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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be a domain with $C^2$ boundary, and let $p \\in \\partial\\Omega$ be a [[strongly pseudoconvex|Z7RVXBA2KB]] point.\nThen there is a [[biholomorphic map|FX92PP3HZR]] $\\Phi$ from an open neighbourhood $U$ of $p$ onto an open subset of $\\C^n$ with this property.\nThe set $\\Phi(U \\cap \\Omega)$ is strictly convex.\nThat is, the hypersurface $\\Phi(U \\cap \\partial\\Omega)$ has a defining function whose real Hessian is positive definite on the real tangent space at each of its points.\n\n## Remarks\n\nThe proof is a computation with the second-order Taylor expansion of a defining function $\\rho$ at $p$.\nAfter a complex-linear change of coordinates, $p = 0$ and $\\rho(z) = 2\\operatorname{Re}\\big(z_n + \\sum_{j,k} a_{jk} z_j z_k\\big) + \\sum_{j,k} \\rho_{j\\bar k}\\, z_j \\bar z_k + o(\\abs{z}^2)$, where $(\\rho_{j \\bar k})$ is the [[Levi form|80MTNXWW80]] matrix.\nThe holomorphic quadratic change of coordinates $z_n' = z_n + \\sum_{j,k} a_{jk} z_j z_k$, $z_j' = z_j$ for $j \u003c n$, removes the pure terms $z_j z_k$, so the new real Hessian at $0$ is given by the Levi form matrix.\nStrong pseudoconvexity makes it positive definite on the complex tangent space $\\set{z_n = 0}$.\nThe real tangent space at $0$ also contains the direction of $\\operatorname{Im} z_n$.\nA further change $z_n' = z_n + c z_n^2 + \\sum_{j \u003c n} b_j z_j z_n$, with suitable constants $c$ and $b_j$, makes the Hessian positive definite on the whole real tangent space.\nPositive definiteness at $p$ then persists at nearby boundary points, by continuity.\nExample of use: local holomorphic peak functions at $p$ are obtained from linear functions in the convex coordinates.\nThis is the first step towards the fact that every boundary point of a strongly pseudoconvex domain is a peak point.\nWithout strong pseudoconvexity the lemma fails.\nIf the boundary contains a complex disc, its image under a biholomorphic map is a complex disc in the new boundary, and a strictly convex hypersurface contains no such disc.\n\n## Explanations\n\n### L1 human\n\nStrong pseudoconvexity is a complex form of strict convexity, and it is preserved by holomorphic changes of coordinates, whereas convexity is not.\nThe lemma says that the two notions agree locally: near each strongly pseudoconvex boundary point, one can bend the coordinates holomorphically so that the domain becomes strictly convex there.\nSo local problems at such points can be solved with the tools of convex geometry.\n\n### L2 human\n\nThe lemma is used to build local [[peak functions|SKE8QTW3YM]] and local holomorphic support functions, and to construct integral kernels, such as the Henkin–Ramírez kernels, near strongly pseudoconvex boundaries.\nIt does not extend to weakly pseudoconvex points.\nKohn and Nirenberg gave a pseudoconvex domain in $\\C^2$ with a boundary point near which no holomorphic coordinate change makes the domain convex.\nGlobally the lemma also fails: a strongly pseudoconvex domain need not be biholomorphic to a convex domain, for instance because it need not be contractible, while convex domains are.\n",
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