On product identities and the Chow rings of holomorphic symplectic varieties

Barros, Ignacio; Flapan, Laure; Marian, Alina; Silversmith, Rob

doi:10.48550/arxiv.1912.13419

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…)}, (15) where p 0 : M σ (v) × M σ (v) i+k+1 → M σ (v) is the projection on the first factor. (Note that this filtration is denoted S in [BFMS20] and that δj = ∆ 0,1 • • • ∆ 0,j+1 by ( 14)). We note that the filtration (15) can in fact be defined for any smooth projective variety X equipped with a unit o ∈ CH 0 (X), and that we then have the obvious inclusion S BFMS k CH i (X) ⊆ R i+k CH i (X) for all i ≥ 0 and all k ≥ −i.…”

Section: Appendix a The Co-radical Filtration On Positive-dimensional...mentioning

confidence: 99%

On the birational motive of hyper-Kähler varieties

Vial¹

2020

Preprint

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We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for certain moduli spaces of stable sheaves on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with respect to the diagonal embedding and to show in the aforementioned cases the existence of a so-called strict grading. Furthermore, we upgrade to rational equivalence Voisin's notion of "surface decomposition" and use this to show that the birational motive of some projective hyper-Kähler manifolds is determined, as a co-algebra object, by the birational motive of a surface. In an appendix, we relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

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Section: Appendix a The Co-radical Filtration On Positive-dimensional...mentioning

confidence: 99%

On the birational motive of hyper-Kähler varieties

Vial¹

2020

Preprint

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“…The existence of c M is ensured by the surjectivity of the projection map from the incident variety R to X [d] . Let us recall the filtrations defined in [BFMS19].…”

Section: Applications: Comparison With Other Filtrationsmentioning

confidence: 99%

“…Finally, let us compare Voisin's filtration with other filtrations. In [BFMS19], Flapan, Marian and Silversmith have constructed two filtrations S small • CH 0 (M ) and S BFMS • CH 0 (M ) from the product point of view, while Vial has found a so called co-radical filtration R • CH 0 (M ) from the co-algebra structure on birational motives. Combined with the results in [BFMS19,Via20], our main theorem indicates that Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

Beauville-Voisin filtrations on zero cycles of moduli space of stable sheaves on K3 surfaces

Li¹,

Zhang²

2022

Preprint

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The Beauville-Voisin conjecture predicts the existence of a filtration on projective hyper-Kähler manifolds opposite to the conjecture Bloch-Beilinson filtration, called the Beauivlle-Voisin filtration. In [Voi16], Voisin has introduced a filtration on zero cycles of an arbitrary projective hyper-Kähler manifold. On moduli space of stable objects of a projective K3 surface, there are other candidates constructed by Shen-Yin-Zhao, Barros-Flapan-Marian-Silversmith in [SYZ20, BFMS19] and more recently by Vial in [Via20] from different point of views. According to the work in [Via20], all of them are proved to be equivalent except Voisin's filtration. In this paper, we show that Voisin's filtration is the same as the other filtrations. As an application, we prove a conjecture in [BFMS19].

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On product identities and the Chow rings of holomorphic symplectic varieties

Cited by 2 publications

References 5 publications

On the birational motive of hyper-Kähler varieties

On the birational motive of hyper-Kähler varieties

Beauville-Voisin filtrations on zero cycles of moduli space of stable sheaves on K3 surfaces

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