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.
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.