PomeLemma 5.41.1.ADD743D2224C Let $I\subset \mathbb R$ be a nonempty open interval, set $S_ I=\{ z\in \mathbb C:\operatorname{Re}z\in I\} $, and let $\pi :X\times S_ I\to X$ be the projection. Let $(u_ t)_{t\in I}$ be a family of functions on $X$ such that $U(x,z)=u_{\operatorname{Re}z}(x)$ is quasi-plurisubharmonic on $X\times S_ I$ and $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}U\geq 0$. For each $x$, the function $t\mapsto u_ t(x)$ is finite and convex or identically $-\infty $. The finite alternative holds for almost every $x$.
Lemma 5.41.1ADD743D2224C
Proof. A local smooth correction making $U$ plurisubharmonic can be chosen to depend only on $x$. Hence the restriction of $U$ to $\{ x\} \times S_ I$ is subharmonic or identically $-\infty $. In the first case its distributional Laplacian is the second derivative of $t\mapsto u_ t(x)$, which is therefore nonnegative. The resulting convex representative is finite on $I$, and by the submean inequality it agrees everywhere with the subharmonic function. Finally, $U$ is locally integrable on the product, and Fubini's theorem, applied on products of coordinate balls with a fixed rectangle in $S_ I$, excludes the identically $-\infty $ alternative for almost every $x$. $\square$
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