$\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma and $p$-Kähler structures on families of solvmanifolds

We provide families of compact $(n+1)$-dimensional complex non Kähler manifolds satisfying the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma, with holomoprhically trivial canonical bundle, carrying a balanced metric and with no $p$-Kähler structures. Such a construction extends to the completely solvable case in any dimension Nakamura’s construction of low-dimensional holomorphically parallelizable solvmanifolds.