Dynamics of the Segre varieties of a real submanifold in complex space
Authors:
M. S. Baouendi, P. Ebenfelt and Linda Preiss Rothschild
Journal:
J. Algebraic Geom. 12 (2003), 81106
DOI:
https://doi.org/10.1090/S1056391102003053
Published electronically:
July 17, 2002
MathSciNet review:
1948686
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Abstract  References  Additional Information
Abstract: For a smooth (or formal) generic submanifold $M$ of real codimension $d$ in complex space $\mathbb {C}^N$ with $0\in M$, we introduce the notion of a formal Segre variety mapping $\gamma : (\mathbb {C}^N\times \mathbb {C} ^{Nd},0)\to (\mathbb {C}^N,0)$ and its iterated Segre mappings at $0$, $v^j:(\mathbb {C}^{(Nd)j},0) \to (\mathbb {C}^N,0)$, $j\ge 1$. The Segre variety mapping $\gamma$ extends the notion of Segre varieties of a realanalytic generic submanifold to the setting of smooth (or formal) submanifolds. One of the main results in this paper is that $M$ is of finite type (in the sense of Kohn and Bloom–Graham) at $0$ if and only if there exists $k_0\le d+1$ such that the (generic) rank of $v^{k_0}$ is $N$. More generally, we prove that $v^{k_0}$ parameterizes the local CR orbit of $M$ at $0$.

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Additional Information
M. S. Baouendi
Affiliation:
Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 920930112
Email:
sbaouendi@ucsd.edu
P. Ebenfelt
Affiliation:
Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 920930112
MR Author ID:
339422
Email:
pebenfel@ucsd.edu
Linda Preiss Rothschild
Affiliation:
Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 920930112
MR Author ID:
151000
Email:
lrothschild@ucsd.edu
Received by editor(s):
August 16, 2000
Published electronically:
July 17, 2002
Additional Notes:
The first and the third authors are partially supported by National Science Foundation grant DMS 9801258. The second author is partially supported by a grant from the Swedish Natural Science Research Council.