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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A remark on a theorem of Vo Van Tan

Author: Mihnea Colţoiu
Journal: Trans. Amer. Math. Soc. 307 (1988), 857-859
MSC: Primary 32F30; Secondary 32F10
MathSciNet review: 940232
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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.

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Article copyright: © Copyright 1988 American Mathematical Society