361 research outputs found
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 be a hyperk\"ahler variety. Beauville has conjectured that a certain
subring of the Chow ring of should inject into cohomology. This note
proposes a similar conjecture for the ring of algebraic cycles on modulo
algebraic equivalence: a certain subring (containing divisors and codimension
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 . 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 be a hyperk\"ahler variety, and let be a group of finite order
non-symplectic automorphisms of . Beauville's conjectural splitting property
predicts that each Chow group of should split in a finite number of pieces.
The Bloch-Beilinson conjectures predict how should act on these pieces of
the Chow groups: certain pieces should be invariant under , while certain
other pieces should not contain any non-trivial -invariant cycle. We can
prove this for two pieces of the Chow groups when is the Hilbert scheme of
a surface and consists of natural automorphisms. This has consequences
for the Chow ring of the quotient .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
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