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On the embedding of 1-convex manifolds with 1-dimensional exceptional set. (English) Zbl 0966.32008
Recall that a 1-convex (also called a strongly pseudoconvex) complex space $$X$$ is a proper modification $$f:X\to Y$$ at finitely many points of a Stein space $$Y$$. The union of all positive dimensional compact subspaces of $$X$$ is the exceptional set of $$X$$. A 1-convex space is called embeddable if it is biholomorphic to a closed analytic subspace of $$\mathbb{C}^N\times \mathbb{C}\mathbb{P}^m$$ for some $$N$$, $$m$$. Vo Van Tan and M. Coltoiu [see M. Coltoiu, Rev. Roum. Math. Pures Appl. 43, No. 1-2, 97-104 (1998; Zbl 0932.32018)] proved that every 1-convex manifold with 1-dimensional exceptional set $$S$$ is embeddable, except possibly if $$S$$ contains $$\mathbb{C}\mathbb{P}^1$$ with three possible normal bundles and in each of these cases there are examples of non-embeddable 1-convex manifolds. In the paper under review the authors prove the following topological criteria for the embeddability of $$X$$.
Theorem 1. Let $$X$$ be a 1-convex manifold with 1-dimensional exceptional set $$S$$. $$X$$ is Kähler if and only if $$S$$ does not contain any effective curve $$C$$ which is a boundary, i.e. such that $$H_2(X,\mathbb{Z})= 0$$.
Theorem 2. Let $$X$$ be a 1-convex manifold with 1-dimensional exceptional set $$S$$. If $$H_2(X,\mathbb{Z})$$ is finitely generated, then $$X$$ is embeddable if and only if it is Kähler.

##### MSC:
 32F10 $$q$$-convexity, $$q$$-concavity 53B35 Local differential geometry of Hermitian and Kählerian structures 32J99 Compact analytic spaces 32C22 Embedding of analytic spaces
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