PomeEA11DD744637Corollary 18.26.10.EA11DD744637 The comparison $\delta _\omega ^{\mathrm{pol}}$ descends to ordinary classes. For a smooth real function $f$ and $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f$, the map
is a bijection between the normalized ordinary models in the two backgrounds, with inverse $q\mapsto P_\omega [q+f]$. If the same Gauduchon forms are used in both backgrounds, $G_ f$ preserves $\delta ^{\mathrm{pol}}$, analytic models and ideal models.
Proof. Boundedly equivalent potentials have boundedly equivalent pullbacks on every modification, with the same pole set. Proposition 5.46.3 therefore gives equality of their polar currents and of their measures $\mu _\pi $ in every coordinate, which proves descent to ordinary classes.
If $p=P_\omega [p]$, the version of proposition 18.5.1 for ordinary envelopes gives
Thus $G_ f(p)\sim p-f$. Since ordinary envelopes depend only on the ordinary type, $P_{\omega '}[G_ f(p)]=P_{\omega '}[p-f]=G_ f(p)$, and similarly $P_\omega [G_ f(p)+f]=P_\omega [p]=p$. The same argument applied to models in background $\omega '$ proves surjectivity and the formula for the inverse. Invariance of the polar coordinates and the gauge identity of proposition 18.26.8 now give
Subtraction of a smooth function and bounded equivalence preserve analytic singularities. If $p=P_\omega [p]_{\mathcal I}$, the gauge estimate for ideal envelopes gives $P_{\omega '}[p-f]_{\mathcal I}\sim p-f\sim G_ f(p)$. By corollary 18.4.4, this ideal envelope is an ordinary model, and two ordinary models of the same ordinary type coincide. Hence it equals $G_ f(p)$, which shows that $G_ f$ maps ideal models to ideal models; the inverse map gives the converse. $\square$
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