We provide new families of compact complex manifolds with no K\"ahler structure carrying symplectic structures satisfying the \textit{ Hard Lefschetz Condition}. These examples are obtained as compact quotients of the solvable Lie group $\mathbb{ C}^{ 2n} \ltimes_{ \rho} \mathbb{ C}^{ 2m}$, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the \textit{ Hard Lefschetz Condition}. Furthermore, we compute the Kodaira dimension of an almost-K\"ahler structure and generators for the de Rham and Dolbeault cohomologies.
Categories: math.DG