Vaisman manifolds with vanishing first Chern class

Compact Vaisman manifolds with vanishing first Chern class split into three categories, depending on the sign of the Bott–Chern class. We show that Vaisman manifolds with non-positive Bott–Chern class admit canonical metrics, are quasi-regular and are stable under deformations. We also show that Calabi–Yau Vaisman manifolds satisfy a version of the Beauville–Bogomolov decomposition and have torsion canonical bundle. Finally, we prove a general result concerning the behaviour of the automorphism group of a complex manifold under deformations.