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Interplay Between CR Geometry and Algebraic Geometry

  • Stephen YauEmail author
  • Huaiqing Zuo
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)

Abstract

By a beautiful theorem of Harvey and Lawson, strongly pseudoconvex connected compact embeddable CR manifolds are the boundaries of subvarieties in \(\mathbb{C}^{N}\) with only normal isolated singularities. This leads to a natural question of how to determine the properties of the interior singularities from the CR manifolds and vice versa. In this paper, we give a survey on the interplay between CR Geometry and Algebraic Geometry for the last 30 years or so.

Keywords

Modulus Space Real Hypersurface Pseudoconvex Domain Complex Euclidean Space Real Analytic Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Akahori, A.: The New Approach to the Local Embedding Theorem of CR-Structures for n ≥ 4. Contemporary Mathematics, vol. 49, pp. 1–10. American Mathematical Society, Providence (1986)Google Scholar
  2. 2.
    Artin, M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 126–136 (1966)Google Scholar
  3. 3.
    Bland, J., Epstein, C.L.: Embeddable CR-structures and deformations of pseudoconvex surfaces Part I: formal deformations. J. Algebr. Geom. 5, 277–368 (1996)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Boutet de Monvel, L.: Intégration des éequations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz (1974–1975)Google Scholar
  5. 5.
    Burns, D., Epstein, C.L.: Embeddability for three dimensional CR-manifolds. J. Am. Math. Soc. 3, 809–841 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Burns, D., Shnider, S., Wells, R.O.: Deformations of strictly pseudoconvex domains. Invent. Math. 46, 237–253 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cartan, E.: La geometrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. IV. Ser. XI, 17–90 (1932)Google Scholar
  8. 8.
    Chern, S.S., Moser, J.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Du, R., Gao, Y., Yau, S.S.-T.: Explicit construction of moduli space of bounded complete Reinhardt domains in \(\mathbb{C}^{n}\). Commun. Anal. Geom. 18(3), 601–626 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Du, R., Yau, S.S.-T.: Higher order Bergman functions and explicit construction of moduli space for complete Reinhardt domains. J. Differ. Geom. 82, 567–610 (2009)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Du, R., Yau, S.S.-T.: Kohn-Rossi cohomology and its application to the complex Plateau problem, III. J. Differ. Geom. 90, 251–266 (2011)MathSciNetGoogle Scholar
  12. 12.
    Falland, G.B., Stein, E.M.: Estimates for the \(\overline{\partial }_{6}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)CrossRefGoogle Scholar
  13. 13.
    Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math. 31, 131–262 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Göbel, M.: Computing bases for rings of permutation-invariant polynomials. J. Symb. Comput. 19(4), 285–291 (1995)zbMATHCrossRefGoogle Scholar
  16. 16.
    Harvey, R., Lawson, B.: On boundaries of complex analytic varieties I. Ann. Math. 102, 233–290 (1975)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Harvey, R., Lawson, B.: Addendum to Theorem 10.4 in: boundaries of complex analytic varieties. arXiv: math/0002195v1 [math CV], 23 Feb 2000Google Scholar
  18. 18.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II. Ann. Math. 79, 109–326 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jacobowitz, H., Treves, F.: Non-realizable CR-structures. Invent. Math. 66, 231–249 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Jacobowitz, H., Treves, F.: Aberrant CR-structures. Hokkaiko Math. J. 12, 276–292 (1983)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of the Conference on Complex Manifolds, Minneapolis. Springer, New York (1965)CrossRefGoogle Scholar
  22. 22.
    Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifolds. Ann. Math. 81, 451–472 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Kuranishi, M.: Strongly pseudoconvex CR structures over small balls I, II, III. Ann. Math. (2) 115, 451–500 (1982); ibid 116 (1982), 1–64; ibid 116 (1982), 249–330Google Scholar
  24. 24.
    Kuranishi, M.: CR geometry and Cartan geometry. Forum math. 7, 147–206 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kuranishi, M.: An approach to the Cartan geometry II: CR manifolds. In: Complex Analysis in Several Variables – Memorial Conference of Kiyoshi Oha’s Centennial Birthday, Kyoto/Nara 2001. Advanced Studies in Pure Mathematics, vol. 42, pp. 165–187 (2004)MathSciNetGoogle Scholar
  26. 26.
    Lawson, H.B., Yau, S.S.-T.: Holomorphic symmetries. Ann. Sci. École Norm. Sup. (4) 20(4), 557–577 (1987)Google Scholar
  27. 27.
    Lempert, L.: Holomorphic invariants, normal forms, and the moduli space of convex domains. Ann. Math. 128, 43–78 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Lempert, L.: Embeddings of three dimensional Cauchy Riemann manifolds. Math. Ann. 300, 1–15 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Luk, H.S., Yau, S.S.-T.: Obstructions to embedding of real (2n − 1)-dimensional compact CR manifolds in \(\mathbb{C}^{n+1}\). Proc. Symp. Pure Math. 52, part 3, 261–276 (1991)Google Scholar
  30. 30.
    Luk, H.S., Yau, S.S.-T.: Complete algebraic CR invariants of real codimension 3 strongly pseudoconvex CR manifold. In: Singularities and Complex Geometry. Studies in Advanced Mathematics, vol. 5, pp. 175–182. AMS, Providence/IP, New York (1997)Google Scholar
  31. 31.
    Luk, H.S., Yau, S.S.-T.: Kohn-Rossi cohomology and its application to the complex Plateau problem II. J. Differ. Geom. 77(1), 135–148 (2007)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Luk, H.S., Yau, S.S.-T., Yu, Y.: Algebraic classification and obstruction to embedding of strongly pseudoconvex compact 3-dimensional CR manifolds in \(\mathbb{C}^{3}\). Math. Nachr. 170, 183–200 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Luk, H.S., Yau, S.S.-T.: Invariant Kohn-Rossi cohomology and obstructions to embedding of compact real (2n − 1)-dimensional CR manifolds in \(\mathbb{C}^{N}\). J. Math. Soc. Jpn. 48, 61–68 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Mather, J., Yau, S.S.-T.: Classification of isolated hypersurfaces singularities by their moduli algebras. Invent. Math. 69, 243–251 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Nirenberg, L.: On a question of Hans Lewy. Russ. Math. Surv. 29, 251–262 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Poincaré, H.: Les fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo 23, 185–220 (1907)zbMATHCrossRefGoogle Scholar
  37. 37.
    Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary. In: Aeppli, A., Calabi, E., Röhrl, H. (eds.) Proceedings of the Conference on Complex Analysis, Minneapolis 1964. Springer, Berlin/Heidelberg/New York (1965)Google Scholar
  38. 38.
    Siu, Y.-T.: Analytic sheaves of local cohomology. Trans. Am. Math. Soc. 148, 347–366 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Straten, D.V., Steenbrink, J.: Extendability of holomorphic differential forms near isolated hypersurface singularities. Abh. Math. Sem. Univ. Hambg. 55, 97–110 (1985)zbMATHCrossRefGoogle Scholar
  40. 40.
    Sunada, T.: Holomorphic equivalence problem for bounded Reinhardt domains. Math. Ann. 235, 111–128 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables. J. Math. Soc. Jpn. 14, 397–429 (1962)zbMATHCrossRefGoogle Scholar
  42. 42.
    Tanaka, N.: A Differential Geometric Study on Strongly Pseudoconvex Manifolds. Lectures in Mathematics, Kyoto University, No. 9. Kinokuniya Book-Store Co., Ltd., Kyoto (1975)Google Scholar
  43. 43.
    Tu, Y.-C., Yau, S.S.-T., Zuo, H.Q.: Nonconstant CR morphisms between compact strongly pseudoconvex CR manifolds and étale covering between resolutions of isolated singularities. J. Differ. Geom. 95, 337–354 (2013)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Webster, S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Xu, Y.-J., Yau, S.S.-T.: Durfee conjecture and coordinate free characterization of homogeneous singularities. J. Differ. Geom. 37, 375–396 (1993)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Yau, S.S.-T.: Kohn-Rossi cohomology and its application to the complex Plateau problem, I. Ann. Math. 113, 67–110 (1981)zbMATHCrossRefGoogle Scholar
  47. 47.
    Yau, S.S.-T.: Various numerical invariants for isolated singularities. Am. J. Math. 104(5), 1063–1110 (1982)zbMATHCrossRefGoogle Scholar
  48. 48.
    Yau, S.S.-T.: Singularities defined by \(sl(2, \mathbb{C})\) invariant polynomials and solvability of Lie algebras arising from isolated singularities. Am. J. Math. 108, 1215–1240 (1986)zbMATHCrossRefGoogle Scholar
  49. 49.
    Yau, S.S.-T.: Solvability of the Lie algebras arising from singularities and non-siolatedness of the singularities defined by the invariant polynomials of \(sl(2, \mathbb{C})\). Am. J. Math. 113, 773–778 (1991)zbMATHCrossRefGoogle Scholar
  50. 50.
    Yau, S.S.-T.: Global invariants for strongly pseudoconvex varieties with isolated singularities: Bergman functions. Math. Res. Lett. 11, 809–832 (2004)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingP. R. China
  2. 2.Yau Mathematical CenterTsinghua UniversityBeijingP. R. China

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