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 $\mathit{\text{Hard}}\mathit{\text{Lefschetz}}\mathit{\text{Condition}}$. These examples are obtained as compact quotients of the solvable Lie group ${\mathbb{C}}^{2n}{⋉}_{\rho }{\mathbb{C}}^{2m}$, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the $\mathit{\text{Hard}}\mathit{\text{Lefschetz}}\mathit{\text{Condition}}$. Furthermore, we compute the Kodaira dimension of an almost-Kähler structure and generators for the de Rham and Dolbeault cohomologies.