Pome91B8C8A3C10ELemma 11.6.3.91B8C8A3C10E Let $\theta $ be a smooth real $(1,1)$-form with $\mathrm{PSH}(X,\theta )\ne \varnothing $. If
for one Gauduchon metric $\gamma $, then $\theta =\mathrm{dd}^{\mathrm{c}}f$ for a smooth real function $f$. Conversely, a form $\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth has zero degree with respect to every Gauduchon metric. The same conclusions hold for $n=1$, with $\gamma ^{n-1}=1$.
Proof. Choose $u\in \mathrm{PSH}(X,\theta )$ and put $S=\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. By proposition 11.5.1, $\int _ XS\wedge \gamma ^{n-1}=0$. Since the trace measure $S\wedge \gamma ^{n-1}$ of the positive current $S$ controls all its coefficient measures, $S=0$. Taking the trace of the equation $\mathrm{dd}^{\mathrm{c}}u=-\theta $ with respect to $\omega $ gives an elliptic equation with smooth right-hand side, so $u$ is smooth by elliptic regularity, and $f=-u$ has the required property. The converse is Stokes' theorem. For $n=1$, the trace measure is $S$ itself and the same argument applies. $\square$
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