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Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a finite number of blow-ups with smooth centers, becomes algebraic projective.

Moishezon manifolds are closely related to Kähler-Einstein metrics with negative Ricci curvature. So this can be the motivation of my question. In fact we have the following characterization of Moishezon manifolds in the language of Kahler-Einstein metric

We know: Let $M$ be any compact complex manifold. Then $M$ is a Moishezon manifold if and only if there is a proper analytic subset $S\subset M$, such that $M\setminus S$ admits a complete Kähler-Einstein metric with negative Ricci curvature.

Question:Does the Gromov-Hausdorff compactification of the moduli space of Kähler-Einstein smooth manifolds of general type (or of negative constant scalar curvature) have the structure of a compact Hausdorff Moishezon space?

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    $\begingroup$ The reason is that when $(X_t, \omega_t)$ converges to the central fiber $(X_0,\omega_0)$, and when fibers are projective, then central fiber is Moishezon, hence for study of Gromov-Hausdorff compactification of the moduli space of Kähler-Einstein smooth manifolds we need to work on Moishezon space instead length space or Alexandrov space $\endgroup$
    – user21574
    Commented Dec 9, 2017 at 11:31

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