Lemma 14.19.14.370AD0860DAD Let $u,v\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$, with $-H\leq v\leq 0$ and $H\geq 1$. Fix $0<\varepsilon <1$, and put
\[ m=\inf _ X(u-(1-\varepsilon )v),\qquad U_ s=\{ u<(1-\varepsilon )v+m+s\} . \]
There are $\delta _\varepsilon >0$, depending only on $(X,\omega ,\varepsilon )$, and $C_ H>0$, depending only on $(n,H)$, such that
F937FDFBF165
14.19.15
\begin{equation} t^ n\operatorname {Cap}(U_ s)\leq C_ H\int _{U_{s+t}}\omega _ u^ n \qquad (0<s,t,\ s+t<\delta _\varepsilon ). \end{equation}
Every $U_ s$, $s>0$, has positive capacity.
Proof.
Let $B_0$ be the constant of theorem 14.19.6, and set
\[ \delta _\varepsilon =\min \{ 1-\varepsilon ,\varepsilon ^3/(128B_0n^2)\} . \]
For a capacity test $0\leq \rho \leq 1$, put
\[ \tau =\frac{t}{2(1-\varepsilon )(H+1)},\quad w=(1-\tau )v+\tau \rho ,\quad m'=\inf _ X(u-(1-\varepsilon )w). \]
Then $0<\tau <1$ and $0\leq (1-\varepsilon )(w-v)\leq t/2$, so $m-t/2\leq m'\leq m$. Consequently
\[ U_ s\subset V:=\{ u<(1-\varepsilon )w+m'+s+t/2\} \subset U_{s+t}. \]
We apply modified comparison to $u,w$ at height $s+t/2<\delta _\varepsilon $; by the choice of $\delta _\varepsilon $, its inverse factor is at most $(8/7)^ n$. Moreover,
\[ \omega _{(1-\varepsilon )w} \geq (1-\varepsilon )\tau \omega _\rho =\frac{t}{2(H+1)}\omega _\rho . \]
Integrating over $U_ s\subset V$, applying modified comparison on $V$, and taking the supremum over $\rho $, we obtain (14.19.15) with $C_ H=[2(H+1)]^ n(8/7)^ n$. Finally, each $U_ s$ is nonempty by the definition of the infimum, so it has positive smooth volume by lemma 5.24.4, and hence positive capacity, as the constant test function shows.
$\square$
Pome
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