A very general prime Fano threefold of genus 7 admits no multiplicative Chow-Künneth decomposition, because the explicit cycle Z_Y on Y×Y is Abel-Jacobi trivial but non-zero in the Chow group.
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Algebraic cycles and Fano threefolds of genus 7
A very general prime Fano threefold of genus 7 admits no multiplicative Chow-Künneth decomposition, because the explicit cycle Z_Y on Y×Y is Abel-Jacobi trivial but non-zero in the Chow group.