Rational dependence of smooth and analytic CR mappings on their jets.

*(English)*Zbl 0942.32027Let \(M\) 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.

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 |