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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be open and let $u \\colon \\Omega \\to \\R$ be of class $C^2$.\nLet $d^c = i(\\bar\\partial - \\partial)$, so that $dd^c = 2i \\partial\\bar\\partial$.\nThe *complex Monge–Ampère operator* applied to $u$ is the $(n, n)$-form\n$$(dd^c u)^n = dd^c u \\wedge \\cdots \\wedge dd^c u \\quad (n \\text{ factors}).$$\nIn coordinates,\n$$(dd^c u)^n = 4^n \\, n! \\, \\det\\left( \\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k} \\right)_{j,k=1}^n dV,$$\nwhere $dV$ is Lebesgue measure on $\\C^n = \\R^{2n}$.\n\n## Remarks\n\nThe constant $4^n n!$ depends on the normalisation of $d^c$; many authors use $d^c = \\frac{i}{4\\pi}(\\bar\\partial - \\partial)$ or write simply $\\det\\left(\\partial^2 u / \\partial z_j \\partial \\bar z_k\\right)$.\nIf $u$ is [[plurisubharmonic|FF2P7ER3QT]], the complex Hessian is positive semidefinite, so $(dd^c u)^n$ is a positive measure.\nExample: for $u(z) = \\abs{z}^2 = \\sum_j \\abs{z_j}^2$ the complex Hessian is the identity, so $(dd^c u)^n = 4^n n! \\, dV$.\nFor $n = 1$, $dd^c u = 4 \\, \\frac{\\partial^2 u}{\\partial z \\partial \\bar z} \\, dV = \\Delta u \\, dV$, the Laplacian.\nNon-example of positivity: for $u = \\abs{z_1}^2 - \\abs{z_2}^2$ on $\\C^2$ the determinant is $-1$, so $(dd^c u)^2$ is a negative measure; this $u$ is not plurisubharmonic.\nThe operator is nonlinear for $n \\ge 2$.\nThe extension of $(dd^c u)^n$ to locally bounded plurisubharmonic functions, where $u$ need not be differentiable, is the [[Bedford–Taylor theory|EEHQSWFHSB]].\n",
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