Hostname: page-component-84b7d79bbc-dwq4g Total loading time: 0 Render date: 2024-07-28T02:07:29.145Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on the Analogue of the Bogomolov Type Theorem on Deformations of CR-Structures

Published online by Cambridge University Press:  20 November 2018

Takao Akahori  and
Kimio Miyajima
Takao Akahori
Affiliation:
Department of Mathematics Himeji Institute of Technology Himeji, 671-22 Japan
Kimio Miyajima
Affiliation:
Department of Mathematics Kagoshima University Kagoshima, 890 Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (M, °T″) be a compact strongly pseudo-convex CR-manifold with trivial canonical line bundle. Then, in [A-M2], a weak version of the Bogomolov type theorem for deformations of CR-structures was shown by an analogy of the Tian- Todorov method. In this paper, we show that: in the very strict sense, there is a counterexample.

Keywords

32G07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 1 , 01 March 1994 , pp. 8 - 12
Copyright
Copyright © Canadian Mathematical Society 1994

References

[Al] Akahori, T., The new estimate for the subbundles Ej and its application to the deformation of the boundaries of strongly pseudo-convex domains, Invent. Math. 63(1981 ), 311–334.Google Scholar
[A2] Akahori, T., The new Neumann operator associated with deformations of strongly pseudo-convex domains and its application to deformation theory, Invent. Math. 68(1982), 317–352.Google Scholar
[A-Ml] Akahori, T. and Miyajima, K., Complex analytic construction of the Kuranishi family on a normal strongly pseudo-convex manifold. II, Publ. Res. Inst. Math. Sci., Kyoto Univ. 16(1980), 811–834.Google Scholar
[A-M2] Akahori, T. and Miyajima, K., An analogy of Tian-Todorov theorem on deformations of CR-structures, Compositio Mathematica 85(1993), 57–85.Google Scholar
[A-M31 Akahori, T. and Miyajima, K., On the CR-version of the Tian-Todorov lemma, preprint, 1992.Google Scholar
[B] Bogomolov, F. A., Hamiltonian Kdhler manifolds, Dokl. Akad. Nauk SSSR 243(1978), 1462–1465.Google Scholar
[B-M] Buchweitz, R. O. and Millson, J. J., CR-Geometry and Deformations of Isolated Singularities, preprint.Google Scholar
[H] Horikawa, E., On deformations of quintic surfaces, Invent. Math. 31(1975), 43–85.Google Scholar
[K] Kuranishi, M., Deformations of isolated singularities and 9£, Columbia University, 1972, preprint.Google Scholar
[L] Lee, J. M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110(1988), 157–178.Google Scholar
[Ml] Miyajima, K., Completion of AkahorVs construction of the versai family ofstrongly pseudo-convex CR structures, Trans. Amer. Math. Soc. (1980), 162-172.Google Scholar
[M2] Miyajima, K., 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.Google Scholar
[S] Schlessinger, M., On rigid singularities. In: Conference on Complex Analysis, Rice University Studies ( 1 ) 59(1972) 147–162.Google Scholar
[T] Tanaka, N., A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Kyoto Univ.Google Scholar
[Ti] Tian, C., Smoothness of the universal deformation space of compact Calahi-Yau manifolds and its Petersson-Weil metric, Mathematical Aspects of String Theory, (éd. S. T. Yau), World Scientific, 1987, 629–646.Google Scholar
[To] Todorov, A. N., The Weil-Petersson geometry 1 of'SU(≥3) (Calahi-Yau) Manifolds I, Comm. Math. Phys. 126(1989), 325–346.Google Scholar