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

Abstract

This article is about hyperkähler fourfolds X admitting 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 some interesting consequences for the Chow ring of the quotient X/ι.(X) should only depend on the subring H * ,0 (X), we arrive at the following conjecture:

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).…”