1999
DOI: 10.1142/s0129167x99000264
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Abelian Varieties of Type Iii and the Hodge Conjecture

Abstract: We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

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Cited by 14 publications
(18 citation statements)
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“…Corollary 3.3. Let A be a 5-dimensional abelian variety whose polarization is given by a hermitian form of signature (3,2), and such that …”
Section: Resultsmentioning
confidence: 99%
See 2 more Smart Citations
“…Corollary 3.3. Let A be a 5-dimensional abelian variety whose polarization is given by a hermitian form of signature (3,2), and such that …”
Section: Resultsmentioning
confidence: 99%
“…Let β be a Riemann form for A. Then there exists a unique k-hermitian form H on [2,6,7,11,14,22] Let A be an abelian variety over C, and k a subfield of End Q (A). Let m be the dimension of V := H 1 (A, Q) considered as a vector space over k. Then m k V is a 1-dimensional vector space over k on which GL(V /k) acts as the determinant.…”
Section: Algebraic Groups and Abelian Varietiesmentioning
confidence: 99%
See 1 more Smart Citation
“…Example 9 (Oguiso-Sakurai [28]) The varieties X 3,1 and X 3,2 constructed in [28, Theorem 3.4] are Calabi-Yau threefolds, obtained as crepant resolutions of quotients E 3 /G, where E is an elliptic curve and G ⊂ Aut(E 3 ) a certain group. 1 Example 10 (Borcea-Voisin) Let S be a K3 surface admitting a non-symplectic involution α which fixes k = 10 rational curves. Let E be an elliptic curve, and let ι : E → E be the involution z → −z.…”
Section: More (Not Necessarily Rigid) Examplesmentioning
confidence: 99%
“…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%