# Commentarii Mathematici Helvetici

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**Volume 90, Issue 2, 2015, pp. 503–511**

**DOI: 10.4171/CMH/362**

Published online: 2015-05-21

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

Qizheng Yin^{[1]}(1) ETH Zürich, Switzerland

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.

*Keywords: *Chow groups, $K3$ surfaces, finite-dimensional motives, Beauville–Voisin conjecture

Yin Qizheng: Finite-dimensionality and cycles on powers of $K3$ surfaces. *Comment. Math. Helv.* 90 (2015), 503-511. doi: 10.4171/CMH/362