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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