A 9-dimensional family of K3 surfaces with finite-dimensional motive

Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite-dimensional motive, in the sense of Kimura. We also establish the Kuga–Satake Hodge conjecture for S, as well as Voisin’s conjecture concerning zero-cycles. As a consequence, we obtain Kimura finite-dimensionality, the Kuga–Satake Hodge conjecture, and Voisin’s conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism.