Several complex variables and analytic spaces CR manifolds Theorem

Chern–Moser normal form

Statement

Let $n \ge 1$ and let $M$ be a real-analytic real hypersurface in $\C^{n+1}$ . Let $p \in M$ , and assume that the Levi form of $M$ at $p$ is nondegenerate, with $r$ positive and $n - r$ negative eigenvalues. Write $\langle z, z \rangle = \sum_{j=1}^{n} \varepsilon_j \abs{z_j}^2$ , where $\varepsilon_j = 1$ for $j \le r$ and $\varepsilon_j = -1$ for $j > r$ , and let $\operatorname{tr} = \sum_{j=1}^{n} \varepsilon_j \frac{\partial^2}{\partial z_j \partial \bar z_j}$ . Then 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 $v = \langle z, z \rangle + \sum_{k, l \ge 2} F_{kl}(z, \bar z, u),$ where:

each $F_{kl}$ is a polynomial of degree $k$ in $z$ and $l$ in $\bar z$ , with coefficients real-analytic in $u$ , and $F_{lk} = \overline{F_{kl}}$ ;
$\operatorname{tr} F_{22} = 0$ , $\operatorname{tr}^2 F_{32} = 0$ and $\operatorname{tr}^3 F_{33} = 0$ .

Remarks

The result is due to S.-S. Chern and J. Moser. The 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. The 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. The normalising coordinates are not unique: they are determined up to the action of the group of local biholomorphisms that preserve the hyperquadric. For $n = 1$ 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. Without the nondegeneracy of the Levi form the quadratic part $\langle z, z \rangle$ is degenerate and this normal form does not apply.

Going deeper

For a student

The normal form solves the local equivalence problem for real-analytic Levi-nondegenerate hypersurfaces. Two such germs are equivalent exactly when their normal forms agree up to the stability group of the hyperquadric. Chern and Moser also gave a geometric approach that builds a Cartan connection on a bundle over $M$ , whose curvature yields the local invariants. These invariants show for instance that a generic small perturbation of the sphere is not locally biholomorphic to the sphere. The method is a complex analogue of the use of normal coordinates in Riemannian geometry.

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CR manifold concept
Levi form concept
Chern–Moser normal form statement

Tag	`FHG1N4EKP4`
URI	`m2://FHG1N4EKP4`
Kind	statement, theorem
Writing	W0 stub
MSC	32V05, 32V40
Status	This copy of the site has no build data. Run `m2 status FHG1N4EKP4`.
About	CR manifold
Uses	Levi form, CR manifold