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Let $X$ be a compact complex manifold with trivial canonical bundle andsatisfying the $\partial\bar{\partial}$-Lemma. We show that the Kuranishi spaceof $X$ is a smooth universal deformation and that small deformations enjoy thesame properties as $X$. If, in addition, $X$ admits a complex symplectic form,then the local Torelli theorem holds and we obtain some information about theperiod map.  We clarify the structure of such manifolds a little by showing that theAlbanese map is a surjective submersion.

$\partial\bar{\partial}$-complex symplectic and Calabi-Yau manifolds: Albanese map, deformations and period maps