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 $0$.

[A69]artin2 Artin, M.: Algebraic approximation of structures over complete local rings. *Inst. Hautes Etudes Sci. Publ. Math.* **36**, 23–58, (1969).
[A68]artin1 Artin, M.: On the solutions of analytic equations. *Invent. Math.* **5**, 277–291, (1968).
[BER99a]BER Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: *Real Submanifolds in Complex Space and Their Mappings*. Princeton Math. Series **47**, Princeton Univ. Press, 1999.
[BER99b]MA Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Rational dependence of smooth and analytic CR mappings on their jets. *Math. Ann.* **315**, 205–249, (1999).
[BER99c]JAMS Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Convergence and finite determination of formal CR mappings, *J. Amer. Math. Soc.*, vol. 13, pp. 697–723, (2000).
[BER98]CAG Baouendi, M. S.; Ebenfelt, P.; Rothschild, L.P.: CR automorphisms of real analytic manifolds in complex space. *Comm. Anal. Geom. 6*, 291–315, (1998).
[BER97]AsianBaouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Parametrization of local biholomorphisms of real-analytic hypersurfaces, *Asian J. Math.* **1**, 1–16, (1997).
[BER96]Acta Baouendi, M.S.; Ebenfelt, P. and Rothschild, L.P.: Algebraicity of holomorphic mappings between real algebraic sets in $\mathbb {C}^n$. *Acta Math.* **177**, 225–273, (1996).
[BHR96]BHR Baouendi, M.S.; Huang, X.; and Rothschild, L.P.: Regularity of CR mappings between algebraic hypersurfaces. *Invent. Math.* **125**, 13–36, (1996).
[BJT85]BJT Baouendi, M. S.; Jacobowitz, H.; and Treves, F.: On the analyticity of CR mappings. *Ann. of Math.* **122**, 365–400, (1985).
[BR88]BR Baouendi, M. S.; Rothschild, L. P.: Germs of CR maps between real analytic hypersurfaces. *Invent. Math.* **93**, 481–500, (1988).
[BRZ00]BRZ Baouendi, M.S.; Rothschild, L.P. and Zaitsev, D. : Equivalences of real submanifolds in complex space, J. Differential Geom. (to appear) http://xxx.lanl.gov/abs/math.CV/0002186.
[BG77]BG Bloom, T.; and Graham, I.: On type conditions for generic real submanifolds of $\mathbb C^n$. *Invent. Math.* **40**, 217–243, (1977).
[CNSW99]CNSW Christ, M.; Nagel, A.; Stein, E.M.; Wainger, S.: Singular and maximal Radon transforms: analysis and geometry. *Ann. of Math.* **150**, 489–577, (1999).
[DF88]DF Diederich, K. and Fornaess, J. E.: Proper holomorphic mappings between real-analytic pseudoconvex domains in ${C}^ n$. *Math. Ann.* **282**, 681–700, (1988).
[DW80]DW Diederich, K. and Webster, S. M.: A reflection principle for degenerate real hypersurfaces. *Duke Math. J.* **47**, 835–843, (1980).
[F89]Forst Forstnerič, F: Extending proper holomorphic mappings of positive codimension. *Invent. Math.* **95**, 31–61, (1989).
[H94]Huang Huang, X.: On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. *Ann. Inst. Fourier (Grenoble)* **44**, 433–463, (1994).
[J86]John John, F: *Partial Differential Equations.* (Fourth Edition, second printing), Springer-Verlag, Series in Applied Mathematical Sciences, 1986.
[K72]Kohn Kohn, J.J.: Boundary behavior of $\bar \partial$ on weakly pseudo-convex manifolds of dimension two. *J. Differ. Geom.* **6**, 523–542, (1972).
[Me99]Merker Merker, J.: Vector field construction of Segre sets using flows of vector fields. http://xxx.lanl.gov/abs/math.CV/9901010, (1999).
[Mi00a]Mirh Mir, N.: Formal biholomorphic maps of real analytic hypersurfaces. *Math. Research Lett.* **7**, 343–359, (2000).
[Mi00b]Mirg Mir, N.: On the convergence of formal mappings. *Comm. Anal. Geom.* (to appear) (2000).
[N66]Nag Nagano., T.: Linear differential systems with singularities and an application to transitive lie algebras. *J. Math. Soc. Japan* **18**, 398–404, (1966).
[W77]Webster Webster, S.M.: On the mapping problem for algebraic real hypersurfaces. *Invent. Math.* **43**, 53–68, (1977).
[Z99]Z2 Zaitsev, D.: Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces. *Acta Math.*, **183**, 273–305, (1999).
[Z97]Z1 Zaitsev, D.: Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets. *Math. Res. Lett.*, **4**, 823–842, (1997).

[A69]artin2 Artin, M.: Algebraic approximation of structures over complete local rings. *Inst. Hautes Etudes Sci. Publ. Math.* **36**, 23–58, (1969).
[A68]artin1 Artin, M.: On the solutions of analytic equations. *Invent. Math.* **5**, 277–291, (1968).
[BER99a]BER Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: *Real Submanifolds in Complex Space and Their Mappings*. Princeton Math. Series **47**, Princeton Univ. Press, 1999.
[BER99b]MA Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Rational dependence of smooth and analytic CR mappings on their jets. *Math. Ann.* **315**, 205–249, (1999).
[BER99c]JAMS Baouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Convergence and finite determination of formal CR mappings, *J. Amer. Math. Soc.*, vol. 13, pp. 697–723, (2000).
[BER98]CAG Baouendi, M. S.; Ebenfelt, P.; Rothschild, L.P.: CR automorphisms of real analytic manifolds in complex space. *Comm. Anal. Geom. 6*, 291–315, (1998).
[BER97]AsianBaouendi, M.S.; Ebenfelt, P.; and Rothschild, L.P.: Parametrization of local biholomorphisms of real-analytic hypersurfaces, *Asian J. Math.* **1**, 1–16, (1997).
[BER96]Acta Baouendi, M.S.; Ebenfelt, P. and Rothschild, L.P.: Algebraicity of holomorphic mappings between real algebraic sets in $\mathbb {C}^n$. *Acta Math.* **177**, 225–273, (1996).
[BHR96]BHR Baouendi, M.S.; Huang, X.; and Rothschild, L.P.: Regularity of CR mappings between algebraic hypersurfaces. *Invent. Math.* **125**, 13–36, (1996).
[BJT85]BJT Baouendi, M. S.; Jacobowitz, H.; and Treves, F.: On the analyticity of CR mappings. *Ann. of Math.* **122**, 365–400, (1985).
[BR88]BR Baouendi, M. S.; Rothschild, L. P.: Germs of CR maps between real analytic hypersurfaces. *Invent. Math.* **93**, 481–500, (1988).
[BRZ00]BRZ Baouendi, M.S.; Rothschild, L.P. and Zaitsev, D. : Equivalences of real submanifolds in complex space, J. Differential Geom. (to appear) http://xxx.lanl.gov/abs/math.CV/0002186.
[BG77]BG Bloom, T.; and Graham, I.: On type conditions for generic real submanifolds of $\mathbb C^n$. *Invent. Math.* **40**, 217–243, (1977).
[CNSW99]CNSW Christ, M.; Nagel, A.; Stein, E.M.; Wainger, S.: Singular and maximal Radon transforms: analysis and geometry. *Ann. of Math.* **150**, 489–577, (1999).
[DF88]DF Diederich, K. and Fornaess, J. E.: Proper holomorphic mappings between real-analytic pseudoconvex domains in ${C}^ n$. *Math. Ann.* **282**, 681–700, (1988).
[DW80]DW Diederich, K. and Webster, S. M.: A reflection principle for degenerate real hypersurfaces. *Duke Math. J.* **47**, 835–843, (1980).
[F89]Forst Forstnerič, F: Extending proper holomorphic mappings of positive codimension. *Invent. Math.* **95**, 31–61, (1989).
[H94]Huang Huang, X.: On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. *Ann. Inst. Fourier (Grenoble)* **44**, 433–463, (1994).
[J86]John John, F: *Partial Differential Equations.* (Fourth Edition, second printing), Springer-Verlag, Series in Applied Mathematical Sciences, 1986.
[K72]Kohn Kohn, J.J.: Boundary behavior of $\bar \partial$ on weakly pseudo-convex manifolds of dimension two. *J. Differ. Geom.* **6**, 523–542, (1972).
[Me99]Merker Merker, J.: Vector field construction of Segre sets using flows of vector fields. http://xxx.lanl.gov/abs/math.CV/9901010, (1999).
[Mi00a]Mirh Mir, N.: Formal biholomorphic maps of real analytic hypersurfaces. *Math. Research Lett.* **7**, 343–359, (2000).
[Mi00b]Mirg Mir, N.: On the convergence of formal mappings. *Comm. Anal. Geom.* (to appear) (2000).
[N66]Nag Nagano., T.: Linear differential systems with singularities and an application to transitive lie algebras. *J. Math. Soc. Japan* **18**, 398–404, (1966).
[W77]Webster Webster, S.M.: On the mapping problem for algebraic real hypersurfaces. *Invent. Math.* **43**, 53–68, (1977).
[Z99]Z2 Zaitsev, D.: Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces. *Acta Math.*, **183**, 273–305, (1999).
[Z97]Z1 Zaitsev, D.: Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets. *Math. Res. Lett.*, **4**, 823–842, (1997).

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.