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On certain non-Kählerian strongly pseudoconvex manifolds. (English) Zbl 0807.32018
This paper investigates the question of which 1-convex complex analytic spaces \(X\) can be embedded biholomorphically into \(\mathbb{C}^ N \times \mathbb{C} P^ m\). In Comment. Math. Helv. 57, 196-201 (1982; Zbl 0555.32012)], the author showed that \(X\) is embeddable if it is of complex dimension two. If \(X\) is three-dimensional, however, it need not be embeddable. In this paper, the author shows that if \(X\) is a 1-convex manifold and if the exceptional subvariety \(S\) of \(X\) is an irreducible compact analytic curve, then, with one possible exception, \(X\) is embeddable. This theorem is proved by showing that the restriction to \(S\) of the canonical bundle of \(X\) is weakly positive. The possible exception is the case in which \(X\) is of dimension three and \(S \cong \mathbb{C} P^ 1\). The author shows that this exception does in fact occur and can be made non-Kähler. He also obtains new examples of non-Kähler Moishezon manifolds of dimension three. Some related problems and questions are discussed.

MSC:
32T99 Pseudoconvex domains
32F10 \(q\)-convexity, \(q\)-concavity
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