Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds

We provide new families of compact complex manifolds with no Kähler structure carrying symplectic structures satisfying the Hard Lefschetz Condition. These examples are obtained as compact quotients of the solvable Lie group C2n⋉ρC2m, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the Hard Lefschetz Condition. Furthermore, we compute the Kodaira dimension of an almost-Kähler structure and generators for the de Rham and Dolbeault cohomologies.