A complex manifold for which the exterior derivative of the fundamental form associated with the given Hermitian metric vanishes, so . In other words, it is a complex manifold with a Kähler structure. It has a Kähler form, so it is also a symplectic manifold. It has a Kähler metric, so it is also a Riemannian manifold.
The simplest example of a Kähler manifold is a Riemann surface, which is a complex manifold of dimension 1. In this case, the imaginary part of any Hermitian metric must be a closed form since all 2-forms are closed on a two real dimensional manifold.