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.
$\square$
Pome
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