Is Kähler current class representable by semipositive forms?

Question

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 $T$?

Answer 1

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 $\varepsilon$.