A Kähler metric is a Riemannian metric g on a complex manifold which gives M a Kähler structure, i.e., it is a Kähler manifold with a Kähler form. However, the term "Kähler metric" can also refer to the corresponding Hermitian metric h = g - i ω, where ω is the Kähler form, defined by ω(X, Y) = g(J X, Y). Here, the operator J is the almost complex structure, a linear map on tangent vectors satisfying J^2 = - I, induced by multiplication by i. In coordinates z_k = x_k + i y_k, the operator J satisfies J(d/dx_k) = d/dy_k and J(d/dy_k) = - d/dx_k.
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