Nordine Mir

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂN are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂN are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface M in complex euclidean space of dimension >= 2 which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p in M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their l(p)-jets at p. As a direct application, we derive a Lie group structure for the topological group Aut(M,p). Furthermore, we also show that the order l(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. As a first consequence, it follows that that given any compact real-analytic hypersurface M in complex euclidean space, there exists an integer k depending only on M such that for every point p in M germs at p of CR diffeomorphisms mapping M into another real-analytic hypersurface in a complex space of the same dimension are uniquely determined by their k-jet at that point. Another consequence is a boundary version of H. Cartans uniqueness theorem. Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

Approximation and convergence of formal CR-mappings

An important step in understanding the existence of analytic objectswith certain properties consists of understanding the same problem at the level of formal power series. The latter problem can be reduced to a sequence of algebraic equations for the coefficients of the unknown power series and is often simpler than the original problem, where the power series are required to be convergent. It is therefore of interest to know whether such power series are automatically convergent or can possibly be replaced by other convergent power series satisfying the same properties. A celebrated result of this kind is Artin’s approximation theorem [1] which states that a formal solution of a system of analytic equations can be replaced by a convergent solution of the same system that approximates the original solution at any prescribed order. In this paper, we study convergence and approximation properties (in the spirit of [1]) of formal (holomorphic) mappings sending real-analytic submanifolds M ⊂ C and M ′ ⊂ C ′ into each other, N,N ′ ≥ 2. In this situation, the above theorem of Artin cannot be applied directly. Moreover, without additional assumptions on the submanifolds, the analogous approximation statement is not even true. Indeed, in view of an example of Moser-Webster [23], there exist real-algebraic surfaces M,M ′ ⊂ C that are formally but not biholomorphically equivalent. However, our firstmain result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C (see Section 2.1 for notation and definitions). Theorem 1.1. Let M ⊂ C be a real-analytic minimal CR-submanifold and M ′ ⊂ C ′ a real-algebraic subset with p ∈ M and p ′ ∈ M ′. Then for any formal (holomorphic)

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

Abstract. We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in

Lie group structures on automorphism groups of real-analytic CR manifolds

\mathcal{C}^{N+1}

ON THE CONVERGENCE OF FORMAL MAPPINGS

,

Formal biholomorphic maps of real analytic hypersurfaces

N\geq 1

Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

. We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild.

Convergence of Formal Embeddings Between Real-Analytic Hypersurfaces in Codimension One

Given any real-analytic CR manifold