Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove this conjecture holds for uniruled 3-folds, 4-folds whose MRCC quotient has dimension ...≤2 and for smooth complete intersections with very small degree.
A smooth projective scheme X over a field k is said to satisfy the Rost nilpotence principle if any endomorphism of X in the category of Chow motives that vanishes on an extension of the base field k ...is nilpotent. We show that an étale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to obtain an elegant proof of Rost nilpotence for surfaces, as well as for birationally ruled threefolds over a field of characteristic 0.
Given a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial ...conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type
c
7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.
This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra ...fourfolds admit a multiplicative Chow-K\"unneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.
Motivated by the generalized Bloch conjecture, we formulate a conjecture about the Chow groups of Plücker hypersurfaces in Grassmannians. We prove weak versions of this conjecture.
In this article, we investigate, for varieties over
with trivial group of 0-cycles, the gap between essential
-dimension 2 and essential
-dimension 0. In particular, we present sufficient (and ...necessary) conditions for a variety with trivial group of 0-cycles and essential
-dimension
to have, in fact, essential
-dimension 0.
Consider three normalized cuspidal eigenforms of weight
$2$
and prime level p. Under the assumption that the global root number of the associated triple product L-function is
$+1$
, we prove that the ...complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve
$X_0(p)$
is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.
Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed ...by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.
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