\(\partial \bar{\partial }\)-lemma and p-Kähler structures on families of solvmanifolds

We provide families of compact \((n + 1)\) $(n+1)$ -dimensional complex non Kähler manifolds satisfying the \(\partial \bar{\partial }\) $\partial \overline{\partial }$ -Lemma, with holomorphically 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.