A remark on a theorem of Vo Van Tan
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- by Mihnea Colţoiu PDF
- Trans. Amer. Math. Soc. 307 (1988), 857-859 Request permission
Abstract:
In this paper we consider the following problem: Let $(X, S)$ be a $1$-convex manifold with $1$-dimensional exceptional set $S$. Does it follow that $X$ is a Kähler manifold? Although this was answered in the affirmative by Vo Van Tan in two papers, we show that his proofs are wrong. It is also shown that the Kähler condition implies that any strongly pseudoconvex domain $D \Subset X$ is embeddable, i.e. can be realized as a closed analytic submanifold in some ${{\mathbf {C}}^N} \times {{\mathbf {P}}_M}$. On the other hand it is known that under some additional assumptions on $S$ ($S$ is not rational or $S \simeq {{\mathbf {P}}^1}$ and $\operatorname {dim} X \ne 3$) it follows that $X$ is embeddable, in particular it is Kählerian.References
- Mihnea Colţoiu, On the embedding of $1$-convex manifolds with $1$-dimensional exceptional set, Comment. Math. Helv. 60 (1985), no. 3, 458–465. MR 814151, DOI 10.1007/BF02567427
- Thomas Peternell, On strongly pseudoconvex Kähler manifolds, Invent. Math. 70 (1982/83), no. 2, 157–168. MR 684170, DOI 10.1007/BF01390726
- Vo Van Tan, Vanishing theorems and Kählerity for strongly pseudoconvex manifolds, Trans. Amer. Math. Soc. 261 (1980), no. 1, 297–302. MR 576877, DOI 10.1090/S0002-9947-1980-0576877-8
- Vo Van Tan, Correction to: “Vanishing theorems and Kählerity for strongly pseudoconvex manifolds” [Trans. Amer. Math. Soc. 261 (1980), no. 1, 297–302; MR0576877 (81f:32042)], Trans. Amer. Math. Soc. 291 (1985), no. 1, 379–380. MR 797067, DOI 10.1090/S0002-9947-1985-0797067-4
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 307 (1988), 857-859
- MSC: Primary 32F30; Secondary 32F10
- DOI: https://doi.org/10.1090/S0002-9947-1988-0940232-6
- MathSciNet review: 940232