Proposition 18.23.1.6235BDB38282 The class $\mathrm{PSH}(X,\theta )$ is nonempty exactly when $A\geq 0$. For $A\geq 0$, the entire ordinary $\theta $-model space is isometric, using the $q$-fiber normalization, to the curve model space in background $\sigma $, whose area is $dA$. Its diameter is $2dA$; it has the analytic closure, ideal retraction, convergence and all-obstacle properties above. Every potential has zero extended top-degree measure and every ordinary type has zero upper and lower singularity volumes. At $A=0$ the model space is a smooth singleton.
If $\kappa _ q^\gamma $ and $\kappa _ p^\gamma $ are the constants defined by the pushforwards of $\gamma ^{n-1}$ to $C'$ and $C$, then
\[ \kappa _ p^\gamma =d\kappa _ q^\gamma . \]
For $A\geq 0$, normalizing the same polar-trace variation by $\kappa _ p^\gamma $ gives $d^{-1}$ times the $q$-normalized metric, with diameter $2A$. This rescaled metric is also independent of $\gamma $. For a model $U=v\circ q$ with polar measure $\mu _ v$,
\[ p_*\lambda _ U^\gamma =\kappa _ q^\gamma r_*\mu _ v. \]
Proof.
Away from the finitely many critical values, $r$ is a covering with $d$ sheets. Change of variables on these sheets gives
\[ \int _{C'}r^*\alpha =d\int _ C\alpha \]
for every smooth real $(1,1)$-form $\alpha $; the branch points and their inverse images have smooth area zero. In particular $\int _{C'}\sigma =dA$. The envelope lemma 18.22.1, applied to $q$, gives nonemptiness exactly for $dA\geq 0$, equivalent to $A\geq 0$, and proves the zero-mass and zero-area assertions. In this nonempty case, apply 18.22.3, 18.22.4 to $q$ and the smooth background $\sigma $, which may vanish at branch points or be indefinite.
The Gauduchon trace lemma for $q$ gives $q_*(\gamma ^{n-1})=\kappa _ q^\gamma $. For a smooth test form $\alpha $ on $C$, the projection formula and the covering calculation yield
\[ \int _ Xp^*\alpha \wedge \gamma ^{n-1} =\kappa _ q^\gamma \int _{C'}r^*\alpha =d\kappa _ q^\gamma \int _ C\alpha . \]
Thus $p_*(\gamma ^{n-1})$ is the constant $d\kappa _ q^\gamma $. The ratio of the two normalizations is $d^{-1}$, giving every metric assertion. Finally $q_*\lambda _ U^\gamma =\kappa _ q^\gamma \mu _ v$, so composing pushforwards proves the last identity.
$\square$
Pome
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