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.