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  "body": "\n## Summary\n\nThe Levi form is a [[Hermitian form|48P4A1CX85]] built from the mixed second derivatives $\\partial^2 \\rho / \\partial z_j \\partial \\bar z_k$ of a real function $\\rho$ on a complex domain.\nFor the boundary of a domain in $\\C^n$ one restricts it to the complex tangent directions of the boundary.\nIts sign decides the convexity notions of [[several complex variables|BH57D2FBNR]]: pseudoconvexity, strong pseudoconvexity and plurisubharmonicity.\n\n## Explanations\n\n### L1 human\n\nOrdinary convexity of a boundary is measured by the real second derivatives of a function that cuts out the domain.\nIn complex analysis only the part of this second derivative that is unchanged by holomorphic changes of coordinates matters, and this part is the Levi form.\nOne also looks only along the directions of the boundary that are complex lines, because these are the directions a holomorphic disc can move in.\nFor the unit ball in $\\C^n$, cut out by $\\rho(z) = \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2 - 1$, the Levi form is $\\abs{w_1}^2 + \\cdots + \\abs{w_n}^2$, which is positive.\nSo the sphere curves \"the right way\" in every complex direction.\n",
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