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Preimages of CR submanifolds with generic holomorphic maps. (English) Zbl 1187.32028

Let \(M \subset \mathbb{C}^{m}\) be a smooth \(CR\) manifold of codimension \(d\) and of \(CR\) dimension \(d_{1}\), let \(f :\mathbb{C}^{n} \rightarrow \mathbb{C}^{m}\) be a holomorphic map, let \(M' = f^{-1}(M)\) and denote by \(\Sigma'\) the set of points \(x \in M'\) where (the germ of) \(f\) is not \(CR\) transversal to \(M\).
If \(f\) is generic, the author investigates the structure of \(\Sigma '\) and shows that it is a smooth subvariety of \(\mathbb{C}^{n}\) of real codimension \( 2(n-m+d_{1} +1)+d\) if \(n+d_{1} \geq m\) or it equals \(M'\) if \(n+d_{1} < m\). Moreover, he describes an interesting Whitney stratification of \(\Sigma '\). Further, he describes the structure of \(M'\) by analyzing what happens in \(M' \setminus \Sigma'\) and in each subset of the stratification of \(\Sigma '\). He also shows that if \(n+d_{1} \geq m\) and \( 2(n-m+d_{1} +1)+d>n\) then a generic holomorphic map is \(CR\) transversal to \(M\). Preliminary statements, necessary for the understanding of the subject are contained in the paper. Interesting examples and comments are also presented.

MSC:

32V05 CR structures, CR operators, and generalizations
32V10 CR functions
32V40 Real submanifolds in complex manifolds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32E10 Stein spaces
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[1] R. Abraham, Transversality in manifolds of mappings, Bull. Amer. Math. Soc. 69 (1963), 470–474. · Zbl 0171.44501 · doi:10.1090/S0002-9904-1963-10969-6
[2] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, NJ, 1999. · Zbl 0944.32040
[3] M. S. Baouendi, L. P. Rothschild and D. Zaitsev, Points in general position in real-analytic submanifolds in \(\mathbb{C}\)n and applications, in Complex Analysis and Geometry (Columbus, OH, 1999), de Gruyter, Berlin, 2001, pp. 1–20. · Zbl 1023.32023
[4] M. S. Baouendi, L. P. Rothschild and D. Zaitsev, Equivalences of real submanifolds in complex space, J. Differential Geometry 59 (2001), 301–351. · Zbl 1037.32030
[5] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, CRC Press, Boca Raton, FL, 1991. · Zbl 0760.32001
[6] E. M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, Dordrecht, 1989. · Zbl 0683.32002
[7] P. Ebenfelt and L. P. Rothschild, Transversality of CR mappings, Amer. J. Math. 128 (2006), 1313–1343. · Zbl 1105.32022 · doi:10.1353/ajm.2006.0039
[8] F. Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), 239–270. · Zbl 1171.32303 · doi:10.1353/ajm.2006.0005
[9] P. Griffiths and J. Harris, Principles of Algebraic Geometry, JohnWiley & Sons, Inc., New York, 1994. · Zbl 0836.14001
[10] M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, Berlin, 1988.
[11] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974.
[12] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. · Zbl 0141.08601
[13] S. Kaliman and M. Zaidenberg, A transversality theorem for holomorphic mappings and stability of Eisenman-Kobayashi measures, Trans. Amer. Math. Soc. 348 (1996), 661–672. · Zbl 0851.32018 · doi:10.1090/S0002-9947-96-01482-1
[14] B. Lamel and N. Mir, Remarks on the rank properties of formal CR maps, Sci. China Ser. A 49 (2006), 1477–1490. · Zbl 1112.32018 · doi:10.1007/s11425-006-2047-8
[15] H. Whitney, Local properties of analytic varieties, in Differential and Combinatorial Topology, A Symposium in Honor of Marston Morse, Princeton University Press, Princeton, NJ, 1965, pp. 205–244. · Zbl 0129.39402
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