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If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over $\operatorname{Spec}(\mathbb{C})$?

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Do you know the characterization of proper as a correspondence between field extensions and specializations along a valuation ring? Apparently this will implies the answer yes. Note that your scheme will be a complete algebraic variety. –  user40276 May 31 at 19:54

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up vote 2 down vote accepted

Yes.

See SGA 1

http://arxiv.org/abs/math/0206203

Exposee XII, Proposition 3.2.(v), page 245 (261 in the pdf).

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