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  "body": "\n## Summary\n\nThe complex Monge–Ampère operator sends a function $u$ on an open subset of $\\C^n$ to the determinant of its complex Hessian.\nThe complex Hessian is the matrix $\\left(\\partial^2 u / \\partial z_j \\partial \\bar z_k\\right)$, and the determinant is viewed as a measure.\nIn the language of forms it is the $n$-th exterior power $(dd^c u)^n$.\nFor $n = 1$ it is the Laplacian, so it is the nonlinear analogue of the Laplacian in pluripotential theory.\nIt is the operator of the complex [[Monge–Ampère equation|14AKAC0S0T]], which appears in the construction of Kähler–Einstein metrics and in the study of [[plurisubharmonic functions|FF2P7ER3QT]].\n\n## Explanations\n\n### L1 human\n\nFor a smooth function of several complex variables, one can form the matrix of its mixed second derivatives in $z_j$ and $\\bar z_k$.\nThe complex Monge–Ampère operator takes the determinant of this matrix.\nFor a plurisubharmonic function the matrix is positive semidefinite, so the determinant is nonnegative and can be read as a density of mass.\nFor example, for $u(z) = \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2$ the matrix is the identity and the determinant is $1$ everywhere.\nFor $u(z) = \\abs{z_1}^2$ in $\\C^2$ the determinant is $0$, because $u$ does not depend on $z_2$.\n",
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