Convergence of formal CR mappings into strongly pseudoconvex Cauchy\u2013Riemann manifolds

Bernhard Lamel,Nordine Mir

doi:10.1007/s00222-017-0743-3

Bernhard Lamel, Nordine Mir

Open Access

https://doi.org/10.1007/s00222-017-0743-3

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Abstract

It is shown that any formal holomorphic mapping sending a real-analytic generic submanifold Msubset mathbb {C}^N of finite type into a real-analytic strongly pseudoconvex CR submanifold M'subset mathbb {C}^{N'} is necessarily convergent. As a consequence, we obtain a positive answer to the long-standing open question of whether formal holomorphic maps sending real-analytic strongly pseudoconvex hypersurfaces into each other are convergent.

Highlights

Some of the particular phenomena which occur in CR geometry are the rigidity and the strong regularity properties of CR mappings between CR manifolds
While such properties have very much been studied for automorphisms, many interesting and delicate questions have remained open over the last decades for arbitrary CR mappings between CR manifolds embedded in complex spaces of different dimension
These questions have been at the center of recent interest in the community as shown e.g. by the recent remarkable work on the smooth regularity of CR maps by Berhanu–Xiao [8]. We tackle one such question regarding the convergence of formal holomorphic transformations and prove what seems to be the first general convergence result for formal CR maps in positive codimension

Summary

Introduction

Some of the particular phenomena which occur in CR geometry are the rigidity and the strong regularity properties of CR mappings between CR manifolds. The following, natural, long-standing question, originating essentially from the works of Huang [14,15] and Forstneric [12] and appearing explicitely e.g. in [25], remained open until now: does Chern–Moser’s above mentioned convergence result hold for formal transformations between arbitrary strongly pseudoconvex real-analytic hypersurfaces M ⊂ CN and M ⊂ CN ? Besides settling by the affirmative, a long standing open question, Theorem 1.1 appears to be the first general convergence result for formal transformations between real-analytic CR submanifolds in complex spaces of different dimension. One of the main novelties in our proof of Theorem 1.1 consists of introducing, for any given formal CR map H : M → M , the notion of “meromorphic infinitesimal deformations” of H These formal objects can be seen as formal meromorphic vector fields tangent to the image of H and are directly related to how degenerate the map H is (see Proposition 4.4).

Power series rings and notations

Segre set mappings and basic properties

Two convergence results for formal CR maps