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

Abstract

Let $X$ be a compact complex manifold with trivial canonical bundle and satisfying the $\partial \bar{\partial }$-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.

22 pages

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