# Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures

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?

Is there any reference related to this question?

• 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 – user21574 Dec 9 '17 at 11:31