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We provide families of compact (n + 1) -dimensional complex non K\"ahler manifolds satisfying the -Lemma, with holomoprhically trivial canonical bundle, carrying a balanced metric and with no p-K\"ahler structures. Such a construction extends to the completely solvable case in any dimension Nakamura's construction of low-dimensional holomorphically parallelizable solvmanifolds.
Cattaneo et al. (Fri,) studied this question.
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