It is known when a compact complex analytic space $X$ is the analytification of a complex projective variety? If $X$ is a manifold, then Kodaira's embedding theorem and Chow's theorem says that $X$ is the analytification of a complex projective variety if and only if $X$ admits a Kaehler metric whose associated cohomology class lies in the image of the map $H^2(X, \mathbb{Z}) \to H^2(X, \mathbb{C})$.



Is there a known criterion for a compact complex analytic space to be projective?
The same Kodaira embedding theorem holds as you stated it for proper Kähler analytic spaces (with the usual notion of Kähler metric on an analytic space), and was proved by [Hans Grauert,](http://dx.doi.org/10.1007/BF01441136) see Satz 3 in section 3.


Note that in that paper he uses a slightly different definition of Kähler metric, and therefore he needs to assume that $X$ is normal, but with the usual definition this assumption is not needed (and his proof works), see e.g. [here](http://dx.doi.org/10.1007/BF01198581), Theorem 1.1 which is the relative version of the result you want.


