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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $M$ be a real-analytic real hypersurface in $\\C^{n+1}$.\nLet $p \\in M$, and assume that the [[Levi form|80MTNXWW80]] of $M$ at $p$ is nondegenerate, with $r$ positive and $n - r$ negative eigenvalues.\nWrite $\\langle z, z \\rangle = \\sum_{j=1}^{n} \\varepsilon_j \\abs{z_j}^2$, where $\\varepsilon_j = 1$ for $j \\le r$ and $\\varepsilon_j = -1$ for $j \u003e r$, and let $\\operatorname{tr} = \\sum_{j=1}^{n} \\varepsilon_j \\frac{\\partial^2}{\\partial z_j \\partial \\bar z_j}$.\nThen there are local holomorphic coordinates $(z, w) \\in \\C^n \\times \\C$ centred at $p$, with $w = u + iv$, in which $M$ is given near $p$ by\n$$v = \\langle z, z \\rangle + \\sum_{k, l \\ge 2} F_{kl}(z, \\bar z, u),$$\nwhere:\n\n1. each $F_{kl}$ is a polynomial of degree $k$ in $z$ and $l$ in $\\bar z$, with coefficients real-analytic in $u$, and $F_{lk} = \\overline{F_{kl}}$;\n2. $\\operatorname{tr} F_{22} = 0$, $\\operatorname{tr}^2 F_{32} = 0$ and $\\operatorname{tr}^3 F_{33} = 0$.\n\n## Remarks\n\nThe result is due to S.-S. Chern and J. Moser.\nThe proof removes, degree by degree, all terms of the defining equation that can be killed by a holomorphic change of coordinates; the remaining terms are the normal form.\nThe model case is the hyperquadric $v = \\langle z, z \\rangle$, where all $F_{kl}$ vanish; for $r = n$ it is biholomorphic to the unit sphere minus a point.\nThe normalising coordinates are not unique: they are determined up to the action of the group of local biholomorphisms that preserve the hyperquadric.\nFor $n = 1$ the trace conditions force $F_{22} = F_{32} = F_{33} = 0$, so the first term that may survive is $F_{24}$, which carries the invariant found earlier by É. Cartan.\nWithout the nondegeneracy of the Levi form the quadratic part $\\langle z, z \\rangle$ is degenerate and this normal form does not apply.\n\n## Explanations\n\n### L1 human\n\nTwo hypersurfaces in complex space may look different but be the same after a holomorphic change of coordinates.\nThe theorem gives a standard way to write such a hypersurface near a point, so that the remaining coefficients are true geometric information.\nThe leading part is a quadratic \"complex paraboloid\", and the corrections are put into a reduced shape by fixed trace conditions.\n\n### L2 human\n\nThe normal form solves the local equivalence problem for real-analytic Levi-nondegenerate hypersurfaces.\nTwo such germs are equivalent exactly when their normal forms agree up to the stability group of the hyperquadric.\nChern and Moser also gave a geometric approach that builds a Cartan connection on a bundle over $M$, whose curvature yields the local invariants.\nThese invariants show for instance that a generic small perturbation of the sphere is not locally biholomorphic to the sphere.\nThe method is a complex analogue of the use of normal coordinates in Riemannian geometry.\n",
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