PreprintCC BY 4.0arXiv
On the Chow ring of very general abelian varieties and a question of Pirola
Abstract
We prove that for a very general abelian variety of dimension ≥ 4, a divisor D∈ CH1(A) that satisfies D2=0 in CH2(A) is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration K=J/± Id→ ℳ4, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.
Related
BibTeX
@misc{voisin2026chow,
title = {{On the Chow ring of very general abelian varieties and a question of Pirola}},
author = {Claire Voisin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.07053},
eprint = {2607.07053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Chow ring of very general abelian varieties and a question of Pirola
AU - Claire Voisin
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2607.07053
UR - https://arxiv.org/abs/2607.07053
AB - We prove that for a very general abelian variety of dimension ≥ 4, a divisor D∈ CH^1(A) that satisfies D^2=0 in CH^2(A) is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration K=J/± Id→ ℳ_4, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.
ER -