Smash nilpotent cycles on varieties dominated by products of curves

Abstract

Voevodsky conjectured that numerical equivalence and smash equivalence coincide on a smooth projective variety. We prove the conjecture for 1-cycles on varieties dominated by products of curves.

Algebraic Cycles on a Generalized Kummer Variety

“…The other one is to show that all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nilpotent. This is achieved by combining Theorem 1.1 with recent results on 1-cycles on abelian varieties in [Seb13].…”

Examples of smash nilpotent cycles on rationally connected varieties

“…As in the preceding proof, we can find Y , obtained by blowing up Y along smooth subvarieties such that there is a dominant morphism Y → X. As a consequence of [18,Theorem 9], Lemma 5, numerical and smash equivalence coincide for 1-dimensional cycles on Y . Using Lemma 6 we get that numerical and smash equivalence coincide for 1-dimensional cycles on X.…”

“…The main theorem in [18] now follows since the motive of a product of curves (× n i=1 C i ) is a summand of the motive of the product of their Jacobians (× n i=1 J(C i )). Let R ⊂ CH * (A) denote the smallest subring of the group of cycles modulo algebraic equivalence on an abelian variety A, which is generated by the cycles in the preceding paragraph and is closed under the Pontryagin product and the Fourier transform.…”

“…In [18] it is proved that for one dimensional cycles on a variety dominated by a product of curves, smash equivalence and numerical equivalence coincide. Let C be a smooth and projective curve.…”

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]).…”