BIRS Workshop Lecture Videos
$\partial\bar\partial$-Complex symplectic and Calabiâ Yau manifolds: Albanese map, deformations and period maps Rollenske, Soenke
Let X be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar\partial$-Lemma. If X is KÃ¤hler then, up to a finite cover, X is product of a simply connected manifold and its Albanese Torus $Alb(X)$, be the Beauville-Bogomolov decomposition theorem. We show that in the more general setting, the Albanese map is still a holomorphic submersion but will in general not split after finite pullback. We also 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. I will also mention some open question. Based on joint work with B. Anthes, A. Cattaneo, A. Tomassini.
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