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$\partial\bar\partial$-Complex symplectic and Calabiâ Yau manifolds: Albanese map, deformations and period maps

Banff International Research Station for Mathematical Innovation and Discovery

Let X be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar\partial$-Lemma. If X is Kähler 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.

Mathematics; Differential geometry; Algebraic geometry

video/mp4

Author affiliation: University of Marburg

2020-04-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74238

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/