Lemma 17.7.3ADE91FED84D6

Proof. Let $D\subset X$ be a connected coordinate chart, identified with a domain in $\mathbb C^ n$, on which $\theta =\mathrm{dd}^{\mathrm{c}}h$ for a smooth function $h$; by lemma 6.14.2, such charts, for instance small coordinate balls, cover $X$. Then $V+h\circ p$ is plurisubharmonic on the domain $D\times S\subset \mathbb C^{n+1}$.

(1) Fix $x\in D$. The complex line $\{ x\} \times \mathbb C$ meets $D\times S$ in $\{ x\} \times S$, so by definition 5.8.1 the function $\zeta \mapsto V(x,\zeta )+h(x)$ is subharmonic on $S$, possibly identically $-\infty $; hence so is $w(\zeta )=g(\operatorname{Re}\zeta )$, where $g(t)=v_ t(x)$. If $w\equiv -\infty $, then $g\equiv -\infty $ is convex. Otherwise $w$ is locally integrable on $S$ by proposition 3.4.8, and $\Delta w\geq 0$ by theorem 3.4.12. For $0<\delta <\frac12$, the mollification $w_\delta =w*\rho _\delta $ is smooth on $S_\delta =\{ \delta <\operatorname{Re}\zeta <1-\delta \} $, and $\Delta w_\delta =(\Delta w)_\delta \geq 0$ there, by proposition 2.5.3(1),(3). Since $w$ is invariant under imaginary translations, so is $w_\delta $; thus $w_\delta (\zeta )=g_\delta (\operatorname{Re}\zeta )$ for the smooth function $g_\delta (t)=w_\delta (t)$ on $(\delta ,1-\delta )$, and $g_\delta ''=\Delta w_\delta \geq 0$, so $g_\delta $ is convex by the mean value theorem. By (3.4.13), $w_\delta $ decreases to $w$ as $\delta \downarrow 0$, so $g_\delta $ decreases to $g$ on $(0,1)$. For $a,b\in (0,1)$ and $0<\lambda <1$, letting $\delta \downarrow 0$ in

shows that $g$ is convex. The rest of (1) is lemma 17.7.2(1).

(2) The plurisubharmonic function $V+h\circ p$ is locally integrable on $D\times S$ (theorem 5.9.1). Let $K\subset D$ be compact and $[a,b]\subset (0,1)$. Since $V$ does not depend on $\operatorname{Im}\zeta $, Tonelli's theorem on $K\times \{ \zeta :a\leq \operatorname{Re}\zeta \leq b,\ 0\leq \operatorname{Im}\zeta \leq 1\} $ gives

Hence, for almost every $x\in K$, the value $v_ t(x)$ is finite for almost every $t\in [a,b]$, and by (1) the function $t\mapsto v_ t(x)$ is then finite on $(0,1)$. Since $X$ is covered by countably many such compact sets $K$, this proves (2).

(3) Fix $t\in (0,1)$. The restriction of $V+h\circ p$ to $D\times \{ t\} $ is the function $v_ t+h$ on $D$. It is upper semicontinuous, and, by definition 5.8.1 applied to the complex lines contained in $\mathbb C^ n\times \{ t\} $, its restriction to every connected component of the intersection of $D$ with a complex line is subharmonic or identically $-\infty $. By (2), $v_ t$ is finite almost everywhere on $D$, so $v_ t+h$ is not identically $-\infty $, and $v_ t+h\in \mathrm{PSH}(D)$. Since such charts cover $X$, $v_ t\in \mathrm{PSH}(X,\theta )$. $\square$