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 $\operatorname{Spec}(\mathbb{C})$?
2026-03-31 04:50:11.1774932611
Algebraic Compact manifold originates from a proper scheme?
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Yes.
See SGA 1
http://arxiv.org/abs/math/0206203
Exposee XII, Proposition 3.2.(v), page 245 (261 in the pdf).