Equivalence of “parameterized by 1-cycles” and geometric coniveau 1 for all smooth sextic hypersurfaces

Determine whether, for every smooth sextic hypersurface X ⊂ P^5, the property that the relevant Hodge substructure is parameterized by 1-cycles is equivalent to the property that it has geometric coniveau 1. This equivalence holds for general sextic hypersurfaces, and it would follow from the Lefschetz standard conjecture, but its validity for all smooth sextics remains to be established.

Background

The paper studies Hodge-theoretic properties of sextic fourfolds invariant under a specific involution and their relation to Voisin’s framework connecting parameterization by cycles and geometric coniveau. For general sextic hypersurfaces in P5, Voisin notes that these two notions coincide.

However, extending this equivalence from a general member to all smooth sextics is not currently known, though it would be implied by the Lefschetz standard conjecture. Establishing the full equivalence would clarify the relationship between Hodge-theoretic and cycle-theoretic conditions across the entire moduli of smooth sextic hypersurfaces.

References

Though the notions parameterized by 1-cycles'' andgeometric coniveau 1'' are equivalent for general sextic hypersurfaces $X \subset \mathbb{P}5$ (as Voisin notes Rem.~0.5), their equivalence for all smooth sextics apparently remains unknown, though it would be implied by the Lefschetz standard conjecture .

— Hodge Structures in Sextic Fourfolds Equipped with an Involution  (2603.29157 - Diamond, 31 Mar 2026) in Introduction