On the Chow groups of some hyperkähler fourfolds with a non-symplectic involution

Abstract

ABSTRACT. This note concerns hyperkähler fourfolds X having a non-symplectic involution ι. The Bloch-Beilinson conjectures predict the way ι should act on certain pieces of the Chow groups of X. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient X/ι.

On Chow groups of some hyperkahler fourfolds with a non-symplectic involution, II

“…To prove theorem 3.5, we exploit the fact that the family of cubics under consideration is sufficiently large for the method of "spread" as developed by Voisin [29], [30] to apply. There is only one other family of cubic fourfolds with a polarized involution that is anti-symplectic on the Fano variety; this other family has been treated in [17], using arguments very similar to those of the present article. The action of polarized symplectic automorphisms on Chow groups of Fano varieties of cubic fourfolds has already been studied by L. Fu [8], similarly using Voisin's method of "spread".…”

“…This conjecture is studied (and proven in some favourable cases) in [14], [15], [16], [17], [18]. The aim of this article is to provide more examples where conjecture 1.1 is verified, by considering Fano varieties of lines on cubic fourfolds.…”

Algebraic cycles on certain hyperkähler fourfolds with an order 3 non-symplectic automorphism II

“…For σ of order 2, there are two families of cubic fourfolds, and these have been treated in [31], [32]. For σ of order 3, there are 4 families [10, Examples 6.4, 6.5, 6.6 and 6.7]; the first is treated in [33], the others in theorems 3.1 and 4.1. where b i ∈ A 2 (X) σ and D i ∈ A 1 (X).…”

“…Theorem 4.1 is proven by using Voisin's method of "spread", as developed in [48], [49]. The argument is similar to that of [15] (which dealt with symplectic automorphisms on Fano varieties of cubic fourfolds), and that of [31], [32] (which dealt with anti-symplectic involutions on Fano varieties of cubic fourfolds).…”

Algebraic cycles and EPW cubes

“…This conjecture is studied, and proven in some particular cases, in [21][22][23][24][25]. The aim of this note is to provide some more examples where Conjecture 1.1 is verified, by considering "double EPW cubes" in the sense of [14] (cf.…”