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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be an open set.\nA function $u \\colon \\Omega \\to [-\\infty, \\infty)$ is *plurisubharmonic* if:\n\n1. $u$ is [[upper semicontinuous|RN4G8SGP6P]];\n2. for every $a \\in \\Omega$ and $b \\in \\C^n$ such that the set $\\set{a + \\lambda b : \\lambda \\in \\C, \\abs{\\lambda} \\le 1}$ lies in $\\Omega$,\n$$u(a) \\le \\frac{1}{2\\pi} \\int_0^{2\\pi} u(a + e^{i\\theta} b) \\, d\\theta .$$\n\n## Remarks\n\nClause 2 says that for all $a, b$ the function $\\lambda \\mapsto u(a + \\lambda b)$ is [[subharmonic|Z1MHMGFNFW]] or identically $-\\infty$ on each component of its domain.\nMany authors also require that $u$ is not identically $-\\infty$ on any connected component of $\\Omega$.\nA function $u$ of class $C^2$ is plurisubharmonic if and only if\n$$\\sum_{j,k=1}^n \\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k}(z) \\, w_j \\bar w_k \\ge 0 \\quad \\text{for all } z \\in \\Omega,\\ w \\in \\C^n .$$\nExample: $u(z) = \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2$ has complex Hessian equal to the identity matrix, which is positive definite, so $u$ is plurisubharmonic.\nNon-example: $u(z_1, z_2) = \\abs{z_1}^2 - \\abs{z_2}^2$ is smooth, but on the line $\\lambda \\mapsto (0, \\lambda)$ it is $-\\abs{\\lambda}^2$, which is not subharmonic, so clause 2 fails at $a = 0$, $b = (0, 1)$.\nPlurisubharmonic functions are subharmonic as functions on $\\R^{2n}$, but the converse fails, as the non-example shows.\nThe abbreviation \"psh\" is common.\n",
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