A closed two-form ω on a complex manifold M which is also the negative imaginary part of a Hermitian metric h = g - i ω is called a Kähler form. In this case, M is called a Kähler manifold and g, the real part of the Hermitian metric, is called a Kähler metric. The Kähler form combines the metric and the complex structure, indeed g(X, Y) = ω(X, J Y),
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