# Moishezon projectivity criterion for Moishezon spaces with canonical singularites

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 1-rational singularities( Note that varieies with 1-rational singularities my do not have rational singularities in general but for algebraic surface these two notions are same ). So it is true that a Moishezon space with canonical singularites (in the sense of the minimal model program) is projective if and only if it is Kähler.

Is the following statement true in general?:

Let $M$ be any compact complex variety with 1-rational 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 singular 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 for 1-rational singularites

The motivation is the mild singular version of my recent question

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 $M$.

Definition: An algebraic variety $X$ is said to have 1-rational singularities, if the following two conditions holds true

(1) $X$ is normal,

(2) for every resolution $f : \tilde X \to X$ of $X$ we have $R^1f_∗\mathcal O_{\tilde X} = 0$

• Canonical singularities (or even klt singularities) are rational. So if the answer is known for rational singularities...... – Hacon Jul 24 '17 at 4:42
• Ahh, this Kähler-Einstein metric again drunk me I wrote silly question, I apologize. The answer(in edited version) must be correspond to relation between semi log canonical singularities and 1-rational singularities. If you know, please let me know. – user21574 Jul 24 '17 at 7:37