A new proof shows Weil classes on abelian sixfolds of discriminant -1 are algebraic, which implies the Hodge conjecture for abelian fourfolds.
Algebra Number Theory 3 (2009), no.6, 653–695
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Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds
A new proof shows Weil classes on abelian sixfolds of discriminant -1 are algebraic, which implies the Hodge conjecture for abelian fourfolds.