Distinguished cycles on varieties with motive of abelian type and the Section Property

Fu, Lie; Vial, Charles

doi:10.1090/jag/729

Cited by 29 publications

(84 citation statements)

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“…In other words, φ provides a section (as graded vector spaces) of the natural projection CH * (X) ։ CH * (X). In [14], we found sufficient conditions on the marking φ for DCH * φ (X) to define a Q-subalgebra of CH * (X) :…”

Section: 2mentioning

confidence: 99%

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On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

Laterveer¹,

Vial²

2019

Can. J. Math.-J. Can. Math.

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This note is about certain locally complete families of Calabi-Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow-Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology, via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk-Hulek Calabi-Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk-Hulek Calabi-Yau varieties provide examples of Calabi-Yau varieties with "degenerate" motive.2 ), the Hodge diamond

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Section: 2mentioning

confidence: 99%

“…Definition 2.6 (Definition 3.7 in [14]). We say that the marking φ : h(X) ≃ −→ M satisfies the condition (⋆) if the following two conditions are satisfied :…”

Section: 2mentioning

confidence: 99%

On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

Laterveer¹,

Vial²

2019

Can. J. Math.-J. Can. Math.

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“…A symplectomorphism of (X, ω) is an automorphism f : X → X such that f * ω = ω. If X is irreducible symplectic, it is expected as part of the Bloch conjectures that symplectomorphisms act unipotently on the Chow group of 0-cycles, and, due to the probable distinguishedness of symplectomorphisms in the sense of [23], it is in fact expected that symplectomorphisms act as the identity on the Chow group of 0-cycles. Most notably, this was established for symplectic involutions on K3 surfaces by Voisin [54] and extended to finite-order symplectomorphisms on K3 surfaces by Huybrechts [28].…”

Section: 5mentioning

confidence: 99%

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

Vial¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let γ be a correspondence between two abelian varieties A and B that can be written as a linear combination of products of symmetric divisors. Assume that A is isogenous to the product of an abelian variety of totally real type with the power of an abelian surface. We show that γ satisfies the conclusion of Bloch's conjecture. A key ingredient consists in establishing a strong form of the generalized Hodge conjecture for Hodge sub-structures of the cohomology of A that arise as sub-representations of the Lefschetz group of A. As a by-product of our method, we use a strong form of the generalized Hodge conjecture established for powers of abelian surfaces to show that every finite-order symplectic automorphism of a generalized Kummer variety acts as the identity on the zero-cycles. γ * CH 0 (X) = 0. The Bloch-Srinivas argument [14] implies that γ * H * (X, Q) is supported on a divisor, which in turn implies that γ * H i,0 (X) = 0 for all integers i. The Bloch conjecture stipulates that, conversely, should γ * H i,0 (X) vanish for all integers i, then γ acts nilpotently on CH 0 (X). (In fact, the conjecture predicts that γ should act as zero on the graded pieces of the conjectural Bloch-Beilinson filtration on CH 0 (X).)More generally, if γ * CH r (X) = 0 for all r < n, then the Bloch-Srinivas argument implies that γ * H * (X, Q) is supported on a subscheme of codimension n, which in turn implies that γ * H i,j (X) = 0 for all integers i and j < n. The generalized Bloch conjecture is the following converse assertion :Conjecture 1 (Generalized Bloch conjecture). Let X be a smooth projective complex variety of dimension d, and let γ ∈ CH d (X × X) be a correspondence. Suppose that γ * H i,j (X) = 0 for all j < n, or, equivalently in terms of the Hodge coniveau filtration, γ * H * (X, Q) ⊆ N n H H * (X, Q). Then γ * acts nilpotently on CH r (X) for all r < n.The conjecture is wide open, but has notably been established for surfaces with H 2,0 = 0 not of general type [13], for certain surfaces with H 2,0 = 0 of general type [52,55], and for finite-order symplectic automorphisms of K3 surfaces [54,28].Conjecture 1 follows from the combination of (a) the generalized Hodge conjecture (for smooth projective varieties, and not just for X) and (b) the existence of the conjectural Bloch-Beilinson filtration. Indeed, if γ * H i,j (X) = 0 for all j < n, then the generalized Hodge conjecture for X implies that γ * H * (X, Q) is supported on a closed subscheme X of codimension n. By [7], the

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“…Remark 5.8. My initial hope was to establish that Cancian-Frapporti surfaces have a multiplicative Chow-Künneth decomposition (in the sense of [30]), and satisfy the condition ( * ) of [10]. This proved to be unfeasibly difficult, however.…”

Section: It Follows That the Summands Of Type Hmentioning

confidence: 99%