Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold.  



1. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\mathbb Z)$, then $X$ is projective.
2. (Moishezon condition) If the Kähler manifold $X$ is also a Moishezon manifold, then $X$ is projective.


In summary, we may write:


1. Kähler+integral=projective,
2. Kähler+Moishezon=projective.


And my question is that: what's the relationship between **integral** condition and **Moishezon** condition?
  
It includes: which one is stronger? Or whether one can deduced from the other？ Or are they actually equivalent?
  
Can we unify these 2 conditions together to get only one single condition? Or can we abstract the *essence* of both these 2 conditions?  

I guess the Moishezon condition is another version of "integral" condition, this view comes from a [post](https://math.stackexchange.com/questions/198841/how-should-one-think-of-non-projective-compact-manifolds-and-moishezon-manifolds/206996#206996) from @Gunnar Þór Magnússon



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> A Moishezon manifold can now be characterized as a modification of a projective manifold, so there exists a generically 1-1 meromorphic map $X\to Y$, where $Y$ is projective. This is equivalent to $X$ admitting an integral Kähler current, which is roughly a closed (1,1)-form $T$ that has distribution coefficients (in local coordinates) whose cohomology class is integral.
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So can we unify these 2 conditions together to just one condition: $X$ admits an integral Kähler current?
  
 Let me repeat my question again: what's the relationship between **integral** condition and **Moishezon** condition?


Added later:
  
I know $X$ admits an integral Kähler class if and only if $X$ has a positive line bundle, later I learned a compact complex manifold is Moishezon if and only if it carries a big line bundle. So in the view of "line bundles", we may write:


1. Kähler+positive=projective
2. Kähler+big=projective


And then it turns to the relationship between **big** and **positive**.  

So what's the essence that makes a compact Kähler manifold projective? I apologize that my question seems a little bit vague, so any comment clarifying it is welcome!



What makes a Kähler manifold projective?
If I understand the question correctly, I think that the answer is given by the main result in


S. Ji: [Currents, metrics and Moishezon manifolds](http://dx.doi.org/10.2140/pjm.1993.158.335), *Pac. J. Math.* **158**, No. 2, 335-351 (1993). [ZBL0785.32011](https://zbmath.org/?q=an:0785.32011).


Essentially, the existence of a bimeromorphic modification that is projective gives a $d$-closed (1, 1)-current on our manifold (the pushforward of the Kähler form on the modification) that satisfies three properties. Conversely, the existence of such a current implies that the manifold is Moishezon.


Here is the rewriting in LaTex of the first page (**warning**: there is a misprint in the second line of the Introduction paragraph, the projective algebraic manifold is $\widetilde{M}$, not $M$).



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> **Abstract:** A compact complex manifold $M$ is Moishezon if and only if there exists an integral closed positive $(1, 1)$-current $\omega$ such that $\omega \ge \epsilon\sigma$ and $\omega$ is smooth outside an analytic subvariety.
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> **1. Introduction.** Given a Moishezon manifold $M$, it is well known (cf. [**Mo**], [**W**]) that there is a bimeromorphic morphism $\pi : \widetilde M \to M$ such that the manifold $M$ is projective algebraic. Let $\tilde\omega$ be Kähler form on $\widetilde M$ with $[\tilde\omega] \in H^2(\widetilde M, \mathbf Z)$. Then the pushforward current $\omega = \pi\_\*\tilde\omega$ is a $d$-closed current on $M$ such that
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> (i) $\omega \in H^2(M, \mathbf Z)$;
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> (ii) $\omega$ is smooth on $M - S$, where $S$ is some proper analytic subset in $M$;
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> (iii) $\omega \ge \epsilon\sigma$ in the sense of currents, where $\epsilon > 0$ is some real number and $\sigma$ is a fixed positive definite $(1, 1)$-form (not necessarily $d$-closed) on $M$.
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> Conversely, we shall prove the following
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> **Theorem 1.1.** Let $M$ be a compact complex manifold of dimension $n$. Then $M$ is Moishezon if and only if there exists a $d$-closed $(1, 1)$-current $\omega$ on $M$ such that the conditions (i), (ii) and (iii) above are satisfied.
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