Abstract
We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstnerič.
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Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Math. Series, vol. 47. Princeton University Press, Princeton (1999)
Baouendi M.S., Ebenfelt P., Rothschild L.P.: Local geometric properties of real submanifolds in complex space. Bull. Am. Math. Soc. (N.S.) 37(3), 309–336 (2000)
Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Transversality of holomorphic mappings between real hypersurfaces in different dimensions, preprint (2007). http://arxiv.org/abs/math.CV/0701432
Baouendi, M.S., Ebenfelt, P., Huang, X.: On CR embeddings into hyperquadrics of Levi nondegenerate hypersurfaces with low CR complexity, in preparation
Chern S.S, Moser J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)
D’Angelo, J.: Intersection theory and the \({\bar \partial}\) -Neumann problem. Complex analysis of several variables (Madison, 1982), pp. 51–58. In: Proceedings of Symposium on Pure Mathematics, vol. 41. American Mathematical Society, Providence (1984)
D’Angelo J.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)
Ebenfelt P., Rothschild L.P.: Transversality of CR mappings. Am. J. Math. 128(5), 1313–1343 (2006)
Faran J.J.: The nonimbeddability of real hypersurfaces in spheres. Proc. Am. Math. Soc. 103(3), 902–904 (1988)
Forstnerič F.: Embedding strictly pseudoconvex domains into balls. Trans. Am. Math. Soc. 295(1), 347–368 (1986)
Forstnerič F.: Most real analytic Cauchy-Riemann manifolds are nonalgebraizable. Manuscripta Math. 115(4), 489–494 (2004)
Gaussier H., Merker J.: Nonalgebraizable real analytic tubes in \({\mathbb {C}^n}\). Math. Z. 247(2), 337–383 (2004)
Huang X., Ji S., Yau S.S.T.: An example of a real analytic strongly pseudoconvex hypersurface which is not holomorphically equivalent to any algebraic hypersurface. Ark. Mat. 39(1), 75–93 (2001)
Lempert L.: Imbedding strictly pseudoconvex domains into a ball. Am. J. Math. 104(4), 901–904 (1982)
Lempert L.: Imbedding Cauchy–Riemann manifolds into a sphere. Int. J. Math. 1(1), 91–108 (1990)
Mumford D.: The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1988)
Webster S.M.: On the mapping problem for algebraic real hypersurfaces. Invent. Math. 43, 53–68 (1977)
Webster S.M.: Some birational invariants for algebraic real hypersurfaces. Duke Math. J. 45(1), 39–46 (1978)
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The author was supported in part by the RCBS grant of Trinity College Dublin and by the Science Foundation Ireland.
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Zaitsev, D. Obstructions to embeddability into hyperquadrics and explicit examples. Math. Ann. 342, 695–726 (2008). https://doi.org/10.1007/s00208-008-0253-0
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DOI: https://doi.org/10.1007/s00208-008-0253-0