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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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##### References:
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