Décomposition de motifs abéliens

Ancona, Giuseppe

doi:10.1007/s00229-014-0708-4

Cited by 31 publications

(113 citation statements)

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“…The symmetrically distinguished cycles form a Q-subalgebra Remark 2.25. For discussion and applications of the notion of symmetrically distinguished cycles, in addition to [37] we refer to [43,Section 7], [50], [3], [20].…”

Section: These Two Filtrations Are Expected To Coincide; Indeed Vialmentioning

confidence: 99%

Algebraic cycles on a very special EPW sextic

Laterveer

2018

Rend. Sem. Mat. Univ. Padova

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Section: These Two Filtrations Are Expected To Coincide; Indeed Vialmentioning

confidence: 99%

Algebraic cycles on a very special EPW sextic

Laterveer

2018

Rend. Sem. Mat. Univ. Padova

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“…(b) If G = O 2n , then S λ V = 0 if and only if the sum of the lengths of the first two columns of the Young tableau λ is at most 2n. 6 • If λ = (λ 1 ≥ · · · ≥ λ n = 0), then S λ V is the irreducible representation of so 2n with highest weight λ 1 ε 1 + · · · + λ n ε n . • If λ = (λ 1 ≥ · · · ≥ λ n > 0), then S λ V is the direct sum of the two irreducible representations of so 2n with highest weight λ 1 ε 1 + · · · + λ n−1 ε n−1 + λ n ε n , and λ 1 ε 1 + · · · + λ n−1 ε n−1 − λ n ε n .…”

Section: 2mentioning

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 2.9. For discussion and applications of the theory of symmetrically distinguished cycles, in addition to [20] we refer to [24,Section 7], [26], [1], [10], [11]. Proposition 2.10.…”

Section: Symmetrically Distinguished Cyclesmentioning

confidence: 99%