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Let A ⊂ C N be an irreducible real algebraic set. Assume that there exists p 0 ∈ A such that A is a minimal, generic, holomorphically nondegenerate submanifold at p 0 . We show here that if H is a germ at p 1 ∈ A of a holomorphic mapping from C N into itself, with Jacobian H not identically 0, and H(A) contained in a real algebraic set of the same dimension as A, then H must extend to all of C N (minus a complex algebraic set) as an algebraic mapping. Conversely, we show that for any "model case" (i.e., A given by quasi-homogeneous real polynomials), the conditions on A are actually necessary for the conclusion to hold.

O. IntroductionThis paper consists of three parts:Part I presents results on local embedding of CR structures. We consider an abstract CR manifold whose structure is invariant under a transversal Lie group action. We show that such a manifold can always be locally embedded in complex space as a generic submanifold. The proof is based on selection of canonical coordinates and repeated use of the Newlander-Nirenberg theorem [13]. When the Lie group is abelian the embedding can be given a particularly simple form. Let l~ 1 be the codimension of our submanifold (called M throughout the paper); it is then convenient to denote by n + I the dimension of the ambient complex space and by zl,..., z,, w 1 .... , w t the complex coordinates; we shall systematically write z= (zl ..... z,), w=(wl, ..., wO. One can then arrange that an equation of the embedded submanifold M be given by an equation Imw =~b(z, z-). (0.1) Our viewpoint will be strictly local, about a central point of M which we take to be the origin. Thus ~b = 0 at 0. It is also convenient to assume that the tangent space to M at 0 is the (real) vector subspace Imw = 0, which means that d~b = 0 at 0. We have chosen to call rigid any CR structure that admits an embedding of the kind (0.1).Let us underline the fact that the codimension I can be arbitrary. Parts II &III are devoted to the study of local properties of CR functions or distributions on a rigid CR manifold M. Our first step is to define an adapted FBI (Fourier-Bros-Iagolnitzer) transform of such functions. Our definition is a * During the completion of this work, M.S.

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