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Let $X$ be a compact complex manifold, we say $X$ is in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a holomorphic modification) $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. Another characterizaion is $X$ admits a Kähler current, that is a closed $(1,1)$-current $T$ satisfying $T≥εω$ for some real number $ε>0$ and some positive Hermitian form $ω$ (see for example Demailly-Paun 04, p.1263).

Let $X$ be a manifold in Fujiki class $\mathcal C$, and $T$ be a Kähler current of $X$. Then $T$ determines a class $[T]\in H^{1,1}(X,\mathbb R)$, if we choose a smooths $d$-closed $(1,1)$-form $\tau$ representing the same class, i.e. $[\tau]=[T]\in H^{1,1}(X,\mathbb R)$, then what preperties does $\tau$ possess? Obviously, $\tau$ should not be positive, otherwise, $\tau$ is a Kähler form. Then what other properties does $\tau$ have?

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    $\begingroup$ Suppose that $X$ is Kähler, and $[T]$ is a Kähler class. What can you say from $\tau $, apart that it is cohomologous to a Kähler form? $\endgroup$
    – abx
    Commented Aug 8 at 5:54
  • $\begingroup$ @abx, we can say $\tau+i\partial\bar\partial f$ is positive for some real smooth function $f$ on $X$, but for the non-Kähler case, we can't find such an $f$. $\endgroup$
    – Tom
    Commented Aug 8 at 8:21
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    $\begingroup$ Well, you can say that $\tau +d\omega $ is positive for some real smooth 1-form $\omega $. I don't see what else you can hope for. $\endgroup$
    – abx
    Commented Aug 8 at 8:29

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