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Some examples of $$1$$-convex non-embeddable threefolds. (English) Zbl 1174.32014
A constructive approach leads to a family of $$1$$-convex threefolds with exceptional curve $$C$$ of type $$(0, -2)$$, which are not embeddable in $$\mathbb{C}^{m}\times\mathbb{CP}_{n}$$. A real $$3$$-dimensional chain $$A$$ whose boundary is the complex curve $$C$$ is emphasized in order to prove that these threefolds are not Kähler. Particular cases are to be found in [M. Colţoiu, Rev. Roum. Math. Pures Appl. 43, No. 1-2, 97–104 (1998; Zbl 0932.32018)].

##### MSC:
 32Q40 Embedding theorems for complex manifolds 32F10 $$q$$-convexity, $$q$$-concavity
##### Keywords:
embeddable variety; exceptional curve; threefold
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