Proposition 13.20.6.F4C607C7C503 There is $C>0$, depending only on $\theta $ and $\omega $, such that, for every $u\in \mathrm{PSH}(X,\theta )$, every $\varepsilon \in [0,1]$ and every smooth real $(1,1)$-form $\eta $ with $\theta \leq \eta \leq \theta +\varepsilon \omega $,
\[ 0\leq \underline{\operatorname {vol}}_\eta (u) -\underline{\operatorname {vol}}_\theta (u)\leq C\varepsilon . \]
If $\underline{\operatorname {vol}}_\theta (u)>0$, there is also $\delta >0$, depending on $u$, such that, for every $\varepsilon \geq 0$ and every smooth $\eta $ with $\theta \leq \eta \leq \theta +\varepsilon \omega $, both singularity volumes satisfy
\[ \operatorname {vol}_\theta (u) \leq \operatorname {vol}_\eta (u) \leq (1+\varepsilon /\delta )^ n\operatorname {vol}_\theta (u). \]
Proof.
Choose a Hermitian form $\Omega $ with $\Omega \geq \theta $ and $\Omega \geq \omega $, and set $C=n2^ n v_+(\Omega )$. For every $v\in \mathrm{PSH}(X,\theta )$ and $1\leq k\leq n$, the mixed-mass estimate in the proof of lemma 13.20.1, applied to $s=v$ and $t=0$, together with $\omega \leq \Omega $, gives
\[ \binom nk\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}v)^{n-k}\wedge \omega ^ k \leq 2^ n v_+(\Omega ). \]
Expanding in the smooth positive perturbation and using $\varepsilon ^ k\leq \varepsilon $ yields
\[ \int _ X\mathrm{MA}_\eta (v) \leq \int _ X\mathrm{MA}_{\theta +\varepsilon \omega }(v) \leq \int _ X\mathrm{MA}_\theta (v)+C\varepsilon . \]
Each $v\sim u$ in $\mathrm{PSH}(X,\theta )$ is also a competitor for the lower volume in background $\eta $. Taking the infimum over these $v$ proves the upper bound for $\underline{\operatorname {vol}}_\eta (u)-\underline{\operatorname {vol}}_\theta (u)$. The lower bound is background monotonicity (theorem 13.17.1).
Now assume that $u$ has positive lower volume, and choose $w\leq u$ and $\delta >0$ with $\theta +\mathrm{dd}^{\mathrm{c}}w\geq \delta \omega $ by corollary 13.20.3. The case $\varepsilon =0$ is immediate. For $\varepsilon >0$ put $t=\varepsilon /\delta $. The potential $tw$ belongs to $\mathrm{PSH}(X,t(\theta -\delta \omega ))$, and the form $\eta +t(\theta -\delta \omega )$ is big, since adding $\mathrm{dd}^{\mathrm{c}}((1+t)w)$ to it gives a form at least $(1+t)\delta \omega -t\delta \omega =\delta \omega $. Addition and monotonicity give
\[ \begin{split} \operatorname {vol}_\eta (u) & \leq \operatorname {vol}_{\eta +t(\theta -\delta \omega )}(u+tw)\\ & \leq \operatorname {vol}_{(1+t)\theta }((1+t)u) =(1+t)^ n\operatorname {vol}_\theta (u). \end{split} \]
Here we used $\eta +t(\theta -\delta \omega )\leq (1+t)\theta $, which holds because $\eta \leq \theta +\varepsilon \omega $, and $u+tw\leq (1+t)u$. The lower bound is again background monotonicity.
$\square$
Pome
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