Del-delbar-complex symplectic and Calabi-Yau manifolds: Albanese map, deformations and period maps

Let X be a compact complex manifold with trivial canonical bundle and satisfying the ∂∂¯ -Lemma. We show that the Kuranishi space of X is a smooth universal deformation and that small deformations enjoy the same properties as X. If, in addition, X admits a complex symplectic form, then the local Torelli theorem holds and we obtain some information about the period map. We clarify the structure of such manifolds a little by showing that the Albanese map is a surjective submersion.