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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a domain and let $w \\in \\Omega$.\nLet $\\mathcal{F}_w$ be the set of [[plurisubharmonic functions|FF2P7ER3QT]] $u$ on $\\Omega$ such that:\n\n1. $u \u003c 0$ on $\\Omega$;\n2. there are a neighbourhood $U$ of $w$ and a constant $C$ with $u(\\zeta) \\le \\log\\abs{\\zeta - w} + C$ for $\\zeta \\in U$.\n\nThe *pluricomplex Green function* of $\\Omega$ with pole at $w$ is the function $g_\\Omega(\\cdot, w) \\colon \\Omega \\to [-\\infty, 0]$ given by\n$$g_\\Omega(z, w) = \\sup\\set{u(z) : u \\in \\mathcal{F}_w} .$$\n\n## Remarks\n\nExample: for the unit ball $\\Omega = \\set{\\abs{z} \u003c 1}$ and $w = 0$, $g_\\Omega(z, 0) = \\log\\abs{z}$.\nCheck: $\\log\\abs{z}$ belongs to $\\mathcal{F}_0$.\nConversely, let $u \\in \\mathcal{F}_0$ and let $\\xi$ be a unit vector.\nThe function $\\lambda \\mapsto u(\\lambda \\xi) - \\log\\abs{\\lambda}$ is subharmonic on $0 \u003c \\abs{\\lambda} \u003c 1$ and bounded above near $0$, so it extends across $0$.\nIts boundary values are at most $0$, so $u(\\lambda \\xi) \\le \\log\\abs{\\lambda}$ by the maximum principle.\nNon-example: $u(z) = \\abs{z}^2 - 1$ is negative and plurisubharmonic on the ball but is bounded near $0$, so it fails clause 2.\nFor $n = 1$ and a bounded regular domain, $g_\\Omega(z, w)$ is the classical Green function with the sign convention $g_\\Omega(z, w) \\sim \\log\\abs{z - w}$ near $w$.\nHolomorphic maps decrease it: if $f \\colon \\Omega_1 \\to \\Omega_2$ is holomorphic, then $g_{\\Omega_2}(f(z), f(w)) \\le g_{\\Omega_1}(z, w)$.\nIf $\\Omega$ is bounded, then $g_\\Omega(\\cdot, w)$ is plurisubharmonic and belongs to $\\mathcal{F}_w$.\n",
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