Huang, Xiaojun On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. (English) Zbl 0803.32011 Ann. Inst. Fourier 44, No. 2, 433-463 (1994). We prove that if \(M_ 1\) and \(M_ 2\) are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if \(f\) is a holomorphic mapping defined near a neighborhood of \(M_ 1\) so that \(f(M_ 1) \subset M_ 2\), then \(f\) is also algebraic, i.e., the field generated by adding each element of \(f\) to the rational functions field is of finite extension. Results in this direction originate in the work of Poincaré; modern results were obtained by Webster in case \(M_ 1\) and \(M_ 2\) lie in the same complex space. When \(M_ 1\) and \(M_ 2\) are spheres, then the result is due to Webster, Faran, Cima-Suffridge, and Forstneric (in this specific case, \(f\) is rational, algebraic of degree 1).The proof of our result is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument can also be used to prove a reflection principle, which answers a problem of Forstneric and which in turn allows our main result to be stated for mappings that are holomorphic on one side and \(C^{k+1}\) smooth up to \(M_ 1\) where \(k\) is the codimension. Reviewer: X.Huang (St.Louis) Cited in 2 ReviewsCited in 34 Documents MSC: 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 14E05 Rational and birational maps 14H05 Algebraic functions and function fields in algebraic geometry 32J99 Compact analytic spaces Keywords:algebraic real hypersurfaces; holomorphic mapping; reflection principle × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [Al] , Holomorphic mappings from ball and polydisc, Math. Ann., 209 (1974), 245-256. · Zbl 0272.32006 [2] [BBR] , , and , Mappings of three-dimensional CR manifolds and their holomorphic extension, Duke Math. J., 56 (1988), 503-530. · Zbl 0655.32015 [3] [BR] and , Germs of CR maps between real analytic hypersurfaces, Invent. Math., 93 (1988), 481-500. · Zbl 0653.32020 [4] [Be] , Proper holomorphic mappings, Bull. Amer. Math. Soc., 10 (1984), 157-175. · Zbl 0534.32009 [5] [BN] and , Proper holomorphic mappings of complex spaces, EMS 69, Several Complex Variables VI (edited by W. Barth and R. Narasimhan), Springer-Verlag, 1990. · Zbl 0733.32021 [6] [BM] and , Several Complex Variables, Princeton University Press, 1948. · Zbl 0041.05205 [7] [CS1] and , A reflection principle with applications to proper holomorphic mappings, Math Ann., 265 (1983), 489-500. · Zbl 0525.32021 [8] [CKS] , , and , A reflection principle for proper holomorphic mappings of strictly pseudoconvex domains and applications, Math. Z., 186 (1984), 1-8. · Zbl 0518.32009 [9] [DF1] and , Proper holomorphic mappings between real-analytic domains in Cn, Math. Ann., 282 (1988), 681-700. · Zbl 0661.32025 [10] [DF2] and , Applications holomorphes propres entre domaines à bord analytique réel, C.R.A.S., Ser.I-Math., 307, No7 (1988), 321-324. · Zbl 0656.32013 [11] [Fa1] , A reflection principle for proper holomorphic mappings and geometric invariants, Math. Z., 203 (1990), 363-377. · Zbl 0664.32021 [12] [Fa2] , Maps from the two ball to the three ball, Invent Math., 68 (1982), 441-475. · Zbl 0519.32016 [13] [Fe] , The Bergman kernel and biholomorphic mappings pseudo-convex domains, Invent. Math., 26 (1974), 1-65. · Zbl 0289.32012 [14] [Fr1] , Extending proper holomorphic mappings of positive codimension, Invent. Math., 95 (1989), 31-62. · Zbl 0633.32017 [15] [Fr2] , A survey on proper holomorphic mappings, Proceeding of Year in SCVs at Mittag-Leffler Institute, Math. Notes 38, Princeton, NJ : Princeton University Press, 1992. [16] [Le] , On the boundary behavior of holomorphic mappings, Acad. Naz., Lincei, 3 (1977), 1-8. [17] [Kr] , Function Theory of Several Complex Variables, 2nd Ed., Wadsworth Publishing, Belmont, 1992. · Zbl 0776.32001 [18] [Pi] , On analytic continuation of biholomorphic mappings, Mat. USSR Sb., 105 (1978), 574-593. [19] [Po] , Les fonctions analytiques de deux variables et la représentation conforme, Ren. Cire. Mat. Palermo, II. Ser. 23 (1907), 185-220. · JFM 38.0459.02 [20] [Ta] , On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan, 14 (1962), 397-429. · Zbl 0113.06303 [21] [We1] , On the mapping problem for algebraic real hypersurfaces, Invent. Math., 43 (1977), 53-68. · Zbl 0348.32005 [22] [We2] , On mappings an (n + 1)-ball in the complex space, Pac. J. Math., 81 (1979), 267-272. · Zbl 0379.32018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.