Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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), 81-106
DOI: https://doi.org/10.1090/S1056-3911-02-00305-3
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 $\gamma : (\mathbb {C}^N\times \mathbb {C} ^{N-d},0)\to (\mathbb {C}^N,0)$ and its iterated Segre mappings at $0$, $v^j:(\mathbb {C}^{(N-d)j},0) \to (\mathbb {C}^N,0)$, $j\ge 1$. The Segre variety mapping $\gamma$ extends the notion of Segre varieties of a real-analytic 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 92093-0112
Email: sbaouendi@ucsd.edu

P. Ebenfelt
Affiliation: Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 92093-0112
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 92093-0112
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 98-01258. The second author is partially supported by a grant from the Swedish Natural Science Research Council.