A Moishezon manifold is projective if and only if it is Kähler. This is no longer true for a singular Moishezon space. Moishezon proved a projectivity criterion for Moishezon spaces with isolated singularities. It is also known for Moishezon spaces with rational singularities.

> **So it is natural to ask: Is it true that a Moishezon space with canonical singularites (in the sense of the minimal model programme) is projective if and only if it is Kähler?**

Is  the following statement true in general?:

> Let $M$ be any compact complex variety with canonical singularities.
> Then $M$ is a Moishezon space if and only if there is a proper
> analytic subset $S⊂M$, such that $M\setminus S$ admits a complete
> Kähler-Einstein metric with negative Ricci curvature.

In fact the existence of a Kähler-Einstein metric has been verified for varieties of general type with  mild singularites, so this question might be natural to ask.

The motivation is the mild singular version of [my recent question][1] 


**Definition**: A compact complex space $M$ is Moishezon if and only if there exists a weakly positive coherent $\mathcal O_M$-module of rank 1 on $M$.

  [1]: https://mathoverflow.net/questions/275930/compactification-of-the-moduli-space-of-k%C3%A4hler-manifolds-with-negative-constant