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A compact complex manifold is called 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 characterization is that if and only if $X$ admits a Kähler current, that is a closed (1,1) current $T$ satisfying $T\ge\varepsilon\omega$ for some real number $\varepsilon>0$ and some positive Hermitian form $\omega$ (see for example Demailly-Paun 04, p.1263).

As we know the de Rham class $[T]$ of the Kähler current $T$ is also representable by a smooth form $\alpha$, such that $[\alpha]=[T]\in H^{1,1}(X,\mathbb R)$, then what property does $\alpha$ have? Of course it should not be positive, otherwise the manifold is already Kähler, but except that, what other properties does $\alpha$ have, can we always find a semi-positive $\alpha$ to represent the class $[T]$ of a Kähler current $T$?

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This is an answer to your last question. In general we can't represent the class of a Kähler current by a semi-positive smooth form $\alpha$. Consider the blowup $\tilde{S} \to S$ of a compact Kähler surface at a point and let $E \subset \tilde{S}$ be the exceptional divisor. Then $[E] + \varepsilon [\omega] \in H^{1,1}(\tilde{S},\mathbb{R})$ is the class of a Kähler current for every Kähler form $\omega$ on $\tilde{S}$ and $\varepsilon > 0$. If $[E] + \varepsilon [\omega]$ is represented by a semi-positive smooth form $\alpha$, then $$-1 + \varepsilon \int_{E}\omega = [E] \cdot ([E] + \varepsilon [\omega]) = \int_{E} \alpha \ge 0,$$ which is impossible for small $\varepsilon$.

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  • $\begingroup$ In your case, both $S$ and $\tilde S$ are Kähler?But actually what I want to know is that for a non-Kähler manifold in Fujiki class $\mathcal C$, the class of a Kähler current can always be represented by a semipositive form? Sorry for the confusion. $\endgroup$
    – Tom
    May 15, 2022 at 4:38
  • $\begingroup$ You can replace $S$ by any manifold $X$ you like. If $\omega$ is a Kähler current on the blowup $\tilde{X}$, then $[E] + \varepsilon [\omega]$ for $\varepsilon > 0$ is represented by a Kähler current where $E \subset \tilde{X}$ is the exceptional divisor. Again for small $\varepsilon$, we can't represent $[E] + \varepsilon [\omega]$ by a smooth semipositive form because $[\ell] ([E] + \varepsilon [\omega]) < 0$ where $\ell$ is a line in $E$. You can construct non-Kähler $\tilde{X}$ by e.g. blowing up Hironaka's example at a general point. $\endgroup$
    – HYL
    May 15, 2022 at 9:38
  • $\begingroup$ For the holomorphic modification $\mu:\tilde X\to X$, we say $X$ is in Fujiki class $\mathcal C$ if $\tilde X$ is a compact Kähler manifold. Then the Kähler current $T$ should be on $X$ not $\tilde X$, right? $\endgroup$
    – Tom
    May 15, 2022 at 11:39
  • $\begingroup$ If $X$ is in the Fujiki class $\mathcal{C}$, so is $\tilde{X}$. Hironaka's example is Moishezon, in particular in the Fujiki class $\mathcal{C}$. $\endgroup$
    – HYL
    May 16, 2022 at 2:54
  • $\begingroup$ Dear @HYL, sorry for my late reply. The Hironaka's example you mentioned should be a complete non-projective (but Moishezon) 3-fold, right? Following your statement, it may be better to find a non-projective Moishezon surface (which must process singularities worse than rational singularities), since you talk about the intersectional number $[l][E]$? $\endgroup$
    – Invariance
    Aug 26, 2023 at 0:38

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