Let X be a reduced paracompact complex analytic space of dimension n. It is proved that if X has no compact irreducible branches of dimension n, then X is n-complete, and if X has finitely many compact irreducible branches of dimension n, X is n-pseudoconvex, both in the sense of Andreotti-Grauert. Applying Andreotti-Grauert's finiteness theorem (resp. Barlet's theorem on the Steinness of cycle spaces) we deduce from our result Siu's theorem on analytic sheaf cohomology groups of noncompact complex spaces (resp. the Steinness of the space of (n—1) cycles of analytic spaces whose n-dimensional irreducible branches are non-compact).
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Takeo Ohsawa, Completeness of Noncompact Analytic Spaces . Publ. Res. Inst. Math. Sci. 20 (1984), no. 3, pp. 683–692DOI 10.2977/PRIMS/1195181418