CR transversality of holomorphic mappings between generic submanifolds in complex spaces

Ebenfelt, Peter; Son, Duong

doi:10.1090/S0002-9939-2011-11091-8

CR transversality of holomorphic mappings between generic submanifolds in complex spaces
HTML articles powered by AMS MathViewer

by Peter Ebenfelt and Duong Ngoc Son PDF

Proc. Amer. Math. Soc. 140 (2012), 1729-1738 Request permission

Abstract:

We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in 2006. Another one of our main results, in the special setting of hypersurfaces, was also explicitly stated as a conjecture by B. Lamel and N. Mir in a paper from 2006.

References

Angle, R., Geometric properties and related results for holomorphic Segre preserving maps. arXiv:0810.2570v1
E. M. Chirka and C. Rea, Normal and tangent ranks of CR mappings, Duke Math. J. 76 (1994), no. 2, 417–431. MR 1302319, DOI 10.1215/S0012-7094-94-07614-X
E. M. Chirka and C. Rea, Differentiable CR mappings and CR orbits, Duke Math. J. 94 (1998), no. 2, 325–340. MR 1638591, DOI 10.1215/S0012-7094-98-09415-7
M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Algebraicity of holomorphic mappings between real algebraic sets in $\textbf {C}^n$, Acta Math. 177 (1996), no. 2, 225–273. MR 1440933, DOI 10.1007/BF02392622
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. MR 1651418, DOI 10.4310/CAG.1998.v6.n2.a3
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. MR 1668103, DOI 10.1515/9781400883967
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. MR 1721797, DOI 10.1007/s002080050365
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. MR 1948686, DOI 10.1090/S1056-3911-02-00305-3
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. MR 2379806
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. MR 1362966, DOI 10.4310/MRL.1995.v2.n6.a6
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. MR 1037411
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. MR 1198813, DOI 10.1007/BF01231291
Eric Bedford and Steve Bell, Proper self-maps of weakly pseudoconvex domains, Math. Ann. 261 (1982), no. 1, 47–49. MR 675205, DOI 10.1007/BF01456408
Peter Ebenfelt and Linda P. Rothschild, Transversality of CR mappings, Amer. J. Math. 128 (2006), no. 5, 1313–1343. MR 2262176
John Erik Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), no. 2, 529–569. MR 422683, DOI 10.2307/2373900
John Erik Fornaess, Biholomorphic mappings between weakly pseudoconvex domains, Pacific J. Math. 74 (1978), no. 1, 63–65. MR 481111
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. MR 737190, DOI 10.1007/978-3-642-61798-0
Xiaojun Huang and Yifei Pan, Proper holomorphic mappings between real analytic domains in $\textbf {C}^n$, Duke Math. J. 82 (1996), no. 2, 437–446. MR 1387236, DOI 10.1215/S0012-7094-96-08219-8
Bernhard Lamel and Nordine Mir, Remarks on the rank properties of formal CR maps, Sci. China Ser. A 49 (2006), no. 11, 1477–1490. MR 2288209, DOI 10.1007/s11425-006-2047-8
S. I. Pinčuk, Analytic continuation of mappings along strictly pseudo-convex hypersurfaces, Dokl. Akad. Nauk SSSR 236 (1977), no. 3, 544–547 (Russian). MR 0481115