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  "body": "\n## Statement\n\nLet $M$ be a compact smooth [[CR manifold|ZC78JKZX9S]] of hypersurface type, of real dimension $2n + 1$ with $n \\ge 2$.\nAssume that $M$ is strongly pseudoconvex, that is, its [[Levi form|80MTNXWW80]] is definite at every point.\nThen there is a smooth CR embedding of $M$ into $\\C^N$ for some $N$.\n\n## Remarks\n\nThe theorem is due to L. Boutet de Monvel.\nThe proof uses Kohn's analysis of the tangential Cauchy–Riemann operator $\\bar\\partial_b$ and of the [[Kohn Laplacian|ZBQ6TVNZZ8]].\nOn a compact strongly pseudoconvex manifold of real dimension at least $5$, this analysis gives good solvability of $\\bar\\partial_b$.\nThis produces many global [[CR functions|JFKYYK2CEB]], and finitely many of them give an embedding.\nThe boundary of a bounded strongly pseudoconvex domain in $\\C^{n+1}$ with smooth boundary is an example where the embedding is given from the start.\nThe bound on the dimension is necessary: in real dimension $3$ there are compact strongly pseudoconvex CR manifolds with no CR embedding into any $\\C^N$, such as [[Rossi's example|1TJXANASFE]].\n\n## Explanations\n\n### L1 human\n\nA CR manifold carries part of the structure of a hypersurface in complex space, but it is defined abstractly, without a surrounding space.\nAssume the manifold is compact, curves in the strongly convex way, and has dimension at least five.\nThen the theorem says that it really sits inside some complex space.\nSo in these dimensions the abstract notion gives nothing new.\n\n### L2 human\n\nThe result is a global embedding theorem, in contrast with local realisation questions, where Kuranishi and others worked on embedding small pieces.\nCombined with results on filling, it shows that such $M$ bound complex spaces with isolated singularities.\nThe three-dimensional case behaves very differently and is an active subject: there embeddability is a delicate property that is not preserved under small deformations.\n",
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