Some remarks concerning Voevodsky’s nilpotence conjecture

Bernardara, Marcello; Marcolli, Matilde; Tabuada, Gonçalo

doi:10.1515/crelle-2015-0068

Cited by 6 publications

(11 citation statements)

References 33 publications

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“…Therefore, we can consider the direct summand e E.Y / of E.Y / associated to the idempotent e 2 R. / 1=n . In the case where the category D is l -linear for a field l which contains the n th roots of unity and 1=n 2 l , the _ -action on E.Y / may be translated back 6 into a . __ D /-grading on E.Y /.…”

Section: Remark 120 (Mckay Correspondence)mentioning

confidence: 99%

Additive invariants of orbifolds

Tabuada

Bergh

2018

Geom. Topol.

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Using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely "fixedpoint data". As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic K-theory, of Baranovsky concerning cyclic homology, of the second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned computation is new in the literature. As an application, we verify Grothendieck's standard conjectures of type C C and D, as well as Voevodsky's smash-nilpotence conjecture, in the case of "low-dimensional" orbifolds. Finally, we establish a result of independent interest concerning nilpotency in the Grothendieck ring of an orbifold. 14A15, 14A20, 14A22, 19D55 1. Introduction 3004 2. Preliminaries 3014 3. Action of the representation ring 3021 4. Decomposition of the representation ring 3022 5. G 0-motives over an orbifold 3025 6. Proofs: decomposition of orbifolds 3034 7. Proofs: smooth quotients 3036 8. Proofs: equivariant Azumaya algebras 3038 9. Grothendieck and Voevodsky's conjectures for orbifolds 3041 Appendix: Nilpotency in the Grothendieck ring of an orbifold 3043 References 3045

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Section: Remark 120 (Mckay Correspondence)mentioning

confidence: 99%

Additive invariants of orbifolds

Tabuada

Bergh

2018

Geom. Topol.

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“…The left and right kernels of χ(−, −) are the same. In [6] Bernardara, Marcolli and Tabuada conjectured the following statement:…”

Section: 4mentioning

confidence: 98%

“…State of art. Conjecture V was proven for curves, surfaces, abelian threefolds, uniruled threefolds, quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals (see [2], [13], [22], [28], [29], [6]).…”

Section: 3mentioning

confidence: 99%

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Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

Ornaghi

Pertusi

2019

J. Noncommut. Geom.

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In the first part of this paper we will prove the Voevodsky's nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing a τ -plane Gr(2, 3) and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.Theorem (BMT). Let X be a smooth projective k-scheme. The conjecture V(X) is equivalent to the conjecture V nc (perf dg (X)).Here, perf dg (X) denotes the unique enhancement of the derived category of perfect complexes on X. Making use of noncommutative motives, Voevodsky's conjecture was proven for example for quadric fibrations, intersection of quadrics, linear sections of Grassmannians, etc. (see [6] and [7] for details).The first result of this paper is the proof of Voevodsky's conjecture for cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. We recall that a cubic fourfold is a smooth complex hypersurface of degree 3 in P 5 , while a Gushel-Mukai fourfold is a smooth and transverse intersection of the form Cone(Gr(2, V 5 )) ∩ Q, where Q is a quadric hypersurface in P 8 .Theorem (A). Let X be a cubic fourfold or an ordinary generic Gushel-Mukai fourfold; then the conjecture V (X) holds. 3

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“…Remark 4.14. In [9, Theorem 1.7], the special case of Corollary 4.13 where n = 6 or n = 7 was proven (in [9], the restriction to n = 6, 7 is necessary because the argument relies on HPD for Grassmannians via non-commutative motives). The argument proving Corollary 4.13 is more elementary, in that we do not rely on derived category arguments at all.…”

Section: Fano Varieties With Finite-dimensional Motivementioning

confidence: 99%

Motives and the Pfaffian–Grassmannian equivalence

Laterveer

2021

J. London Math. Soc.

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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.

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Some remarks concerning Voevodsky’s nilpotence conjecture

Cited by 6 publications

References 33 publications

Additive invariants of orbifolds

Additive invariants of orbifolds

Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

Motives and the Pfaffian–Grassmannian equivalence

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