Subtracting a smooth function transports a curve to a new background, and projection restores the envelope normalization. The gauge estimate is uniform in the perturbation, so this transport also respects the limiting curve. The limiting Monge–Ampère mass behaves differently: in a nonclosed background, it can change even for the system all of whose slices are zero.
Proposition 16.29.13.5B2E6F75BA98 Let $f\in C^\infty (X,\mathbb R)$ and $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$. For $\Gamma \in \mathscr T_{a,c}(\theta )$ and $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$, the formulas
\[ (\mathsf G_ f\Gamma )_\tau =P_{\theta '}[\Gamma _\tau -f]_{\mathcal I}, \qquad (\mathsf G_ f\boldsymbol \Gamma )^\eta _\tau =P_{\theta '+\eta }[\Gamma ^\eta _\tau -f]_{\mathcal I} \]
define bijections $\mathsf G_ f:\mathscr T_{a,c}(\theta )\to \mathscr T_{a,c}(\theta ')$ and $\mathsf G_ f:\mathscr S_{a,c}(\theta )\to \mathscr S_{a,c}(\theta ')$. They satisfy the cocycle law $\mathsf G_ g\mathsf G_ f=\mathsf G_{f+g}$, where $\mathsf G_ g$ is formed in the background $\theta '$, and they commute with $\mathsf E$ and $\mathsf R$.
Proof.
Subtracting $f$ changes the background from $\theta $ to $\theta '$ and preserves concavity, decreasingness and constancy on the left half-line. Slicewise projection preserves these properties by the argument of proposition 16.29.5. Since $f$ is smooth, every new slice has the multiplier ideals of the original one. Applying the construction with $-f$ therefore gives a $P_\theta [\cdot ]_{\mathcal I}$-fixed slice with the same ideals as the original slice, hence the original slice itself. This proves that $\mathsf G_{-f}$ is the inverse of $\mathsf G_ f$; the same uniqueness of fixed representatives proves the cocycle law.
Compatibility of the transformed system and commutation with $\mathsf E$ follow by comparing multiplier ideals in the common final background. For commutation with $\mathsf R$, the gauge estimate of proposition 18.5.1, applied in each background $\theta +\eta $, gives
\[ \Gamma ^\eta _\tau +\min _ Xf \leq (\mathsf G_ f\boldsymbol \Gamma )^\eta _\tau +f \leq \Gamma ^\eta _\tau +\max _ Xf, \]
with constants independent of $\eta $. Pass to the decreasing limits along a cofinal sequence. The new limit has the same ordinary singularity type as the old limit minus $f$, and hence the same multiplier ideals. It is $P_{\theta '}[\cdot ]_{\mathcal I}$-fixed by theorem 16.29.9, so it equals $P_{\theta '}[(\mathsf R\boldsymbol \Gamma )_\tau -f]_{\mathcal I}$. This proves commutation with $\mathsf R$.
$\square$
For closed backgrounds on a compact Kähler manifold, 22.1.1 The definition of non-Archimedean metrics indexes such systems by Kähler perturbations, as above, and defines their volume as a limit of the masses of the initial potentials. Its components are general test curves fixed by the ideal envelope, with initial potentials of positive mass, whereas our curves are constant on a prescribed left half-line and our system spaces impose no mass condition. For nonclosed positive backgrounds, the corresponding limit of masses is not invariant under the gauge bijections.
Proposition 16.29.14.3B3109886F9F There are a compact complex surface $X$, Hermitian forms $\omega ,h$ on $X$, and a smooth real function $f$ with $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f>0$, with the following property. For every $a<c$ there is a system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\omega )$ such that the limits
\[ \lim _{\epsilon \downarrow 0} \int _ X\mathrm{MA}^{\mathrm{ext}}_{\omega +\epsilon h} (\Gamma ^{\epsilon h}_ a), \qquad \lim _{\epsilon \downarrow 0} \int _ X\mathrm{MA}^{\mathrm{ext}}_{\omega '+\epsilon h} ((\mathsf G_ f\boldsymbol \Gamma )^{\epsilon h}_ a) \]
both exist and are different.
Proof.
Take the torus $X$ and the smooth perturbation of proposition 18.5.2, so that $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f$ is Hermitian and $\int _ X(\omega ')^2\neq \int _ X\omega ^2$, and fix any Hermitian form $h$. In every background $\omega +\eta $, let every slice with $\tau <c$ be zero. This is an envelope system for every $a<c$, since zero is the multiplier-ideal envelope of the unit-ideal class in a positive background. Its first limit is
\[ \lim _{\epsilon \downarrow 0}\int _ X(\omega +\epsilon h)^2 =\int _ X\omega ^2. \]
The gauge bijection sends every slice to $P_{\omega '+\eta }[-f]_{\mathcal I}=0$, since $f$ is smooth and the new background is positive. The second limit is therefore $\int _ X(\omega ')^2$, which is different.
$\square$
Pome
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