CR transversality of holomorphic mappings between generic submanifolds in complex spaces. (English) Zbl 1244.32011

Ebenfelt, Peter; Duong Ngoc Son

Proc. Am. Math. Soc. 140, No. 5, 1729-1738 (2012).

A holomorphic mapping \(H\) from a neighbourhood \(U\) of a point \(p\) in \(\mathbb C^N\) is said to be CR transversal at \(p\) to a generic submanifold \(M'\) that contains \(p'=H(p)\) if \(T_{p'}^{1,0}M' + dH(T_p^{1,0}\mathbb C^N)= T_{p'}^{1,0}\mathbb C^N\).
The main result of the paper is Theorem 1.1: Let \(M,M'\subset\mathbb C^N\) be smooth generic submanifolds of the same dimension through \(p\) and \(p'\) respectively, and let \(H: (\mathbb C^N,p) \to (\mathbb C^N,p')\) be a germ of a holomorphic mapping such that \(H(M)\subset M'\). Assume that \(M\) is of finite type at \(p\) and \(\text{Jac\,} H\not\equiv 0\). Then \(H\) is CR transversal to \(M'\) at \(p'\).
This result and its corollaries affirmatively answer questions posed by L. P. Rothschild and the first author [“Transversality of CR mappings”, Am. J. Math. 128, No. 5, 1313–1343 (2006; Zbl 1105.32022)] whether the conditions assumed in earlier versions of CR transversality theorems could be relaxed.

Reviewer: Gerd Schmalz (Armidale)

MSC:

32H02	Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32V20	Analysis on CR manifolds
32V40	Real submanifolds in complex manifolds

Keywords:

CR transversality; Hopf lemma; finite type

Citations:

Zbl 1105.32022

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