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 $\mathsf R$.
$\square$
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