PomeProposition 5.44.4.A89FF7CB0129 Let $\varphi \in \mathrm{PSH}(X,\theta )$ be unbounded below, and for each $\lambda >0$ let $u_\lambda $ be a quasi-plurisubharmonic function with $u_\lambda \sim \varphi $. Then, for each $\lambda >0$, the function $\lambda u_\lambda $ has infinite distance in $L^\infty (\mathrm{d}V)$ from every constant. In particular, $\lambda u_\lambda $ cannot converge uniformly on $X$ to a finite constant as $\lambda \downarrow 0$.
5.44.3 Vanishing exponents7442B45F8567
Proof. Since $u_\lambda \leq \varphi +C$ for a constant $C$, the function $u_\lambda $ is unbounded below. Every nonempty strict sublevel set $\{ u_\lambda <t\} $ of the upper semicontinuous function $u_\lambda $ is open, hence of positive smooth volume. Thus $u_\lambda $, and with it $\lambda u_\lambda $, is essentially unbounded below, and both conclusions follow. $\square$
A vanishing exponent gives convergence of the exponentials to one, even when the normalized potentials vary.
Lemma 5.44.5.D260E972F57C Let $v_ j\in \mathrm{PSH}(X,\theta )$ satisfy $\sup _ Xv_ j=0$, and let $\lambda _ j>0$ decrease to zero. For every finite non-pluripolar measure $\mu $,
\[ \int _ X\left|\mathrm{e}^{\lambda _ jv_ j}-1\right|\, \mathrm{d}\mu \longrightarrow 0. \]
If $f\in L^ p(\mathrm{d}V)$ for some $p>1$, then more precisely
\[ \int _ X\left|\mathrm{e}^{\lambda _ jv_ j}-1\right|\, |f|\, \mathrm{d}V \leq C_ p\lambda _ j\| f\| _{L^ p(\mathrm{d}V)}. \]
The constant depends on the fixed background, $p$ and $\mathrm{d}V$.
Proof. Choose a Hermitian form $\Omega \geq \theta $, so that $v_ j\in \mathrm{PSH}(X,\Omega )$. By lemma 5.29.2, for every $T>1$,
\[ \operatorname {Cap}_\Omega (\{ v_ j<-T\} )\leq C/T \]
uniformly in $j$. Since $v_ j\leq 0$ and $1-\mathrm{e}^{-x}\leq x$ for $x\geq 0$, splitting the integral at this sublevel set gives
\[ \int _ X(1-\mathrm{e}^{\lambda _ jv_ j})\, \mathrm{d}\mu \leq \lambda _ jT\mu (X)+\mu (\{ v_ j<-T\} ). \]
By lemma 5.29.18, the second term is uniformly small when $T$ is large, and for this fixed $T$ the first term tends to zero as $j\to \infty $. This proves the general assertion; here the exponential is given the value zero at a pole, and pole sets have zero $\mu $-measure.
For the quantitative assertion, lemma 5.26.2, applied in the background $\Omega $, gives a uniform $L^ q$ bound for $-v_ j$, where $q=p/(p-1)$. Hölder's inequality and the bound
\[ 0\leq 1-\mathrm{e}^{\lambda _ jv_ j}\leq \lambda _ j|v_ j| \]
give the claimed estimate for every $j$. $\square$
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