Algebraic cycles on hyper-Kähler varieties of generalized Kummer type

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-Kähler variety of generalized Kummer type. For an arbitrary variety X of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational cohomology generated by \(H^2(X,\mathbb {Q})\) $H^2(X,\mathbb{Q})$ are algebraic.