Corollary 17.5.12F89C0E4EBAC2

Proof. By proposition 10.3.1 there is a constant $C$ with $q_ j,q\leq C$. We show that the functions $\mathrm e^{\lambda q_ j}$ and $\mathrm e^{\lambda q}$, with value zero on the pole sets, are $C_1\omega $-psh for a constant $C_1$ independent of $j$; the preceding lemma then applies to them.

Choose $C_0\geq 0$ with $\theta \leq C_0\omega $, and a finite cover of $X$ by coordinate balls $W'\Subset W$ such that the smaller balls $W'$ still cover $X$, together with smooth functions $h_ W$ near $\overline W$ with $\mathrm{dd}^{\mathrm{c}}h_ W=\theta $. The function $q_ j+h_ W$ is plurisubharmonic on $W$, so for every $\delta >0$ smaller than the distance from $W'$ to the boundary of $W$, its regularization $g_\delta =(q_ j+h_ W)*\rho _\delta $ by the radial mollifier is a smooth plurisubharmonic function on the ball $W'$, and $g_\delta $ decreases to $q_ j+h_ W$ as $\delta \downarrow 0$, by proposition 5.9.2(2). Put $q_{j,\delta }=g_\delta -h_ W$. Then $\mathrm{dd}^{\mathrm{c}}q_{j,\delta }\geq -\theta \geq -C_0\omega $. Moreover $g_\delta (x)$ is an average of values of $q_ j+h_ W\leq C+h_ W$ at points within distance $\delta $ of $x$, so by uniform continuity of $h_ W$ on a compact neighborhood of $\overline{W'}$ in $W$ we have $q_{j,\delta }\leq C+1$ on $W'$ for all small $\delta $, independently of $j$.

For a smooth real function $s$, the chain rule $\partial \bar\partial \, \mathrm e^{\lambda s}=\lambda \mathrm e^{\lambda s}\partial \bar\partial s+\lambda ^2\mathrm e^{\lambda s}\partial s\wedge \bar\partial s$ and the identity $\frac{\mathrm i}{2\pi }\partial s\wedge \bar\partial s=\mathrm ds\wedge \mathrm d^{\mathrm c}s$, which follows from (5.2.10), give

because $\mathrm dq_{j,\delta }\wedge \mathrm d^{\mathrm c}q_{j,\delta }$ is a semipositive $(1,1)$-form. Thus $\mathrm e^{\lambda q_{j,\delta }}$ is $C_1\omega $-psh on $W'$ with $C_1=\lambda C_0\mathrm e^{\lambda (C+1)}$, and so is its decreasing limit $\mathrm e^{\lambda q_ j}$ as $\delta \downarrow 0$, by theorem 5.10.1 applied after adding a smooth local potential of $C_1\omega $; the limit is not identically $-\infty $, since it is nonnegative. Since the balls $W'$ cover $X$, the function $\mathrm e^{\lambda q_ j}$ belongs to $\mathrm{PSH}(X,C_1\omega )$, and the same argument applies to $\mathrm e^{\lambda q}$.

The exponential is Lipschitz with constant $\lambda \mathrm e^{\lambda C}$ on $(-\infty ,C]$, so $\mathrm e^{\lambda q_ j}\to \mathrm e^{\lambda q}$ in $L^1(X)$. These functions take values in $[0,\mathrm e^{\lambda C}]$, and lemma 17.5.11, with the background $C_1\omega $, gives the assertion. $\square$