Finite-dimensionality and cycles on powers of $K3$ surfaces

Abstract

For a $K3$ surface $S$, consider the subring of $\mathrm{CH}(S^n)$ generated by divisor and diagonal classes (with $\mathbb{Q}$-coefficients). Voisin conjectures that the restriction of the cycle class map to this ring is injective. We prove that Voisin's conjecture is equivalent to the finite-dimensionality of $S$ in the sense of Kimura-O'Sullivan. As a consequence, we obtain examples of $S$ whose Hilbert schemes satisfy the Beauville-Voisin conjecture.