Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Thank you for this. It seems that Beck and Schicho, at the end of the proof of their Proposition 9, are just quoting the result from page 91 of Fulton, Introduction to toric varieties, so perhaps Fulton's book is the best reference. I haven't checked all the hypotheses, but I think Fulton's proof, which is partly based on work of Danilov, does prove the general result.
@YangMills: The Douady space classifies the compact analytic subspaces of a given analytic space. Thus to complete an argument along these lines, it seems that one would still need to know that there are countably many "ambient" analytic spaces that contain all the compact complex manifolds. But this does not seem obvious, even if one wants to contain only the compact Kähler manifolds.
I think Step 1 must have been known long before the papers you cite. For instance, Nash, "Real algebraic manifolds", Annals of Math. in 1952 proved that every compact differentiable real manifold is diffeomorphic to a component of a real algebraic variety; there are only countably many families of real algebraic varieties; and the diffeomorphism type is locally constant within each family, by Ehresmann's theorem proved in 1950.
