Rational dependence of smooth and analytic CR mappings on their jets. (English) Zbl 0942.32027

Baouendi, M. S.; Ebenfelt, P.; Rothschild, Linda Preiss

Math. Ann. 315, No. 2, 205-249 (1999).

Let \(M\) and \(M'\) be smooth generic submanifolds through \(p_0\in\mathbb C^N\) and \(p_0'\in\mathbb C^{N'},\) respectively, such that \(M\) is of finite type at \(p_0\) and \(M'\) is \(\ell_0\)-nondegenerate at \(p_0'\) for some integer \(\ell_0> 0.\)
The main theorem of the paper gives rational dependence of the associated CR submersive mapping, and more generally of a formal CR submersive power series mapping \((M,p_0)\to (M',p_0')\) on its jet of a predetermined order. The applications of this theorem state that a holomorphic mapping sending \(M\) into \(M'\) which is CR submersive at \(p_0\) is uniquely determined by finitely many derivatives at \(p_0\) and give sufficient conditions for all CR submersive formal mappings between real-analytic generic submanifolds to be convergent. Smooth perturbations of generic submanifolds, satisfying the appropriate conditions are considered.
The paper concludes with an application of the authors’s methods to the study of algebraicity of holomorphic mappings which map one real algebraic submanifold into another.

Reviewer: A.V.Chernecky (Odessa)

MSC:

32V40	Real submanifolds in complex manifolds
32H02	Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables

Keywords:

rational dependence; formal power series mapping; jet; uniqueness; convergence; CR submersive mapping; smooth perturbations of generic submanifolds; algebraicity of holomorphic mappings