Geodesic
Hodge Classes and Abelian Varieties of Quaternionic Type
Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 247-256

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We obtain coniugacy classes (with respect to a \(\mathfrak{s}l_2\) action) in the space of Hodge cycles in the middle cohomology of an Abelian variety of quaternionic type. The existence of such a class simplifies the study of the Hodge conjecture. 
Lombardo, Giuseppe. Hodge Classes and Abelian Varieties of Quaternionic Type. Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 247-256. http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_1_a12/
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     title = {Hodge {Classes} and {Abelian} {Varieties} of {Quaternionic} {Type}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {247--256},
     year = {2006},
     volume = {Ser. 8, 9B},
     number = {1},
     zbl = {1150.14001},
     mrnumber = {2204910},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_1_a12/}
}
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