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  "body": "\n## Definition\n\nLet $E \\subseteq \\C^n$.\nThe set $E$ is *pluripolar* if each point $a \\in E$ has an open neighbourhood $U$ with the following property.\nThere is a [[plurisubharmonic function|FF2P7ER3QT]] $u$ on $U$, not identically $-\\infty$ on any component of $U$, such that\n$$E \\cap U \\subseteq \\set{z \\in U : u(z) = -\\infty}.$$\nThe set $E$ is *complete pluripolar* in an open set $\\Omega$ if $E = \\set{z \\in \\Omega : u(z) = -\\infty}$ for some function $u$ of the following kind.\nThe function $u$ is plurisubharmonic on $\\Omega$ and not identically $-\\infty$ on any component of $\\Omega$.\n\n## Remarks\n\nSome authors call the sets of the first definition locally pluripolar, and call $E$ globally pluripolar if one function $u$ plurisubharmonic on all of $\\C^n$, not identically $-\\infty$, has $E \\subseteq \\set{u = -\\infty}$.\nBy [[Josefson's theorem|V6XSDT5C21]] the two notions agree.\nExample: $E = \\set{z \\in \\C^2 : z_1 = 0}$ is pluripolar; take $U = \\C^2$ and $u(z) = \\log\\abs{z_1}$, which is plurisubharmonic and equals $-\\infty$ exactly on $E$.\nNon-example: $\\R^n \\subset \\C^n$ is not pluripolar.\nFor $n = 1$: if $u$ is subharmonic near a point of $\\R$ and $u = -\\infty$ on an interval, then $u \\equiv -\\infty$ near that point, since intervals are not [[polar|C95QJ7YH5T]].\nApplying this on complex lines parallel to the coordinate axes, one coordinate at a time, gives the general case.\nFor $n = 1$, pluripolar sets are exactly polar sets.\n",
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