A brief note concerning hard Lefschetz for Chow groups

We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.Comment: 15 pages. Comments welcome ! To appear (in slightly different version) in Canadian Math. Bulleti

A family of cubic fourfolds with finite-dimensional motive

We prove that cubic fourfolds in a certain 10-dimensional family have finite-dimensional motive. The proof is based on the van Geemen-Izadi construction of an algebraic Kuga-Satake correspondence for these cubic fourfolds, combined with Voisin's method of "spread". Some consequences are given.Comment: 19 pages, to appear in Journal Math. Soc. Japan, comments vey welcom

On Voisin's conjecture for zero-cycles on hyperkaehler varieties

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi-Yau varieties. We reformulate Voisin's conjecture in the setting of hyperk\"ahler varieties, and we prove this reformulated conjecture for one family of hyperk\"ahler fourfolds.Comment: 10 pages, to appear in Journal Korean Math. Soc., comments very welcome !. arXiv admin note: text overlap with arXiv:1704.01083, arXiv:1703.0399

Some elementary examples of quartics with finite-dimensional motive

This small note contains some easy examples of quartic hypersurfaces that have finite-dimensional motive. As an illustration, we verify a conjecture of Voevodsky (concerning smash-equivalence) for some of these special quartics.Comment: 5 pages, to appear in Annali dell'Universita di Ferrara, comments welcome ! arXiv admin note: substantial text overlap with arXiv:1611.0881

A remark on Beauville's splitting property

Let $X$ be a hyperk\"ahler variety. Beauville has conjectured that a certain subring of the Chow ring of $X$ should inject into cohomology. This note proposes a similar conjecture for the ring of algebraic cycles on $X$ modulo algebraic equivalence: a certain subring (containing divisors and codimension $2$ cycles) should inject into cohomology. We present some evidence for this conjecture.Comment: 8 pages, to appear in Manuscr. Math., comments very welcom

Bloch's conjecture for Enriques varieties

Enriques varieties have been defined as higher-dimensional generalizations of Enriques surfaces. Bloch's conjecture implies that Enriques varieties should have trivial Chow group of zero-cycles. We prove this is the case for all known examples of irreducible Enriques varieties of index larger than $2$. The proof is based on results concerning the Chow motive of generalized Kummer varieties.Comment: 16 pages, to appear in Osaka J. Math., comments welcom

About Chow groups of certain hyperk\"ahler varieties with non-symplectic automorphisms

Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms. This has consequences for the Chow ring of the quotient $X/G$.Comment: 16 pages, to appear in Vietnam J. Math., comments welcom

Correspondences and singular varieties

What is generally known as the "Bloch--Srinivas method" consists of decomposing the diagonal of a smooth projective variety, and then considering the action of correspondences in cohomology. In this note, we observe that this same method can also be extended to singular and quasi--projective varieties. We give two applications of this observation: the first is a version of Mumford's theorem, the second is concerned with the Hodge conjecture for singular varieties.Comment: 11 pages. Comments welcome ! To appear in Monatsh. Math. (in slightly different version). arXiv admin note: text overlap with arXiv:1507.0448