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Proposition 5.23.1237930AECE45

Proposition 5.23.1.237930AECE45 Let $f$ be a smooth real function on $X$, and set $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$. Then

\[ \mathrm{PSH}(X,\theta )\longrightarrow \mathrm{PSH}(X,\theta '), \qquad u\longmapsto u-f \]

is a bijection. It preserves the positive current and changes each potential by a bounded function. Moreover, $\mathrm{d}\theta '=\mathrm{d}\theta $.

Proof. For every $u\in \mathrm{PSH}(X,\theta )$,

\[ \theta '+\mathrm{dd}^{\mathrm{c}}(u-f)=\theta +\mathrm{dd}^{\mathrm{c}}u. \]

The inverse map is $v\mapsto v+f$, and $f$ is bounded because $X$ is compact. The last assertion follows from $\mathrm{d}\mathrm{dd}^{\mathrm{c}}f=0$. $\square$

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