Versal dg deformation of Calabi–Yau manifolds

For a higher dimensional Calabi–Yau manifold, we prove that deformations and taking dg categories of perfect complexes intertwine in a suitable sense. As a consequence, the two associated deformation theories are canonically equivalent. Moreover, the dg category of perfect complexes on a versal deformation of the original manifold provides a versal Morita deformation of its dg category of perfect complexes. Besides the classical uniqueness up to étale neiborhood of the base, we prove another sort of uniqueness of versal Morita deformations.