Lemma 13.26.1.D5C96B9222A4 For every $\delta >0$ there are $C_\delta >0$ and $\varepsilon _0>0$ with the following property. Let $\beta ,\gamma $ be smooth real $(1,1)$-forms with
\[ \theta -\varepsilon h\leq \beta \leq \gamma \leq \theta +\varepsilon h, \qquad 0\leq \varepsilon \leq \varepsilon _0, \]
and let $u\in \mathrm{PSH}(X,\beta )$ satisfy $\underline{\operatorname {vol}}_\beta (u)\geq \delta $. Then both singularity volumes satisfy
\[ 0\leq \operatorname {vol}_\gamma (u)-\operatorname {vol}_\beta (u) \leq C_\delta \varepsilon . \]
For two such potentials $u,v$, one also has
\[ \left|d_\gamma (P_\gamma [u],P_\gamma [v])-d_\beta (u,v)\right| \leq 2C_\delta \varepsilon . \]
Proof.
Put $\theta _0=\theta -\kappa h/2$ and $h_0=(1+\kappa /2)h$. Since $\theta _0+\mathrm{dd}^{\mathrm{c}}\rho \geq \kappa h/2$, the form $\theta _0$ is big, and $\rho $ is a strict analytic potential for it with the rational margin $\kappa /(2+\kappa )$ relative to $h_0$. Let $\varepsilon \leq \min (\kappa /2,1/2)$. Then $\theta _0\leq \beta \leq \theta _0+h_0$ and $\beta \leq \gamma \leq \beta +2\varepsilon h_0$, so proposition 13.25.2 applies with $\theta _0$, $h_0$, $\rho $ and a fixed Hermitian form $\Omega _0\geq \theta _0+2h_0$ in place of $\theta $, $h$, $\rho $ and $\Omega $. These choices do not depend on $\beta $ and $\gamma $. Since the volumes in background $\beta $ are at most $v_+(\Omega _0)$, the multiplicative estimate of the proposition implies the additive estimate, with a constant $C_\delta $ independent of $\beta $, $\gamma $ and $u$. Applying it to $u$, $v$ and $\max (u,v)$, the three-error calculation in the proof of corollary 13.25.3 and projection invariance in background $\gamma $ give the second assertion.
$\square$
Pome
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