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), where J is the almost complex structure induced by multiplication by i. Since the Kähler form comes from a Hermitian metric, it is preserved by J, i.e., since h(X, Y) = h(J X, J Y). The equation d ω = 0 implies that the metric and the complex structure are related. It gives M a Kähler structure, and has many implications.
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