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

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.

### 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

Zbl 1105.32022
Full Text:

### References:

 [1] Angle, R., Geometric properties and related results for holomorphic Segre preserving maps. arXiv:0810.2570v1 · Zbl 1254.32004 [2] E. M. Chirka and C. Rea, Normal and tangent ranks of CR mappings, Duke Math. J. 76 (1994), no. 2, 417 – 431. · Zbl 0819.32008 [3] E. M. Chirka and C. Rea, Differentiable CR mappings and CR orbits, Duke Math. J. 94 (1998), no. 2, 325 – 340. · Zbl 0949.32016 [4] M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Algebraicity of holomorphic mappings between real algebraic sets in \?$$^{n}$$, Acta Math. 177 (1996), no. 2, 225 – 273. · Zbl 0890.32005 [5] M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, CR automorphisms of real analytic manifolds in complex space, Comm. Anal. Geom. 6 (1998), no. 2, 291 – 315. · Zbl 0982.32030 [6] M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. · Zbl 0944.32040 [7] M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann. 315 (1999), no. 2, 205 – 249. · Zbl 0942.32027 [8] M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Dynamics of the Segre varieties of a real submanifold in complex space, J. Algebraic Geom. 12 (2003), no. 1, 81 – 106. · Zbl 1037.32011 [9] M. S. Baouendi, Peter Ebenfelt, and Linda P. Rothschild, Transversality of holomorphic mappings between real hypersurfaces in different dimensions, Comm. Anal. Geom. 15 (2007), no. 3, 589 – 611. · Zbl 1144.32005 [10] M. S. Baouendi, Xiao Jun Huang, and Linda Preiss Rothschild, Nonvanishing of the differential of holomorphic mappings at boundary points, Math. Res. Lett. 2 (1995), no. 6, 737 – 750. · Zbl 0847.32018 [11] M. S. Baouendi and Linda Preiss Rothschild, Geometric properties of mappings between hypersurfaces in complex space, J. Differential Geom. 31 (1990), no. 2, 473 – 499. · Zbl 0702.32014 [12] M. S. Baouendi and Linda Preiss Rothschild, A generalized complex Hopf lemma and its applications to CR mappings, Invent. Math. 111 (1993), no. 2, 331 – 348. · Zbl 0781.32021 [13] Eric Bedford and Steve Bell, Proper self-maps of weakly pseudoconvex domains, Math. Ann. 261 (1982), no. 1, 47 – 49. · Zbl 0499.32016 [14] Peter Ebenfelt and Linda P. Rothschild, Transversality of CR mappings, Amer. J. Math. 128 (2006), no. 5, 1313 – 1343. · Zbl 1105.32022 [15] John Erik Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), no. 2, 529 – 569. · Zbl 0334.32020 [16] John Erik Fornaess, Biholomorphic mappings between weakly pseudoconvex domains, Pacific J. Math. 74 (1978), no. 1, 63 – 65. · Zbl 0353.32026 [17] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001 [18] Xiaojun Huang and Yifei Pan, Proper holomorphic mappings between real analytic domains in \?$$^{n}$$, Duke Math. J. 82 (1996), no. 2, 437 – 446. · Zbl 0853.32030 [19] Bernhard Lamel and Nordine Mir, Remarks on the rank properties of formal CR maps, Sci. China Ser. A 49 (2006), no. 11, 1477 – 1490. · Zbl 1112.32018 [20] S. I. Pinčuk, Analytic continuation of mappings along strictly pseudo-convex hypersurfaces, Dokl. Akad. Nauk SSSR 236 (1977), no. 3, 544 – 547 (Russian).
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