the Chow group的产品。这些套装最近是由Voisin引入的，她表明$ \ dim v_k（a）\ leq k-1 $和$ v_k（a）$对于非常普遍的阿贝尔（Abelian）尺寸至少至少2k-1 $是可数的。 我们特别研究了基因座$ \ MATHCAL V_ {G，2} $在固定极化的Abelian尺寸$ G $的模量空间中，其中$ v_2（a）$是正维。 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很锋利。
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.
