PomeC288B8AF0D4ELemma 20.4.10.C288B8AF0D4E Replacing smooth forms and smooth functions in this quotient by real currents and real distributions gives the same space. In particular, every closed positive $(1,1)$-current $T$ on $Z$ can be written
with $\beta $ smooth and closed and $w$ a global quasi-plurisubharmonic function.
Proof. The decomposition of lemma 5.38.4 shows that the natural map from the smooth quotient to the current quotient is surjective, and it gives a quasi-plurisubharmonic potential for positive currents. The same lemma shows that two distributional potentials in such decompositions differ by a smooth function; equivalently, a distribution whose $\mathrm{dd}^{\mathrm{c}}$ is smooth is itself smooth. Hence two smooth forms that are identified in the current quotient are already identified in the smooth quotient, and the map is injective. $\square$
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