Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism

TL;DR

This paper investigates algebraic cycles on specific hyperk"ahler fourfolds with a non-symplectic automorphism of order 3, verifying Bloch's conjecture for their quotients.

Contribution

It proves Bloch's conjecture for Fano varieties of lines on special cubic fourfolds with a non-symplectic automorphism of order 3.

Findings

Chow group of 0-cycles is trivial for the quotients

Verification of Bloch's conjecture in new cases

Analysis of automorphisms on hyperk"ahler varieties

Abstract

Let $X$ be a hyperk\"ahler variety, and assume $X$ has a non-symplectic automorphism $\sigma$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<\sigma>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.