Questions about Kähler manifolds and Kähler metrics.
A hermitian metric on a complex manifold is a Kähler metric if the associated $(1, 1)$-form (sometimes called a Kähler form) is closed. A complex manifold equipped with a Kähler metric is called a Kähler manifold.
Wikipedia: Kähler manifold
nLab: Kähler manifold