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  "body": "\n## Summary\n\nA pluriharmonic function is a real function on an open subset of $\\C^n$ whose restriction to every complex line is [[harmonic|0DEB0JMN0W]].\nEquivalently, all its mixed derivatives $\\partial^2 u / \\partial z_j \\partial \\bar z_k$ vanish.\nLocally, pluriharmonic functions are exactly the real parts of [[holomorphic functions|507C34W0ZT]].\nThey are the functions $u$ such that both $u$ and $-u$ are [[plurisubharmonic|FF2P7ER3QT]].\n\n## Explanations\n\n### L1 human\n\nIn one complex variable, harmonic functions are the real parts of holomorphic functions.\nIn several variables, the real parts of holomorphic functions satisfy more equations than the Laplace equation: they are harmonic on each complex line.\nThese functions are called pluriharmonic.\nFor example, $u(z_1, z_2) = \\operatorname{Re}(z_1 z_2)$ is pluriharmonic on $\\C^2$.\nThe function $\\abs{z_1}^2 - \\abs{z_2}^2$ is harmonic on $\\R^4$ but not pluriharmonic, since on the line $z_2 = 0$ it is $\\abs{z_1}^2$, which is not harmonic.\n",
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