## Formalized Name Chern-Moser Normal Form Theorem for Strictly Pseudoconvex Real-Analytic Hypersurfaces ## Formalized Statement Let $n\ge 1$, let $M\subset \mathbb C^{n+1}$ be a real-analytic embedded real hypersurface, and let $p\in M$. Assume that $M$ is strictly pseudoconvex at $p$, meaning that, after choosing the pseudoconvex side, the Levi form of a $C^2$ local defining function is positive definite on $T_pM\cap iT_pM$. Then there exist an open neighbourhood $U\subset\mathbb C^{n+1}$ of $p$ and a biholomorphic map $\Phi:U\to \Phi(U)\subset\mathbb C^n\times\mathbb C$ such that $\Phi(p)=0$ and, writing the target coordinates as $(z,w)=(z_1,\dots,z_n,u+iv)$, the hypersurface germ $\Phi(M\cap U)$ at $0$ is given by a convergent real-analytic equation \begin{align*} v=|z|^2+\sum_{k,l\ge 2}F_{kl}(z,\bar z,u). \end{align*} Here $|z|^2=\sum_{j=1}^n |z_j|^2$, and each coefficient $F_{kl}:\mathbb C^n\times\mathbb C^n\times\mathbb R\to\mathbb R$ is real-analytic near $(0,0,0)$, homogeneous of degree $k$ in $z$, and homogeneous of degree $l$ in $\bar z$. For a polynomial $P(z,\bar z,u)$ in $z$ and $\bar z$ with real-analytic dependence on $u$, define the Chern-Moser trace operator by \begin{align*} \operatorname{tr}P=\sum_{j=1}^n \frac{\partial^2 P}{\partial z_j\,\partial \bar z_j}. \end{align*} Let $\operatorname{tr}^m$ denote the $m$-fold iterate of $\operatorname{tr}$. The coefficients satisfy the Chern-Moser trace conditions \begin{align*} \operatorname{tr}F_{22}=0, \end{align*} \begin{align*} \operatorname{tr}^2F_{23}=0, \end{align*} \begin{align*} \operatorname{tr}^3F_{33}=0. \end{align*} Let \begin{align*} Q=\{(z,w)\in\mathbb C^n\times\mathbb C:\operatorname{Im}(w)=|z|^2\}. \end{align*} If $\Phi_1$ and $\Phi_2$ are two biholomorphic coordinate germs at $p$ satisfying the preceding normal-form conclusions, then the transition germ $\Phi_2\circ\Phi_1^{-1}$ is the germ at $0$ of a biholomorphic automorphism of $Q$ fixing $0$. Conversely, composing any normalizing coordinate germ on the left with an element of $\operatorname{Aut}(Q)_0$ gives another admissible normalizing coordinate germ exactly when the resulting equation is again in the displayed Chern-Moser normal form. Thus the residual freedom in normalized coordinates is precisely the isotropy action of $\operatorname{Aut}(Q)_0$; after fixing one representative of this isotropy, the normalizing coordinate germ and the coefficients $F_{kl}$ are unique. ## Proof [proofplan] The proof reduces the hypersurface germ to the hypotheses of the classical Chern-Moser normal-form theorem, then invokes that theorem in its precise real-analytic strictly pseudoconvex form. The preliminary coordinate choice fixes the point, tangent hyperplane, sign convention, and Levi form so that the model quadric is $\operatorname{Im}(w)=|z|^2$. The external Chern-Moser theorem then supplies the convergent normalizing coordinates, the trace-normalized expansion, and the uniqueness statement modulo the isotropy group of the model quadric. [/proofplan] [step:Choose coordinates with standard positive Levi form] Let $\rho_0:V_0\to\mathbb R$ be a real-analytic local defining function for $M$ on an open neighbourhood $V_0\subset\mathbb C^{n+1}$ of $p$, chosen so that its Levi form is positive definite on the complex tangent space after the prescribed pseudoconvex-side convention. Since $M$ is embedded and real-analytic, $d\rho_0$ is nonzero on $M\cap V_0$. Choose complex affine coordinates $(\tilde z,\tilde\zeta)\in\mathbb C^n\times\mathbb C$ centred at $p$ such that \begin{align*} T_0M=\{(\tilde z,\tilde\zeta):\operatorname{Re}(\tilde\zeta)=0\}. \end{align*} Multiplying the defining function by a positive real-analytic unit if necessary, we may write \begin{align*} \rho_1(\tilde z,\tilde\zeta)=2\operatorname{Re}(\tilde\zeta)+L(\tilde z,\overline{\tilde z})+R_3(\tilde z,\overline{\tilde z},\operatorname{Im}(\tilde\zeta),\operatorname{Re}(\tilde\zeta)), \end{align*} where $L:\mathbb C^n\times\mathbb C^n\to\mathbb R$ is the positive definite Hermitian Levi form at $p$, and $R_3$ has Euclidean order at least $3$ in the real variables. Choose an invertible complex-linear map $A:\mathbb C^n\to\mathbb C^n$ such that \begin{align*} L(Az,\overline{Az})=|z|^2. \end{align*} In the coordinates $(z,\zeta)$ defined by $\tilde z=Az$ and $\tilde\zeta=\zeta$, the defining equation has the form \begin{align*} \rho_2(z,\zeta)=2\operatorname{Re}(\zeta)+|z|^2+R_3(z,\bar z,\operatorname{Im}(\zeta),\operatorname{Re}(\zeta)). \end{align*} Define the holomorphic normal coordinate $w=u+iv$ by \begin{align*} w=2i\zeta. \end{align*} Then $v=2\operatorname{Re}(\zeta)$. Hence the equation $\rho_2=0$ is equivalent to \begin{align*} v+|z|^2+R_3(z,\bar z,u/2,v/2)=0. \end{align*} Replacing the defining function by its negative before the preceding normalization, if necessary, gives the sign convention \begin{align*} v=|z|^2+E(z,\bar z,u), \end{align*} where $E:W\to\mathbb R$ is real-analytic on a neighbourhood $W\subset\mathbb C^n\times\mathbb C^n\times\mathbb R$ of $(0,0,0)$ and has no terms of weighted degree at most $2$ for the weights $\operatorname{wt}(z_j)=\operatorname{wt}(\bar z_j)=1$ and $\operatorname{wt}(u)=2$. [guided] We first put the hypersurface into the coordinate convention used by Chern and Moser. Let $\rho_0:V_0\to\mathbb R$ be a real-analytic defining function for $M$ near $p$. Because $M$ is an embedded real hypersurface, $d\rho_0$ is nonzero on $M$. The strict pseudoconvexity hypothesis says that, after choosing the pseudoconvex side, the Levi form of such a defining function is positive definite on the complex tangent space. Choose complex affine coordinates $(\tilde z,\tilde\zeta)\in\mathbb C^n\times\mathbb C$ centred at $p$ and adapted to the real tangent hyperplane, so that \begin{align*} T_0M=\{(\tilde z,\tilde\zeta):\operatorname{Re}(\tilde\zeta)=0\}. \end{align*} After multiplying the defining function by a positive real-analytic unit, the first-order normal term is $2\operatorname{Re}(\tilde\zeta)$. Its second-order part in complex tangential variables is the Levi form. Thus there is a positive definite Hermitian form $L:\mathbb C^n\times\mathbb C^n\to\mathbb R$ and a real-analytic remainder $R_3$ of Euclidean order at least $3$ such that \begin{align*} \rho_1(\tilde z,\tilde\zeta)=2\operatorname{Re}(\tilde\zeta)+L(\tilde z,\overline{\tilde z})+R_3(\tilde z,\overline{\tilde z},\operatorname{Im}(\tilde\zeta),\operatorname{Re}(\tilde\zeta)). \end{align*} Positive definiteness is used exactly here: it gives an invertible complex-linear map $A:\mathbb C^n\to\mathbb C^n$ satisfying \begin{align*} L(Az,\overline{Az})=|z|^2. \end{align*} After the change $\tilde z=Az$, the Levi form is the standard Hermitian form. Now define $w=u+iv$ by \begin{align*} w=2i\zeta. \end{align*} Since $2i\zeta=2i\operatorname{Re}(\zeta)-2\operatorname{Im}(\zeta)$, its imaginary part is \begin{align*} v=2\operatorname{Re}(\zeta). \end{align*} Thus the sign of the model equation is controlled by the sign of the defining function. If the preceding expression gives $v=-|z|^2$ as the quadratic model, replace the defining function by its negative before making the same normalization. This replacement reverses the defining-function sign and is allowed because the pseudoconvex side has already fixed which sign of the Levi form is to be positive. With that convention, the hypersurface is written as \begin{align*} v=|z|^2+E(z,\bar z,u). \end{align*} The map $E:W\to\mathbb R$ is real-analytic because the defining equation and the coordinate changes are real-analytic and the coefficient of $v$ in the defining equation is nonzero at the origin, so the real-analytic implicit function theorem solves for $v$ as a real-analytic function of $(z,\bar z,u)$. The tangent-plane normalization removes weighted degrees $0$ and $1$, and the Levi normalization makes the entire weighted-degree-$2$ part equal to $|z|^2$. Hence $E$ has no weighted terms of degree at most $2$. [/guided] [/step] [step:Invoke the real-analytic Chern-Moser normalization theorem] We use the following external standard theorem of Chern and Moser in the strictly pseudoconvex real-analytic case. If a real-analytic hypersurface germ in $\mathbb C^{n+1}$ is written near $0$ as \begin{align*} v=|z|^2+E(z,\bar z,u), \end{align*} where $E$ is real-valued, real-analytic, and has no weighted terms of degree at most $2$, then there is a biholomorphic coordinate germ $\Psi:(\mathbb C^{n+1},0)\to(\mathbb C^{n+1},0)$ with invertible complex Jacobian at $0$ such that the transformed hypersurface has a convergent equation \begin{align*} v=|z|^2+\sum_{k,l\ge 2}F_{kl}(z,\bar z,u). \end{align*} For every $k,l\ge 2$, the coefficient $F_{kl}:\mathbb C^n\times\mathbb C^n\times\mathbb R\to\mathbb R$ is real-analytic near $(0,0,0)$, homogeneous of degree $k$ in $z$, and homogeneous of degree $l$ in $\bar z$. Moreover the coefficients satisfy \begin{align*} \operatorname{tr}F_{22}=0, \end{align*} \begin{align*} \operatorname{tr}^2F_{23}=0, \end{align*} \begin{align*} \operatorname{tr}^3F_{33}=0. \end{align*} The theorem also asserts convergence of the normalizing transformation in the real-analytic category. The hypotheses of this external theorem are exactly the conclusions of the previous step: the hypersurface germ is real-analytic, the Levi form is the standard positive Hermitian form $|z|^2$, and the graphing error has no weighted terms of degree at most $2$. Therefore the theorem gives a biholomorphic germ $\Psi$ at $0$ with the displayed normal-form equation. Composing $\Psi$ with the preliminary coordinate map from the previous step gives a biholomorphic map $\Phi:U\to\Phi(U)$ on a sufficiently small neighbourhood $U\subset\mathbb C^{n+1}$ of $p$. This proves the asserted existence of convergent Chern-Moser normal coordinates and the stated trace conditions. [/step] [step:Apply the Chern-Moser uniqueness theorem to identify the residual freedom] Let $\Phi_1$ and $\Phi_2$ be two biholomorphic coordinate germs at $p$ satisfying the normal-form conclusions. Define the transition germ \begin{align*} H=\Phi_2\circ\Phi_1^{-1}. \end{align*} Then $H:(\mathbb C^{n+1},0)\to(\mathbb C^{n+1},0)$ is a biholomorphic germ sending one Chern-Moser normal-form hypersurface germ with standard Levi form $|z|^2$ to another such germ. We use the uniqueness part of the external Chern-Moser theorem in the following precise form. If a biholomorphic germ $H:(\mathbb C^{n+1},0)\to(\mathbb C^{n+1},0)$ sends one convergent Chern-Moser normal form with standard positive Levi form to another convergent Chern-Moser normal form with the same standard Levi form, then there exists a unique germ $A\in\operatorname{Aut}(Q)_0$ such that $H=A$. Equivalently, after the residual isotropy parameter $A$ is fixed, the normalizing germ and all normal-form coefficients are unique. The hypotheses are satisfied because both $\Phi_1(M)$ and $\Phi_2(M)$ are real-analytic hypersurface germs in the displayed Chern-Moser normal form with model quadric \begin{align*} Q=\{(z,w)\in\mathbb C^n\times\mathbb C:\operatorname{Im}(w)=|z|^2\}. \end{align*} Thus $H\in\operatorname{Aut}(Q)_0$. Conversely, the same uniqueness theorem says that the kernel left by the normalizing equations is exactly this isotropy group: composing a normalizing germ with an element of $\operatorname{Aut}(Q)_0$ is precisely the residual coordinate freedom, subject to the requirement that the resulting equation is again in the displayed normal form. Once one representative of this isotropy has been fixed, the transition germ is the identity, and the coefficients $F_{kl}$ agree. This proves the residual-ambiguity and uniqueness assertions. [/step]