Proposition 18.26.8.CB31D6224569 Every $A_\pi $ is positive. The comparison (18.26.7) is a pseudometric on $\mathrm{PSH}(X,\omega )$, bounded by two, and its restriction to normalized ordinary models with analytic singularities is a metric. If $X$ is a curve, then for all ordinary models $p,q$,
\[ \delta _\omega ^{\mathrm{pol}}(p,q) =\frac{\| \mu _{\operatorname {id}_ X}(p)-\mu _{\operatorname {id}_ X}(q)\| _{\mathrm{TV}}}{\int _ X\omega }, \]
so $\delta _\omega ^{\mathrm{pol}}$ induces the complete topology of the polar metric on curve models, in which the closure of the analytic models is exactly the set of ideal models.
Let $f$ be a smooth real function and $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f$. If $\delta _{\omega '}^{\mathrm{pol}}$ is defined by the same formula with the same Gauduchon forms, then its denominators are positive and
\[ \delta _{\omega '}^{\mathrm{pol}}(u-f,v-f) =\delta _\omega ^{\mathrm{pol}}(u,v) \qquad (u,v\in \mathrm{PSH}(X,\omega )). \]
Proof.
The form $\pi ^*\omega $ is semipositive, and it is positive on the nonempty open set where $\pi $ is an isomorphism, so $A_\pi >0$. By proposition 5.43.18, $u\circ \pi \in \mathrm{PSH}(Y_\pi ,\pi ^*\omega )$. Since $\mathrm{dd}^{\mathrm{c}}(\eta _\pi ^{n-1})=0$, distributional Stokes gives
\[ \int _{Y_\pi }\bigl (\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )\bigr ) \wedge \eta _\pi ^{n-1}=A_\pi . \]
Thus $\mu _\pi (u)$ is a positive measure of mass at most $A_\pi $, and the normalized total variation distance of two such measures is at most two. Symmetry and the triangle inequality persist under the supremum, which proves the pseudometric assertion.
Let $p,q$ be ordinary models with analytic singularities, choose a common log resolution $\pi $ in $\mathscr R$, and suppose that $\delta _\omega ^{\mathrm{pol}}(p,q)=0$. Then their measures in this coordinate agree. Let $E_ i$ be the prime components of the resolved ideals on $Y_\pi $, and $a_ i,b_ i$ the coefficients of $p$ and $q$ along them. The divisorial restriction argument of lemma 5.38.1 gives
\[ \mu _\pi (p)=\sum _ i a_ i[E_ i]\wedge \eta _\pi ^{n-1},\qquad \mu _\pi (q)=\sum _ i b_ i[E_ i]\wedge \eta _\pi ^{n-1}. \]
Indeed, a smooth local correction makes the complex Hessian closed and positive without changing its coefficient mass on a divisor. Distinct components do not share trace mass on their intersections, and each component has strictly positive trace mass. Equality of these measures therefore implies $a_ i=b_ i$ for every $i$. The criterion of lemma 5.38.11, which uses only the finitely many divisors $E_ i$, applied in both directions gives $p\sim q$. An ordinary model is the ordinary envelope of every potential of its type, so ordinary models with the same type are equal. This proves separation.
On a curve, every smooth modification is an isomorphism and $\eta _\pi ^{n-1}=1$. Hence $A_\pi =\int _ X\omega $, and pullback preserves the total variation of polar curvature. The formula on curves and the assertions following it are therefore consequences of corollary 13.28.7, proposition 13.28.11.
Finally, $u-f$ has, in background $\omega '$, the same pulled-back curvature current and the same pole set as $u$ in background $\omega $, and Stokes' theorem gives $A_{\pi ,\omega '}=A_{\pi ,\omega }$. The gauge formula follows. It holds for the potentials themselves, before any ordinary projection.
$\square$
Pome
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