Questions on the Chow ring of complete intersections

抄録

We state several questions, and prove some partial results, about the Chow ring 𝐴^*(𝑋) of complete intersections in projective space. For one thing, we prove that if 𝑋 is a general Calabi–Yau hypersurface, the intersection product 𝐴²(𝑋) ⋅ 𝐴^𝑖(𝑋) is one-dimensional, for any 𝑖 > 0. We also show that quintic threefolds have a multiplicative Chow–Künneth (MCK) decomposition. We wonder whether all Calabi–Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.