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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 ). 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 $M$.

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 $R^1f_∗\mathcal O_{\tilde X} = 0$

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. 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 $M$.

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 ). 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 $M$.

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 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 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

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 $M$.

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. 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 $M$.