Proposition 13.25.2.F274299A5A71 Let $\beta $ be a smooth real $(1,1)$-form with $\theta \leq \beta \leq \theta +h$, and let $u\in \mathrm{PSH}(X,\beta )$ satisfy $\underline{\operatorname {vol}}_\beta (u)\geq \delta >0$. For every $\varepsilon \in [0,1]$ and every smooth real $(1,1)$-form $\gamma $ with $\beta \leq \gamma \leq \beta +\varepsilon h$, both singularity volumes satisfy
\[ \operatorname {vol}_\beta (u) \leq \operatorname {vol}_\gamma (u) \leq (1+\varepsilon /\kappa _\delta )^ n\operatorname {vol}_\beta (u). \]
Proof.
Normalize $u\leq 0$, so that $u\leq V_\beta $, and note that $\rho \leq V_\beta $. Since $\Omega \geq \beta $ and $\underline{\operatorname {vol}}_\beta (u)\geq \delta $, lemma 13.20.1, applied in the big background $\beta $, shows that $H=Q_{b_\delta }(u,V_\beta )$ is defined. The potential
\[ w=b_\delta ^{-1}H+(1-b_\delta ^{-1})\rho \]
satisfies $w\leq u$ and $\beta +\mathrm{dd}^{\mathrm{c}}w\geq (1-b_\delta ^{-1})(\theta +\mathrm{dd}^{\mathrm{c}}\rho )\geq \kappa _\delta h$. The addition argument in the proof of proposition 13.20.6, with $h$, $\kappa _\delta $ and $t=\varepsilon /\kappa _\delta $ in place of $\omega $, $\delta $ and $t$, therefore gives the upper bound, with constants independent of $\beta $ and $u$. The lower bound is volume monotonicity.
$\square$
Pome
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