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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Omega \\subset \\C^n$ be a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] with $C^\\infty$ boundary.\nLet $\\rho$ be a $C^\\infty$ defining function of $\\Omega$: a real [[smooth function|0V33BNKZ9S]] on a neighbourhood of $\\overline{\\Omega}$ with $\\Omega = \\set{\\rho \u003c 0}$ and $d\\rho \\ne 0$ on $\\partial\\Omega$.\nLet $K(z) = K(z, z)$ be the [[Bergman kernel|8RWDHW0CXC]] of $\\Omega$ on the diagonal.\nThen there are functions $\\varphi, \\psi \\in C^\\infty(\\overline{\\Omega})$, with $\\varphi \u003e 0$ on $\\partial\\Omega$, such that\n$$K(z) = \\frac{\\varphi(z)}{(-\\rho(z))^{n+1}} + \\psi(z) \\log(-\\rho(z)) \\quad \\text{for all } z \\in \\Omega.$$\n\n## Remarks\n\nThe value of $\\varphi$ on $\\partial\\Omega$ is an explicit expression in the first and second derivatives of $\\rho$, built from the Levi form.\nHörmander had found this leading behaviour before Fefferman.\nFefferman's proof compares $\\Omega$ near each boundary point with a model domain osculating it, and controls the error with estimates for the $\\bar\\partial$-Neumann problem.\nBoutet de Monvel and Sjöstrand later gave a microlocal proof that describes the kernel off the diagonal as well.\n\nExample: for the unit ball $B \\subset \\C^n$, $K(z) = \\dfrac{n!}{\\pi^n (1 - \\abs{z}^2)^{n+1}}$.\nWith $\\rho(z) = \\abs{z}^2 - 1$ this is the expansion with $\\varphi = n!/\\pi^n$ and $\\psi = 0$.\nThe logarithmic term is in general nonzero, and its coefficient $\\psi$ on $\\partial\\Omega$ carries geometric information about the boundary.\n\nStrong pseudoconvexity is needed for the exponent $n + 1$.\nFor the unit polydisc in $\\C^2$, $K(z) = \\pi^{-2}(1 - \\abs{z_1}^2)^{-2}(1 - \\abs{z_2}^2)^{-2}$.\nLet $\\delta$ be the distance to the boundary.\nNear a point with $\\abs{z_1} = 1$ and $\\abs{z_2} \u003c 1$, the kernel grows only like $\\delta^{-2}$, not like $\\delta^{-3}$.\n\n## Explanations\n\n### L1 human\n\nThe Bergman kernel measures how large a holomorphic function of unit size can be at a point.\nNear the boundary of the domain it blows up.\nFefferman's theorem says exactly how it blows up on a smooth strongly pseudoconvex domain: like a power of the distance to the boundary, plus a correction with a logarithm.\nThe leading part is the same as for the ball, the simplest such domain.\n\n### L2 human\n\nThe expansion is the analytic input of [[Fefferman's theorem on the extension of biholomorphisms|N95NQ0G3A6]], through the boundary behaviour of the [[Bergman metric|BB7S4NTZ6V]].\nIt shows that the [[Bergman projection|6J1X63Z7QZ]] and the Bergman metric of a strongly pseudoconvex domain look, near the boundary, like those of the ball.\nThe coefficients $\\varphi$ and $\\psi$ restricted to $\\partial\\Omega$ are local invariants of the boundary under biholomorphic maps.\nThis led Fefferman to a programme of computing them from the CR geometry of $\\partial\\Omega$.\n",
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