Proposition 11.5.1.828FAD1A35BF Let $\theta $ be a smooth real $(1,1)$-form, let $u$ be a $\theta $-plurisubharmonic function, and let $\gamma $ be a Gauduchon metric. Then
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11.5.2
\begin{equation} \int _ X(\theta +\mathrm{dd}^{\mathrm{c}}u)\wedge \gamma ^{n-1} =\int _ X\theta \wedge \gamma ^{n-1}. \end{equation}
Proof.
Since $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$, distributional integration by parts gives
\[ \langle \mathrm{dd}^{\mathrm{c}}u,\gamma ^{n-1}\rangle =\langle u,\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})\rangle =0. \]
The product on the left of (11.5.2) is a positive measure, whose total mass is therefore the finite number on the right.
$\square$
Pome
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