The Rational Hodge Conjecture for All Powers of Varieties of Generalized Kummer Type

Let Y be a smooth projective complex variety of generalized Kummer deformation type and dimension 2N−2, with N at least 3. We prove the rational Hodge conjecture for every finite self-power of Y. Torsion line bundles on the sum-zero abelian cover give explicit vector bundles through the reverse McKay equivalence. Their full degree-two Hochschild obstruction maps have rank one, and their projective bundles admit smooth hyperholomorphic transport. A punctual intersection calculation identifies normalized Chern-character combinations with every exact-order bottom cohomology packet. Combining these cycles with the invariant-motive and cup-generation theorems of Floccari and Fu and the known all-powers theorem for discriminant-one Weil fourfolds proves the result. The argument also treats mixed products Y^a × J_Y^b, where J_Y is the associated fourfold, and gives selected Tate summands in the rational Chow motive. The additional deformation-type range beyond the prior results discussed in the paper consists of dimensions 10, 14, 18, and higher dimensions congruent to 2 modulo 4.