Hodge structures on abelian varieties of CM-type

Abdulali, Salman

doi:10.1515/crll.2001.034

Cited by 6 publications

(14 citation statements)

References 19 publications

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“…In the introduction we listed 7 cases where the general Hodge conjecture for all A i × E j is known, the results of Corollary 3.3 being Cases (6) and (7). Cases (1), (4), and, (5) can be proved similarly. In Cases (2) and (3), the algebraicity of the Weil cycles is not known for a general fiber of the Weil family, but Schoen [19] obtains the general Hodge conjecture in these cases by directly proving the algebraicity of the Weil cycles in A × E w .…”

Section: Resultsmentioning

confidence: 73%

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Hodge structures on abelian varieties of type IV

Abdulali

2004

Mathematische Zeitschrift

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Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties. (2000): Primary 14C30, 14K20. Mathematics Subject Classification

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Section: Resultsmentioning

confidence: 73%

“…In a series of papers [1][2][3][4][5] we have shown that the general Hodge conjecture for certain abelian varieties A is implied by the usual Hodge conjecture for a class of abelian varieties, and used this to prove the general Hodge conjecture in some cases. In this paper we extend these results to certain abelian varieties of type IV.…”

Section: Introductionmentioning

confidence: 99%

Hodge structures on abelian varieties of type IV

Abdulali

2004

Mathematische Zeitschrift

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“…Let us start with the case where our abelian varieties have no factor of totally real type. The following theorem is due to Abdulali [1, Examples 2 & 3] : Theorem 3.10 (Abdulali [1], strong GHC for powers of CM abelian surfaces and certain products of CM elliptic curves). Let A be an abelian variety that is isogenous to either Proof.…”

Section: 5mentioning

confidence: 99%

“…Nonetheless, the generalized Hodge conjecture was established by Abdulali [1] for powers of a simple abelian surface of CM type (see Theorem 3.10). Abdulali's proof yields a strong form of the generalized Hodge conjecture (as in Conjecture 1.6) for powers of abelian varieties of dimension ≤ 2 (see Corollary 3.13).…”

Section: Introductionmentioning

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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“…In the second case, if M is a torus and hence D M consists of a single point, we say H is of CM type. By [1], every (effective) Hodge structure of CM type occurs in the cohomology of a CM abelian variety. In the case of an elliptic curve E = C/Λ, the property of being CM means that there is a complex number α which is not an integer such that α(Λ) ⊂ Λ, and hence multiplication by α induces a map E → E. The Mumford-Tate conjecture for abelian varieties of CM type was proven by Pohlmann [56].…”

Section: Remark 26 a Morphism Of Hodge Structuresmentioning

confidence: 99%

Book Review: Mumford–Tate groups and domains: their geometry and arithmetic

Pearlstein¹

2015

Bull. Amer. Math. Soc.

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Hodge structures on abelian varieties of CM-type

Cited by 6 publications

References 19 publications

Hodge structures on abelian varieties of type IV

Hodge structures on abelian varieties of type IV

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

Book Review: Mumford–Tate groups and domains: their geometry and arithmetic

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