A remark on a theorem of Vo Van Tan

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.