PomeLemma 5.38.2.4296DE167374 Let $u\leq v$ be quasi-plurisubharmonic on a connected compact complex manifold $X$, and let $Z\subsetneq X$ be analytic. If $\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}v$ on $X\setminus Z$, then $v=u+C$ for a real constant $C$.
Lemma 5.38.24296DE167374
Proof. The current $S=\mathrm{dd}^{\mathrm{c}}(v-u)$ is real, closed, of order zero and supported on $Z$. By the support theorem in [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG],
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5.38.3
\begin{equation} S=\sum _{j=1}^ m c_ j[Z_ j],\qquad c_ j=\nu (v,Z_ j)-\nu (u,Z_ j)\leq 0, \end{equation}
where $Z_ j$ are the prime divisorial components of $Z$. Indeed, the support theorem first gives real constants $c_ j$ such that $S-\sum _ jc_ j[Z_ j]$ is supported in codimension at least two; this closed current of order zero then vanishes by the same theorem. Near a generic smooth point of $Z_ j$, with local equation $z_1=0$, the function $v-u-c_ j\log |z_1|^2$ has zero complex Hessian and is therefore smooth and pluriharmonic. This identifies $c_ j$ with the difference of the generic Lelong numbers, and $c_ j\leq 0$ because $u\leq v$.
Thus $\mathrm{dd}^{\mathrm{c}}(u-v)=-S\geq 0$. Hence the locally integrable distribution $u-v$ has a global plurisubharmonic representative, which is constant by the maximum principle on the compact connected manifold $X$. Since quasi-plurisubharmonic functions that agree almost everywhere agree everywhere, $v=u+C$ for a constant $C$. $\square$
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