CR transversality of holomorphic mappings between generic submanifolds in complex spaces
Authors:
Peter Ebenfelt and Duong Ngoc Son
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1729-1738
MSC (2010):
Primary 32H02, 32V20, 32V40
Posted:
September 9, 2011
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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
Bibliography
[A08] Geometric properties and related results for holomorphic Segre preserving maps . arXiv:0810.2570v1
[CR94] E.
M. Chirka and C.
Rea , Normal and tangent ranks of CR mappings , Duke Math. J.
76 (1994), no. 2, 417–431. MR 1302319
(95j:32041) , http://dx.doi.org/10.1215/S0012-7094-94-07614-X
[CR98] E.
M. Chirka and C.
Rea , Differentiable CR mappings and CR orbits , Duke Math. J.
94 (1998), no. 2, 325–340. MR 1638591
(99f:32015) , http://dx.doi.org/10.1215/S0012-7094-98-09415-7
[BER96] M.
S. Baouendi , P.
Ebenfelt , and L.
P. Rothschild , Algebraicity of holomorphic mappings between real
algebraic sets in 𝐶ⁿ , Acta Math. 177
(1996), no. 2, 225–273. MR 1440933
(99b:32030) , http://dx.doi.org/10.1007/BF02392622
[BER98] 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
(99i:32024)
[BER99a] 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
(2000b:32066)
[BER99b] 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
(2001b:32075) , http://dx.doi.org/10.1007/s002080050365
[BER03] 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
(2004f:32050) , http://dx.doi.org/10.1090/S1056-3911-02-00305-3
[BER07] 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
(2009a:32019)
[BHR95] 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
(97d:32025)
[BR90] 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
(91g:32032)
[BR93] 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
(95c:32017) , http://dx.doi.org/10.1007/BF01231291
[BB82] Eric
Bedford and Steve
Bell , Proper self-maps of weakly pseudoconvex domains , Math.
Ann. 261 (1982), no. 1, 47–49. MR 675205
(84c:32026) , http://dx.doi.org/10.1007/BF01456408
[ER06] Peter
Ebenfelt and Linda
P. Rothschild , Transversality of CR mappings , Amer. J. Math.
128 (2006), no. 5, 1313–1343. MR 2262176
(2008b:32024)
[Forn76] John
Erik Fornaess , Embedding strictly pseudoconvex domains in convex
domains , Amer. J. Math. 98 (1976), no. 2,
529–569. MR 0422683
(54 #10669)
[Forn78] John
Erik Fornaess , Biholomorphic mappings between weakly pseudoconvex
domains , Pacific J. Math. 74 (1978), no. 1,
63–65. MR
0481111 (58 #1257)
[GT83] 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
(86c:35035)
[HP96] Xiaojun
Huang and Yifei
Pan , Proper holomorphic mappings between real analytic domains in
𝐶ⁿ , Duke Math. J. 82 (1996),
no. 2, 437–446. MR 1387236
(97f:32036) , http://dx.doi.org/10.1215/S0012-7094-96-08219-8
[LM06] 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
(2007k:32049) , http://dx.doi.org/10.1007/s11425-006-2047-8
[P77] 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
(58 #1261)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
32H02 ,
32V20 ,
32V40
Retrieve articles in all journals
with MSC (2010):
32H02 ,
32V20 ,
32V40
Additional Information
Peter Ebenfelt
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email:
pebenfel@math.ucsd.edu
Duong Ngoc Son
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email:
snduong@math.ucsd.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11091-8
PII:
S 0002-9939(2011)11091-8
Received by editor(s):
September 25, 2010
Received by editor(s) in revised form:
January 25, 2011
Posted:
September 9, 2011
Additional Notes:
The first author was partly supported by NSF grant DMS-0701121.Communicated by:
Franc Forstneric
Article copyright:
© Copyright 2011 American Mathematical Society
