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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$.\nLet $h$ be a Hermitian metric on $X$: a smooth family of positive definite [[Hermitian forms|48P4A1CX85]] $h_p$ on the holomorphic tangent spaces $T^{1,0}_p X$.\nIn local holomorphic coordinates write $h = \\sum_{j,k} h_{j\\bar k}\\, dz_j \\otimes d\\bar z_k$, where $(h_{j\\bar k})$ is a positive definite Hermitian matrix of smooth functions.\nThe *fundamental form* of $h$ is the real $(1,1)$-[[form|0E3G0GRPZ6]]\n$$\\omega = \\frac{i}{2} \\sum_{j,k} h_{j\\bar k}\\, dz_j \\wedge d\\bar z_k.$$\nThe metric $h$ is *Kähler* if $\\omega$ is [[closed|0ZV14XRNF0]], that is, $d\\omega = 0$.\nA *Kähler manifold* is a complex manifold together with a Kähler metric; $\\omega$ is then called its Kähler form.\n\n## Remarks\n\nIn coordinates, $d\\omega = 0$ holds if and only if $\\dfrac{\\partial h_{j\\bar k}}{\\partial z_l} = \\dfrac{\\partial h_{l\\bar k}}{\\partial z_j}$ for all $j, k, l$.\nEquivalent forms: locally $\\omega = i\\partial\\bar\\partial\\varphi$ for a real smooth strictly plurisubharmonic function $\\varphi$; or the complex structure is parallel for the Levi-Civita connection of the underlying Riemannian metric $\\operatorname{Re} h$.\nA compact complex manifold is called Kähler if it admits some Kähler metric.\n\nExample: on $\\C^n$, $h_{j\\bar k} = \\delta_{jk}$ gives $\\omega = \\frac{i}{2}\\sum_j dz_j \\wedge d\\bar z_j$, which has constant coefficients and so is closed.\nOn a Riemann surface every Hermitian metric is Kähler, because every $3$-form on a manifold of real dimension $2$ is zero.\nThe Fubini–Study form on $\\mathbb{CP}^n$, given on $\\set{z_0 \\ne 0}$ by $\\omega = \\frac{i}{2}\\partial\\bar\\partial \\log(1 + \\abs{w}^2)$ in the coordinates $w_j = z_j/z_0$, is closed because $d = \\partial + \\bar\\partial$ and $\\partial^2 = \\bar\\partial^2 = 0$, $\\partial\\bar\\partial = -\\bar\\partial\\partial$.\n\nNon-example: on $W = \\C^2 \\setminus \\set{0}$, the form $\\omega = \\frac{i}{2}\\abs{z}^{-2}\\sum_j dz_j \\wedge d\\bar z_j$ is the fundamental form of a Hermitian metric invariant under $z \\mapsto \\lambda z$, so it descends to a [[Hopf surface|PNFNGBXRM3]].\nIt is not closed: $d\\omega = d(\\abs{z}^{-2}) \\wedge \\omega_0$ with $\\omega_0 = \\frac{i}{2}\\sum_j dz_j \\wedge d\\bar z_j$, and the wedge product of a nonzero $1$-form with $\\omega_0$ is nonzero in complex dimension $2$.\nSo this metric is a [[Hermitian metric|CK1DCPC3D3]] that is not Kähler; in fact the Hopf surface has no Kähler metric at all.\n",
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