PomeProposition 5.22.2.887F604607E9 Every real distribution $u$ on $X$ with $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ has a unique $\theta $-plurisubharmonic representative. On each relatively compact coordinate ball with coordinates $z=(z_1,\ldots ,z_ n)$, there is a constant $A>0$ such that $u+A|z|^2$ is plurisubharmonic. Conversely, every quasi-plurisubharmonic function $u$ on $X$ belongs to $\mathrm{PSH}(X,C\omega )$ for some constant $C>0$.
Proposition 5.22.2887F604607E9
Proof. On a coordinate ball with compact closure in a larger chart, smoothness of $\theta $ gives
\[ \theta \leq A\mathrm{dd}^{\mathrm{c}}|z|^2 \]
for a sufficiently large $A$. Consequently,
\[ \mathrm{dd}^{\mathrm{c}}(u+A|z|^2)\geq A\mathrm{dd}^{\mathrm{c}}|z|^2-\theta \geq 0. \]
The local distributional characterization of plurisubharmonic functions, [Chapter I, Theorem 5.8, DemCADG], gives a unique plurisubharmonic representative of $u+A|z|^2$. Subtracting the smooth correction gives a quasi-plurisubharmonic representative of $u$ on the ball. These representatives agree on overlaps: after a common local smooth correction, two of them become plurisubharmonic functions that are equal almost everywhere, hence everywhere. They therefore glue to a unique $\theta $-plurisubharmonic representative.
For the converse, cover $X$ by finitely many open sets $V_ j\Subset U_ j$, where each $U_ j$ is a chart on which $u=p_ j+f_ j$, with $p_ j$ plurisubharmonic and $f_ j$ smooth. On the compact set $\overline V_ j$, the smooth form $\mathrm{dd}^{\mathrm{c}}f_ j$ is bounded below by $-C_ j\omega $ for some constant $C_ j$, so $\mathrm{dd}^{\mathrm{c}}u\geq \mathrm{dd}^{\mathrm{c}}f_ j\geq -C_ j\omega $ on $V_ j$. Any constant $C>0$ larger than all the $C_ j$ gives $\mathrm{dd}^{\mathrm{c}}u\geq -C\omega $ on $X$. $\square$
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