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Mappings of real algebraic hypersurfaces. (English) Zbl 0869.14025

A smooth real hypersurface in \(\mathbb{C}^N\) is called algebraic if it is contained in the zero set of a nonzero real-valued polynomial in \(Z\) and \(\overline Z\). A hypersurface \(M\) in \(\mathbb{C}^N\) is holomorphically degenerate at a point \(p_0 \in M\) if there exists a nonzero germ of a holomorphic vector field tangent to \(M\) in a neighborhood of \(p_0\). The paper contains a complete characterization of algebraic hypersurfaces in \(\mathbb{C}^N\) such that any holomorphic map with non-vanishing Jacobian determinant between two such hypersurfaces is algebraic. Let \(M\) be a connected algebraic hypersurface in \(\mathbb{C}^N\), \(N>1\), and let \(p_0\) be a point of \(M\). Every biholomorphism defined in a neighborhood of \(p_0\) and mapping \(M\) to another algebraic hypersurface in \(\mathbb{C}^N\) is algebraic if and only if \(M\) is not holomorphically degenerate at any point. This is a generalization of a result of S. M. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)].
The authors also describe a relationship between essential finiteness and holomorphic nondegeneracy. If \(M\) is a connected real analytic hypersurface in \(\mathbb{C}^N\), then there exists a point of \(M\) in which \(M\) is essentially finite if and only if \(M\) is not holomorphically degenerate at any point in \(M\).

MSC:

14P99 Real algebraic and real-analytic geometry
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14J70 Hypersurfaces and algebraic geometry
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[1] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277 – 291. · Zbl 0172.05301 · doi:10.1007/BF01389777
[2] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23 – 58. · Zbl 0181.48802
[3] M. S. Baouendi, H. Jacobowitz, and F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985), no. 2, 365 – 400. · Zbl 0583.32021 · doi:10.2307/1971307
[4] M. S. Baouendi and Linda Preiss Rothschild, Germs of CR maps between real analytic hypersurfaces, Invent. Math. 93 (1988), no. 3, 481 – 500. · Zbl 0653.32020 · doi:10.1007/BF01410197
[5] 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. · Zbl 0702.32014
[6] M. S. Baouendi and Linda Preiss Rothschild, Images of real hypersurfaces under holomorphic mappings, J. Differential Geom. 36 (1992), no. 1, 75 – 88. · Zbl 0770.32009
[7] Eric Bedford and Steve Bell, Extension of proper holomorphic mappings past the boundary, Manuscripta Math. 50 (1985), 1 – 10. · Zbl 0583.32044 · doi:10.1007/BF01168824
[8] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. · Zbl 0041.05205
[9] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219 – 271. · Zbl 0302.32015 · doi:10.1007/BF02392146
[10] J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Math. Ann. 267 (1984), no. 2, 213 – 238. · Zbl 0518.12015 · doi:10.1007/BF01579200
[11] K. Diederich and J. E. Fornæss, Proper holomorphic mappings between real-analytic pseudoconvex domains in \?\(^{n}\), Math. Ann. 282 (1988), no. 4, 681 – 700. · Zbl 0661.32025 · doi:10.1007/BF01462892
[12] K. Diederich and S. M. Webster, A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), no. 4, 835 – 843. · Zbl 0451.32008
[13] Franc Forstnerič, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), no. 1, 31 – 61. · Zbl 0633.32017 · doi:10.1007/BF01394144
[14] Michael Freeman, Real submanifolds with degenerate Levi form, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 141 – 147.
[15] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. · Zbl 0141.08601
[16] Xiao Jun Huang, On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 433 – 463 (English, with English and French summaries). · Zbl 0803.32011
[17] J. J. Kohn, Boundary behavior of \? on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523 – 542. Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. · Zbl 0256.35060
[18] Katsumi Nomizu, Lie groups and differential geometry, The Mathematical Society of Japan, 1956. · Zbl 0071.15402
[19] H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo (2) 23 (1907), 185-220. · JFM 38.0459.02
[20] Nancy K. Stanton, Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. J. Math. 117 (1995), no. 1, 141 – 167. · Zbl 0826.32013 · doi:10.2307/2375039
[21] Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of \? complex variables, J. Math. Soc. Japan 14 (1962), 397 – 429. · Zbl 0113.06303 · doi:10.2969/jmsj/01440397
[22] S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), no. 1, 53 – 68. · Zbl 0348.32005 · doi:10.1007/BF01390203
[23] S. M. Webster, On mapping an \?-ball into an (\?+1)-ball in complex spaces, Pacific J. Math. 81 (1979), no. 1, 267 – 272. · Zbl 0379.32018
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