Tate twists of Hodge structures arising from abelian varieties of type IV

Abdulali, Salman

doi:10.1016/j.jpaa.2011.10.005

Cited by 3 publications

(7 citation statements)

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“…where Γ ⊗ γ → Γ * (γ) and where the sum runs over all abelian varieties B. As outlined after the proof of [2,Prop. 4] in the context of Hodge sub-structures, there is a slight subtlety : one needs to use the stronger statement of Theorem 3.7 described in its proof, namely, that for H 1 ⊆ H k1 (A 1 , C) a L(A 1 ) C -sub-representation of level 7 ≤ k 1 − 2n 1 , we have that H 1 is numerically Hodge symmetric and that…”

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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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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“…This sort of problem has been looked at for other Shimura varieties. In particular, the work of [Sat65] and [Abd12] looked at the sub-Shimura varieties of the Shimura varieties associated to Symplectic groups.…”

Section: Introductionmentioning

confidence: 99%

Sub-Shimura Varieties for type O(2,n)

Fiori¹

2018

J. Théor. Nombres Bordeaux

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“…In a series of articles [2][3][4][5][6][7][8] we have shown for a large class of abelian varieties that every effective Tate twist of a Hodge structure contained in the cohomology of one of these abelian varieties is isomorphic to a Hodge structure occurring in the cohomology of some abelian variety. Our earlier results apply to abelian varieties 4210 ABDULALI of type IV in only a few cases-namely, when the Hodge group is semisimple [2], or when the abelian variety is of CM-type [7], or, when the semisimple part of the Hodge group is a product of groups of the form SU p + 1 p [8].…”

Section: Introductionmentioning

confidence: 99%

“…Our earlier results apply to abelian varieties 4210 ABDULALI of type IV in only a few cases-namely, when the Hodge group is semisimple [2], or when the abelian variety is of CM-type [7], or, when the semisimple part of the Hodge group is a product of groups of the form SU p + 1 p [8]. In this article, we extend these results to any abelian variety A such that the semisimple part of the Hodge group of A is a product of copies of SU p 1 for some p > 1; see Theorem 10 for the precise statement.…”

Section: Introductionmentioning

confidence: 99%

“…First, the concept of semidomination (Definition 4) generalizes the concept of weak self-domination introduced in [8], and allows us to work with the semisimple part of the Hodge group of an abelian variety whose Hodge group is neither semisimple nor commutative (see Theorem 7). Second, families of abelian varieties which are not of pel-type play a critical role, even though our main concern is a general member of a pel-family.…”

Section: Introductionmentioning

confidence: 99%

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