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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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