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A note on the Bogomolov-type smoothness
on deformations of the regular parts
of isolated singularities

Author: Kimio Miyajima
Journal: Proc. Amer. Math. Soc. 125 (1997), 485-492
MSC (1991): Primary 32G05; Secondary 14B07, 13D10
MathSciNet review: 1363432
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Abstract | References | Similar Articles | Additional Information

Abstract: We apply the Tian-Todorov method, proving the Bogomolov
smoothness theorem (for deformations of compact Kähler manifolds) to deformations of the regular part of a Stein space with a finite number of isolated singular points. By the argument based on the Hodge structure on a strongly pseudo-convex Kähler domain or on a punctured Kähler space, we obtain an unobstructed subspace of the infinitesimal deformation space.

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  • [Ak-M] Akahori,T. and Miyajima,K., A note on the analogue of the Bogomolov type theorem on deformations of CR-structures, Canadian Math. Bull. 37 (1994), 8-12. MR 95d:32018
  • [Ar] Artin,M., Algebraic approximation of structures over complete local rings, Publ. Math. I.H.E.S. 36 (1969), 23-58. MR 42:3087
  • [B] Bogomolov,F.A., Hamiltonian Kähler manifolds, Dokl. Akad. Nauk. SSSR 243 (1978), 1101-1104; English transl., Soveit Math. Dokl. 19 (1978), 1462-1465. MR 80c:32024
  • [D] Demailly,J.-P., Cohomology of q-convex spaces in top degrees, Math. Z. 204 (1990), 283-295. MR 91e:32014
  • [L] Lempert,L., Algebraic approximations in analytic geometry, Invent. Math. 121 (1995), 335-354.
  • [M] Miyajima,K., A Bogomolov-type smoothness on deformations of quasi-Gorenstein cone singularities of dim $\ge 4$, ``Geometric Complex Analysis'', ed. J. Noguchi et al., World Scientific, 1996, pp. 461-465.
  • [N] Nakano,S., Vanishing theorems for weakly 1-complete manifolds, II, Publ. R.I.M.S., Kyoto Univ. 10 (1974), 101-110. MR 52:3617
  • [O1] Ohsawa,T., A reduction theorem for cohomology groups of very strongly $q$-convex Kähler manifolds, Inv. Math. 63 (1981), 335-354. MR 83a:32015; MR 83k:32032
  • [O2] -, Hodge spectral sequence on compact Kähler spaces, Publ. R.I.M.S., Kyoto Univ. 23 (1987), 265-274. MR 92f:32050
  • [O-T] Ohsawa,T. and Takegoshi,K., Hodge spectral sequence on pseudoconvex domains, Math. Z. 197 (1988), 1-12. MR 89d:32042
  • [S] Schlessinger,M., On rigid singularities, Conference on Complex Analysis, Rice University Studies 59 (1973), no. 1, 147-162. MR 49:9258
  • [Ti] Tian,C., Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical Aspects of String Theory , S.T. Yau ed., World Scientific, 1987, pp. 629-646. MR 89m:81001
  • [To] Todorov,A.N., The Weil-Petersson geometry of the moduli space of $SU(n\ge 3)$ (Calabi-Yau)Manifolds I, Commun. Math. Phys. 126 (l989), 325-346. MR 91f:32022

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Additional Information

Kimio Miyajima
Affiliation: Mathematical Institute, College of Liberal Arts Kagoshima University, Kagoshima-shi 890, Japan

Keywords: Deformation, isolated singularity, Hodge structure
Received by editor(s): August 14, 1995
Additional Notes: Partially supported by The Sumitomo Foundation.
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society

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