Proposition 20.3.73.55D866148BAF For every modification $\pi :Y'\to Y$ and $v\in \mathrm{PSH}(Y,\theta )$ with $v\sim r$,
\[ \underline{\operatorname {vol}}_{\pi ^*\theta }(\pi ^*v) =\underline{\operatorname {vol}}_\theta (v),\qquad \overline{\operatorname {vol}}_{\pi ^*\theta }(\pi ^*v) =\overline{\operatorname {vol}}_\theta (v). \]
These masses and the ideal orders are also unchanged by a smooth gauge $(\theta ,v,r)\mapsto (\theta +\mathrm{dd}^{\mathrm{c}}f,v-f,r-f)$.
Proof.
By proposition 5.43.18, every $\pi ^*\theta $-plurisubharmonic potential descends. Since $\pi $ is surjective, ordinary inequalities are equivalent before and after pullback. Thus the competitors are in bijection. On the locus where $\pi $ is an isomorphism, locality identifies their products. Both extended products give the excluded analytic subsets zero mass, so
\[ \pi _*\mathrm{MA}^{\mathrm{ext}}_{\pi ^*\theta }(\pi ^*v) =\mathrm{MA}^{\mathrm{ext}}_\theta (v). \]
Taking the infimum and supremum proves the identities. Smooth gauge preserves the actual currents and measures; it preserves multiplier ideals because $\mathrm e^{sf}$ is locally bounded above and below.
$\square$
Pome
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