On the embedding of 1-convex manifolds with 1-dimensional exceptional set (Q5928021)

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 \textit{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 \(X\).NEWLINENEWLINENEWLINETheorem 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 \(H_2(X,\mathbb{Z})= 0\).NEWLINENEWLINENEWLINETheorem 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.