## Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions.(English)Zbl 1293.32043

Let $$M\subset {\mathbb C}^{n+1}$$ and $$M' \subset {\mathbb C}^{N+1}$$ be real hypersurfaces, and let $$H: M\to M'$$ be a holomorphic mapping, i.e., a holomorphic mapping defined on some open neighbourhood $$U$$ of $$M$$ in $${\mathbb C}^{n+1}$$ such that $$H(M)\subset M'$$. Such a map is said to be CR transversal to $$M'$$ at a point $$p\in M$$ if $T^{0,1}_{H(p)}M'+dH(T_p^{1,0}{\mathbb C}^{n+1})=T_{H(p)}^{1,0}{\mathbb C}^{N+1},$ where $$T^{0,1}M' = {\mathbb C}TM'\cap T^{0,1}{\mathbb C}^{N+1}$$ denotes the CR bundle on $$M'$$ and $$T^{1,0}M'$$ its complex conjugate. In the case of real hypersurfaces, CR transversality coincides with ordinary transversality, requiring $$T_{H(p)}M'+dH(T_p{\mathbb C}^{n+1})=T_{H(p)}{\mathbb C}^{N+1}.$$
In the strictly pseudoconvex case, transversality follows from the classical Hopf Lemma. In the equidimensional case $$N=n$$, transversality at $$p$$ was proved for maps of full generic rank under the assumption that $$M$$ is of finite type at $$p$$ by P. Ebenfelt and D. N. Son [Proc. Am. Math. Soc. 140, No. 5, 1729–1738 (2012; Zbl 1244.32011)] (see also [M. S. Baouendi et al., Commun. Anal. Geom. 15, No. 3, 589–611 (2007; Zbl 1144.32005)] for transversality results outside a proper, real-analytic subvariety of $$M$$).
In this paper the case of higher codimension $$N>n$$ is considered and conditions are given on the rank of the Levi form of $$M'$$, the CR dimensions $$n$$ and $$N$$ and on the map $$H$$ that guarantee transversality of holomorphic mappings at all points. For $$1\leq s\leq n+1$$, let $$W^s_H=\{z\in U : \text{rk}\,H_z<s\}$$, where $$H_z$$ stands for the Jacobian matrix of $$H$$ at $$z$$.
Theorem. Let $$M\subset {\mathbb C}^{n+1}$$ and $$M' \subset {\mathbb C}^{N+1}$$ be smooth real hypersurfaces through $$p$$ and $$p'$$, and $$H: ({\mathbb C}^{n+1},p) \to ({\mathbb C}^{N+1},p')$$ a germ at $$p$$ of holomorphic mappings such that $$H(M)\subset M'$$. Denote by $$r$$ the rank of the Levi form of $$M'$$ at $$p'$$. If $$~2N-r\leq 2n-2~$$ and the germ at $$p$$ of the analytic variety $$W_H^{n+1}$$ has codimension at least 2, then $$H$$ is transversal to $$M'$$ at $$p$$.
The same conclusion is obtained under the assumptions that $$2N-r\leq 2n-3$$, the hypersurface $$M$$ is of finite type at $$p$$ and $$H$$ is a finite map at $$p$$, or that $$2N-r\leq 2n+s-3$$, for some $$1\leq s\leq n+1$$, the hypersurface $$M$$ is of finite type at $$p$$, the map $$H$$ has generic rank $$n+1$$ and the germ at $$p$$ of the analytic variety $$W_H^{s }$$ has codimension at least 2.

### MSC:

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

### Keywords:

CR hypersurface; holomorphic mappings; CR transversality

### Citations:

Zbl 1244.32011; Zbl 1144.32005
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