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We determine the 6D solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion, generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application, we construct a holomorphic family   $(M,J_a)$   of compact complex manifolds such that   $(M,J_a)$   satisfies the   $\partial {\bar {\partial }}$  -lemma and admits a balanced metric for any   $a\not =0$  , but the central limit neither satisfies the   $\partial {\bar {\partial }}$  -lemma nor admits balanced metrics.

Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle