Author:
Colombo E.,Naranjo J. C.,Pirola G. P.
Abstract
AbstractWe 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.
Funder
Ministerio de Ciencia, Innovación y Universidades
Gnsaga
Ministero dell’Istruzione, dell’Università e della Ricerca
Publisher
Springer Science and Business Media LLC