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  "body": "\n## Definition\n\nLet $U \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]] and let $\\rho \\colon U \\to \\R$ be a function of class $C^2$ (see [[smooth function|0V33BNKZ9S]]).\nThe *Levi form* of $\\rho$ at a point $p \\in U$ is the [[Hermitian form|48P4A1CX85]]\n$$L_\\rho(p; w) = \\sum_{j,k=1}^{n} \\frac{\\partial^2 \\rho}{\\partial z_j \\partial \\bar z_k}(p)\\, w_j \\bar w_k, \\qquad w \\in \\C^n.$$\nIt is also called the complex Hessian of $\\rho$.\n\nNow let $\\Omega \\subseteq \\C^n$ be a domain with $C^2$ boundary.\nA *defining function* of $\\Omega$ is a $C^2$ function $\\rho$ on a neighbourhood $U$ of $\\partial\\Omega$ with $\\Omega \\cap U = \\set{z \\in U : \\rho(z) \u003c 0}$ and $d\\rho \\ne 0$ on $\\partial\\Omega$.\nFor $p \\in \\partial\\Omega$, the complex tangent space of $\\partial\\Omega$ at $p$ is\n$$H_p = \\set{w \\in \\C^n : \\sum_{j=1}^{n} \\frac{\\partial \\rho}{\\partial z_j}(p)\\, w_j = 0}.$$\nThe *Levi form of $\\partial\\Omega$ at $p$* is the restriction of $L_\\rho(p; \\cdot)$ to $H_p$.\n\n## Remarks\n\nExample: for the unit ball, $\\rho(z) = \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2 - 1$ has $\\partial^2 \\rho / \\partial z_j \\partial \\bar z_k = \\delta_{jk}$, so $L_\\rho(p; w) = \\abs{w}^2 \u003e 0$ for every $w \\ne 0$.\nNon-example of positivity: for the bidisc, near a boundary point $p = (1, b)$ with $\\abs{b} \u003c 1$, a defining function is $\\rho(z) = \\abs{z_1}^2 - 1$.\nHere $H_p = \\set{w : w_1 = 0}$, and $L_\\rho(p; (0, w_2)) = 0$, so the Levi form vanishes on $H_p$.\nIf $\\rho$ is replaced by another defining function $h \\rho$ with $h \u003e 0$, the Levi form of $\\partial\\Omega$ at $p$ is multiplied by $h(p)$.\nSo its signature does not depend on the choice.\nThe signature is also unchanged under holomorphic changes of coordinates.\n",
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