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

TL;DR

This paper studies $ ext{ extonehalf} ext{ extonehalf}$-complex symplectic and Calabi-Yau manifolds, showing their deformation spaces are smooth, establishing a local Torelli theorem, and analyzing their Albanese maps and period maps.

Contribution

It proves the smoothness of the Kuranishi space for these manifolds and clarifies the structure of their Albanese maps and period maps.

Findings

Kuranishi space is a smooth universal deformation

Local Torelli theorem holds for manifolds with symplectic form

Albanese map is a surjective submersion

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.