The Michigan Mathematical Journal

Deformation theory of 5-dimentional CR structures and the Rumin complex

Takao Akahori, Peter M. Garfield, and John M. Lee

Full-text: Open access

Article information

Source
Michigan Math. J., Volume 50, Issue 3 (2002), 517-550.

Dates
First available in Project Euclid: 4 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1039029981

Digital Object Identifier
doi:10.1307/mmj/1039029981

Mathematical Reviews number (MathSciNet)
MR2770

Zentralblatt MATH identifier
1065.32018

Subjects
Primary: 32G07: Deformations of special (e.g. CR) structures
Secondary: 32S30: Deformations of singularities; vanishing cycles [See also 14B07] 32V20: Analysis on CR manifolds

Citation

Akahori, Takao; Garfield, Peter M.; Lee, John M. Deformation theory of 5-dimentional CR structures and the Rumin complex. Michigan Math. J. 50 (2002), no. 3, 517--550. doi:10.1307/mmj/1039029981. https://projecteuclid.org/euclid.mmj/1039029981


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References

  • T. Akahori, Intrinsic formula for Kuranishi's $\bar\partial_b^\varphi,$ Publ. Res. Inst. Math. Sci. 14 (1978), 615--641.
  • ------, Complex analytic construction of the Kuranishi family on a normal strongly pseudoconvex manifold, Publ. Res. Inst. Math. Sci. 14 (1978), 789--847.
  • ------, The new estimate for the subbundles $E_j$ and its application to the deformation of the boundaries of strongly pseudoconvex domains, Invent. Math. 63 (1981), 311--334.
  • ------, The new Neumann operator associated with deformations of strongly pseudoconvex domains and its application to deformation theory, Invent. Math. 68 (1982), 317--352.
  • ------, Complex analytic construction of the Kuranishi family on a normal strongly pseudoconvex manifold with real dimension 5, Manuscripta Math. 63 (1989), 29--43.
  • ------, A mixed Hodge structure on a CR manifold, MSRI preprint 1996-026.
  • T. Akahori and K. Miyajima, Complex analytic construction of the Kuranishi family on a normal strongly pseudoconvex manifold II, Publ. Res. Inst. Math. Sci. 16 (1980), 811--834.
  • ------, An analogy of Tian--Todorov theorem on deformations of CR-structures, Compositio Math. 85 (1993), 57--85.
  • ------, A note on the analogue of the Bogomolov type theorem on deformations of CR-structures, Canad. Math. Bull. 37 (1994), 8--12.
  • J. Bland and C. L. Epstein, Embeddable CR-structures and deformations of pseudoconvex surfaces, part I: formal deformations, J. Algebraic Geom. 5 (1996), 277--368.
  • R. Buchweitz and J. Millson, CR-geometry and deformations of isolated singularities, Mem. Amer. Math. Soc. 125 (1997).
  • J. H. Cheng and J. M. Lee, A local slice theorem for 3-dimensional CR structures, Amer. J. Math. 117 (1995), 1249--1298.
  • I. F. Donin, Complete families of deformations of germs of complex spaces, Math. Sb. (N.S.) 89 (1972), 390--399.
  • P. M. Garfield, The Rumin complex on CR manifolds, Ph.D. dissertation, University of Washington, December 2001.
  • H. Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171--198.
  • M. Kuranishi, Application of $\bar\partial_b$ to deformation of isolated singularities, Proc. Sympos. Pure Math., 30, pp. 97--106, Amer. Math. Soc., Providence, RI, 1977.
  • K. Miyajima, Deformations of a complex manifold near a strongly pseudo-convex real hypersurface and a realization of Kuranishi family of strongly pseudo-convex CR structures, Math. Z. 205 (1990), 593--602.
  • ------, Deformations of strongly pseudo-convex CR structures and deformations of normal isolated singularities, Complex analysis (Wuppertal, 1991), Aspects Math., E17, pp. 200--204, Vieweg, Braunschweig, 1991.
  • ------, CR construction of the flat deformations of normal isolated singularities, J. Algebraic Geom. 8 (1999), 403--470.
  • M. Rumin, Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994), 281--330.
  • N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, 9, Department of Mathematics, Kyoto Univ., Kinokuniya, Tokyo, 1975.
  • G. N. Tjurina, Locally semi-universal flat deformations of isolated singularities of complex spaces, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1026--1058.
  • S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25--41.