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**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.

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.

Reviewer: Laura Geatti (Roma)