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I want to prove the following statement.

Let $\Omega$ be a bounded domain, and $\Gamma \subset \text{Aut}(\Omega)$ be the subgroup acting totally discontinuously on $\Omega$ without fixed points such that the quotient space $X = \Omega \setminus \Gamma$ is a compact complex manifold. Then $X$ is a projective manifold.

My idea is to use the Kodaira Emdedding theorem by constructing a positive holomorphic line bundle. However, I am not sure how to do so. Can you provide me with some ideas how to do so? Thank you.

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1 Answer 1

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Projectivity of such quotients was actually proven by Kodaira himself in Theorem 6 in

Kodaira, Kunihiko, On Kähler varieties of restricted type. (An intrinsic characterization of algebraic varieties.), Ann. Math. (2) 60, 28-48 (1954). ZBL0057.14102.

In brief, the canonical bundle of the quotient manifold is positive (one uses a curvature computation for the Bergman metric on $K_\Omega$), hence, Kodaira's embedding theorem applies.

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