Algebraic Cycles on Fano Varieties of Some Cubics

Acta. Math. Sin.-English Ser.

2017

Self Cite

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.Because the cubics X as in theorem 3.1 form a large family, and the correspondence Γ exists for the whole family, one can apply Voisin's method of "spread" [33], [34], [35], [36] to this isomorphism, and obtain a statement on the level of rational equivalence which proves theorem 3.1.

Section: Preliminariesmentioning

confidence: 99%

On the Chow groups of certain cubic fourfolds

Acta. Math. Sin.-English Ser.

2017

Self Cite

“…Remark 4.3. In the set-up of Corollary 4.2, the varieties X and Y are actually birational [47], and the isomorphism of motives can be readily obtained by exploiting the specific form of the birationality [34]. However, the proof given here does not rely on the birationality.…”

Section: Cubic Threefolds and Fano Threefolds Of Genusmentioning

confidence: 99%

Motives and the Pfaffian–Grassmannian equivalence

2021

J. London Math. Soc.

Self Cite

We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds, and verifications of Kimura's finite-dimensionality conjecture, Voevodsky's smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.

On complete intersections in varieties with finite-dimensional motive

The Quarterly Journal of Mathematics

Nagel

Peters

2018

Self Cite

Let X be a complete intersection inside a variety M with finite dimensional motive and for which the Lefschetz-type conjecture B(M ) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps A k (X) → H 2k (X) are injective for k < c. We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.