Rational Dependence of Smooth and Analytic Cr Mappings on Their Jets

Let M ⊂ C and M ′ ⊂ C ′ be two smooth (C) generic submanifolds with p0 ∈ M and p ′ 0 ∈ M . We shall consider holomorphic mappings H : (C , p0) → (C ′ , p0), defined in a neighborhood of p0 ∈ C N , such that H(M) ⊂ M ′ (and, more generally, smooth CR mappings (M, p0) → (M , p0); see below). We shall always work under the assumption that M is of finite type at p0 in the sense of Kohn and Bloom–Graham, and that M ′ is finitely nondegenerate at p0 (see §1 for precise definitions). More precisely, we shall assume that M ′ is l0-nondegenerate at p0, for some integer l0 ≥ 0. (Recall that for a real hypersurface, 1-nondegeneracy at a point is equivalent to Levi nondegeneracy at that point. Also, as is further explained in §2.2, the notion of finite nondegeneracy for real-analytic generic submanifolds is intimately related to the notion of holomorphic nondegeneracy defined below in this section.) We denote by J the complex structure map on TC . Recall that for p ∈ M , T c pM denotes the complex tangent space to M at p, i.e. the largest J-invariant subspace of TpM , the tangent space of M at p. A smooth mapping H : M → M ′ is called CR if its tangent map dH maps T c pM into T c H(p)M ′ for every p ∈ M . A CR mapping H : M → M ′ is called CR submersive at p if dH maps T c pM onto T c H(p)M . A holomorphic mapping H sending M into M ′ is called CR submersive if its restriction to M is. To a smooth CR mapping H : M → M , and p0 ∈ M , one may associate a unique formal (holomorphic) power series mapping