Convergence of the Chern–Moser–Beloshapka normal forms

Abstract

In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of C N of codimension d ≥ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. We show that our techniques can be adapted in the case d = 1 in order to obtain a new and direct proof of Chern-Moser normal form theorem.The typical model for this situation is a hyperquadric, that is, a manifold of the formwhere each J k is a Hermitian n × n matrix, and the conditions of nondegeneracy and full rank are expressed byand the homogeneous parts in z andz of a series Φ(z,z, u) = j,k Φ j,k (z,z, u), where Φ j,k (tz, sz, u) = t j s k Φ j,k (z,z, u); Φ j,k is said to be of type (j, k).We then say that Φ ∈ N CM if it satisfies the following (Chern-Moser) normal form conditions: Φ j,0 = Φ 0,j = 0, j ≥ 0;There are a number of aspects particular to the case d = 1 which allow Chern and Moser to construct, based on these conditions (which arise rather naturally from a linearization of the problem with respect to the ordering by type), a convergent choice of coordinates. In particular, Chern and Moser are able to restate much of their problem in terms of ODEs, which comes from the fact that there is only one transverse variable when d = 1; in particular, existence and regularity of solutions is guaranteed. In higher codimension, this changes dramatically, and we obtain systems of PDEs; neither do we a priori know that those are solvable nor do we know anything about the regularity of their solutions (should they exist). Our normal form has to take this into account. Another aspect of the problem, which really changes dramatically from the case d = 1 to d > 1, is the second line of the normal form conditions above: We cannot impose that

On the Beloshapka's rigidity conjecture for real submanifolds in complex space

“…An incomplete list of authors here includes Ezhov and Schmalz (e.g. [ES94a,ES94b,ES06]), Lamel and Stolovitch [LS17], Garrity-Mizner [GM95,Mi89], Isaev (e.g. [Is08]), Isaev-Kaup [IK12], Shevchenko [She94a,She94b].…”

The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

“…For Levi-degenerate hypersurfaces in C N , N ≥ 3 satisfying certain special conditions (in addition to the Hormander-Kohn bracket-generating condition) normal form constructions were carried out by Ebenfelt [16,15] and by Kossovskiy-Zaitsev [33]. For results on normal forms for real submanifolds of higher codimension as well as CR-singular submanifolds we refer to Ezhov-Schmalz [18], Beloshapka [4], Lamel-Stolovitch [36], Moser-Webster [39], Huang-Yin [20,21], Gong [19], Coffman [13], Burcea [8]. See also Zaitsev [48] for normal forms in the non-integrable setting.…”