Let $\theta $ be any smooth real $(1,1)$-form. Taking the infimum and the supremum of the mass over a singularity type gives two invariants with values in $[0,+\infty ]$. They can differ, because a bounded change of potential need not preserve the mass.
Definition 16.28.58.52560A3D59BD For $u\in \mathrm{PSH}(X,\theta )$, define
\[ \begin{split} \underline{\operatorname {vol}}_\theta (u) & =\inf _{v\in \mathrm{PSH}(X,\theta ),\ v\sim u} \int _ X\mathrm{MA}_\theta (v),\\ \overline{\operatorname {vol}}_\theta (u) & =\sup _{v\in \mathrm{PSH}(X,\theta ),\ v\sim u} \int _ X\mathrm{MA}_\theta (v). \end{split} \]
These are the lower and upper volumes of the singularity type of $u$ in the background $\theta $. When $\mathrm{PSH}(X,\theta )$ is nonempty, we write $\underline{\operatorname {vol}}(\theta )$ and $\overline{\operatorname {vol}}(\theta )$ for their values at the potential $V_\theta $ with minimal singularities.
For semipositive $\theta $, the singularity type of the zero potential consists of the bounded potentials. Extending the lower mass of definition 13.14.14 to this background, we put
\[ v_-(\theta )=\underline{\operatorname {vol}}_\theta (0) =\inf _{v\in \mathrm{PSH}(X,\theta )\cap L^\infty (X)} \int _ X\mathrm{MA}_\theta (v). \]
The condition $v_-(\theta )>0$ is stronger than the non-collapsing property used in bounded domination: it requires a single positive lower bound for the masses of all bounded potentials. Singularity volumes under a bounded mass hypothesis are compared in section 13.16.
Both volumes are invariant under bounded changes of $u$, by definition. They are also invariant under the smooth change of decomposition $(\theta ,u)\mapsto (\theta +\mathrm{dd}^{\mathrm{c}}f,u-f)$, for $f\in C^\infty (X,\mathbb R)$: the map $v\mapsto v-f$ identifies the competitors, their currents and their measures. Scaling gives
\[ \underline{\operatorname {vol}}_{t\theta }(tu) =t^ n\underline{\operatorname {vol}}_\theta (u),\qquad \overline{\operatorname {vol}}_{t\theta }(tu) =t^ n\overline{\operatorname {vol}}_\theta (u)\quad (t>0). \]
Pullback by a modification also identifies the entire ordinary singularity type. The measure identity below gives preservation of both volumes in the exact pulled background.
Proposition 16.28.59.A8ED96D18B4A Let $\mu :Y\to X$ be a modification of connected compact complex manifolds, let $\theta $ be a smooth real $(1,1)$-form on $X$, and let $u\in \mathrm{PSH}(X,\theta )$. Then
\[ \mu _*\mathrm{MA}^{\mathrm{ext}}_{\mu ^*\theta }(u\circ \mu ) =\mathrm{MA}^{\mathrm{ext}}_\theta (u). \]
Both singularity volumes satisfy
\[ \underline{\operatorname {vol}}_{\mu ^*\theta }(u\circ \mu ) =\underline{\operatorname {vol}}_\theta (u),\qquad \overline{\operatorname {vol}}_{\mu ^*\theta }(u\circ \mu ) =\overline{\operatorname {vol}}_\theta (u), \]
including infinite values.
Proof.
Choose a proper analytic set $S\subset X$ such that $\mu $ is an isomorphism outside $D=\mu ^{-1}(S)$. On these complements, bounded truncations commute with composition by $\mu $, and their products commute with the biholomorphic change of coordinates. Passing to the increasing limits gives the measure identity there. By theorem 13.14.3, the two measures give no mass to $S$ and $D$, which proves the global identity and equality of total masses.
Pullback is a bijection between the potentials in the two backgrounds by proposition 5.43.18. Surjectivity of $\mu $ preserves and reflects bounded differences, so this bijection identifies the ordinary-type competitor families. Their masses agree by the first assertion. Taking their infima and suprema proves the volume identities.
$\square$
The background upstairs is the exact pullback $\mu ^*\theta $, which may be degenerate even when $\theta >0$. These identities do not compare volumes for an independently chosen Hermitian form on $Y$.
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