Examples of smash nilpotent cycles on rationally connected varieties

Abstract

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.

Hard Lefschetz for Chow groups of generalized Kummer varieties

“…For abelian varieties, this is proven by Sebastian [30]. For Hilbert schemes of abelian surfaces and for generalized Kummer varieties, one can apply the argument of [34, Theorem 3.17] to prove equality (3).…”

Algebraic Cycles on Fano Varieties of Some Cubics

“…Proposition 20 (Sebastian [34]) Let Y be a smooth projective variety of dimension d, dominated by a product of curves.…”

Some New Examples of Smash-Nilpotent Algebraic Cycles

“…Not a great deal is known about this conjecture of Voevodsky's; most results focus on 1cycles. For instance, Voevodsky's conjecture has been proven for 1-cycles on varieties rationally dominated by products of curves [38], [39,Proposition 2] (this is further generalized by [44,Theorem 3.17]).…”

“…In this note (which is inspired by [38], [39] and particularly [44]), we aim for results for cycles in other dimensions by restricting attention to very special varieties. The main result is as follows:…”