Smash-nilpotent cycles on abelian $3$-folds

Kahn, Bruno; Sebastian, Ronnie

doi:10.4310/mrl.2009.v16.n6.a8

Cited by 22 publications

(20 citation statements)

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“…In [She74], Shermenev proves that, for an abelian variety A, the motive h 1 (A) is oddly finite dimensional. In [KS09], the authors combine this with results in [Kim05], to show that skew cycles (β is called skew if [−1] * β = −β) are smash nilpotent. From this, using results in [Bea86], they prove that on an abelian variety of dimension ≤ 3, any homologically trivial cycle is smash nilpotent.…”

Section: Introductionmentioning

confidence: 99%

Smash nilpotent cycles on varieties dominated by products of curves

Sebastian

2013

Compositio Math.

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Section: Introductionmentioning

confidence: 99%

Smash nilpotent cycles on varieties dominated by products of curves

Sebastian

2013

Compositio Math.

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“…Thanks to the work of Kahn-Sebastian, Matsusaka, Voevodsky, and Voisin (see [15,30,36,37] and also [1, §11.5.2.3]), the above conjecture holds in the case of curves, surfaces, and abelian 3-folds (when k is of characteristic zero).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 96%

Some remarks concerning Voevodsky’s nilpotence conjecture

Bernardara

Marcolli

Tabuada³

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this article we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds. , homological projective duality, Moishezon manifolds, noncommutative algebraic geometry.Sections 3-5 are a revised version of half of an old preprint [29]. The remaining work is a recent collaboration between the first and the last author.

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“…When X is an abelian variety over a finite field, it is proven by [3]. Because smash-nilpotence on algebraic cycles with Q-coefficients implies homological triviality, the results [6] and [21] on equivalence of smash-nilpotence and numerical equivalence for certain cycles with Q-coefficients on certain varieties also imply special cases of the standard conjecture (D).…”

Section: 3mentioning

confidence: 96%

On numerical equivalence for algebraic cobordism

Banerjee

Park

2016

Journal of Pure and Applied Algebra

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We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel-Moore homology theory on schemes and oriented cohomology theory on smooth varieties.We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equivalence, answering negatively the integral cobordism analogue of the standard conjecture (D). For the latter, using Kimura finiteness on cobordism motives, we partially resolve the cobordism analogue of a conjecture by Voevodsky on rational smash-equivalence and numerical equivalence.

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Smash-nilpotent cycles on abelian $3$-folds

Abstract: We show that homologically trivial algebraic cycles on a 3-dimensional abelian variety are smash-nilpotent.

Cited by 22 publications

References 9 publications

Smash nilpotent cycles on varieties dominated by products of curves

Smash nilpotent cycles on varieties dominated by products of curves

Some remarks concerning Voevodsky’s nilpotence conjecture

On numerical equivalence for algebraic cobordism

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