Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 4

Bloch’s conjecture for certain hyperkähler fourfolds

Pages: 639 – 692

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n4.a3


Robert Laterveer (Institut de Recherche Mathématique Avancée, CNRS, Université de Strasbourg, France)


On a hyperkähler fourfold $X$, Bloch’s conjecture predicts that any involution acts trivially on the deepest level of the Bloch–Beilinson filtration on the Chow group of $0$-cycles.We prove a version of Bloch’s conjecture when $X$ is the Hilbert scheme of $2$ points on a generic quartic in $\mathbb{P}^3$, and the involution is the non-natural, non-symplectic involution on $X$ constructed by Beauville. This has interesting consequences for the Chow groups of the quotient.


algebraic cycles, Chow groups, motives, Bloch’s conjecture, Bloch–Beilinson filtration, hyperkähler varieties, $K3$ surfaces, Hilbert schemes, non-symplectic involution, multiplicative Chow–Künneth decomposition, “spread” of algebraic cycles in a family

2010 Mathematics Subject Classification

14C15, 14C25, 14C30

Received 29 September 2017

Published 21 December 2018