Proposition 13.25.1.61209D2B61BD Let $f\in C^\infty (X,\mathbb R)$ and $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$. For every $\delta >0$, the map
\[ G_ f:\mathcal M_\delta (\theta )\longrightarrow \mathcal M_\delta (\theta '), \qquad G_ f(p)=P_{\theta '}[p-f], \]
is a bijection with inverse $q\mapsto P_\theta [q+f]$. It preserves both singularity volumes, and
\[ d_{\theta '}(G_ f(p),G_ f(q))=d_\theta (p,q). \]
It takes analytic models to analytic models and ideal models to ideal models.
Proof.
The map $u\mapsto u-f$ is a bijection from $\mathrm{PSH}(X,\theta )$ to $\mathrm{PSH}(X,\theta ')$ that preserves curvature currents, singularity types and hence both singularity volumes. Also $\max (u-f,v-f)=\max (u,v)-f$, so
\[ d_{\theta '}(u-f,v-f)=d_\theta (u,v). \]
By proposition 18.5.1, $G_ f(p)+f\sim p$, that is, $G_ f(p)\sim p-f$. Thus $G_ f(p)$ has the same volumes as $p$, and it is an ordinary model by theorem 13.19.5; in particular $G_ f(p)\in \mathcal M_\delta (\theta ')$. Since the volume comparison depends only on ordinary types, the displayed identity follows. Taking the ordinary envelope of $G_ f(p)+f\sim p$ gives $P_\theta [G_ f(p)+f]=P_\theta [p]=p$. The same argument with the two backgrounds interchanged proves that the map is bijective with the stated inverse.
Subtraction of a smooth function and bounded changes preserve analytic singularities. If $p=P_\theta [p]_{\mathcal I}$, proposition 18.5.1 also gives
\[ P_{\theta '}[p-f]_{\mathcal I}\sim p-f\sim G_ f(p). \]
The ideal envelope is an ordinary model by corollary 18.4.4, and ordinary models of the same ordinary type coincide; hence $P_{\theta '}[G_ f(p)]_{\mathcal I}=P_{\theta '}[p-f]_{\mathcal I}=G_ f(p)$. The inverse map gives the converse.
$\square$
Pome
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