PomeProposition 16.29.20.D9BF9081FC38 For every $b>0$ and $v\in \mathrm{PSH}(B,b\kappa )$,
\[ P_{bh}[\pi ^*v]_{\mathcal I} =\pi ^*P_{b\kappa }[v]_{\mathcal I}. \]
The envelope on the left uses all ambient competitors.
16.29.19 Recovery through the Hopf fibration6DDF231D4303
Positive lower volume on a closed base can force recovery even when the ambient manifold has unbounded upper mass. We now take
\[ X=(\mathbb C^3\setminus \{ 0\} )/\langle z\mapsto \mathrm e^{-1}z\rangle , \qquad B=\mathbb P^2,\qquad \pi :X\to B, \]
and use the geometry of section 23.1. Write $\kappa =\omega _{\mathrm{FS}}$, with $\int _ B\kappa ^2=1$, and put
\[ s=\log |z|^2,\qquad \alpha =\pi ^*\kappa =\mathrm{dd}^{\mathrm{c}}s,\qquad \beta =\mathrm ds\wedge \mathrm d^{\mathrm c}s,\qquad h=\alpha +\beta . \]
The derivatives of $s$ descend to $X$. Scalar multiplication gives an action of the elliptic group $E=\mathbb C/(\mathbb Z+2\pi \mathrm i\mathbb Z)$, transitive on each fiber of $\pi $ and preserving $h$. By theorem 23.1.1, $h$ has unbounded upper mass. Invariant potentials nevertheless reduce to potentials on $B$ because the horizontal part of $h$ is the pullback of $\kappa $.
Proof. First, an $E$-invariant $q\in \mathrm{PSH}(X,bh)$ descends to an upper semicontinuous function $w$ on $B$, since $\pi $ is proper and $q$ is constant on its fibers. In a product chart, Fubini gives $w\in L^1_{\mathrm{loc}}$, and
\[ bh+\mathrm{dd}^{\mathrm{c}}q=\pi ^*(b\kappa +\mathrm{dd}^{\mathrm{c}}w)+b\beta . \]
Lift a smooth base vector to a vector annihilated by $\partial s$. Its pairing with $\beta $ is zero. Testing the positive current above against these lifts and a nonnegative smooth fiber density of integral one proves $b\kappa +\mathrm{dd}^{\mathrm{c}}w\geq 0$ distributionally. A local potential of $b\kappa $ gives its canonical plurisubharmonic representative. Its pullback agrees almost everywhere with $q$, hence everywhere by lemma 5.24.3. Thus $w\in \mathrm{PSH}(B,b\kappa )$. Conversely, every such $w$ lifts to a $bh$-plurisubharmonic function by the displayed identity.
Now set $q=P_{bh}[\pi ^*v]_{\mathcal I}$. Translation by any $g\in E$ preserves its complete competitor family: it preserves the background, the obstacle zero, and the ideals of the invariant input. The envelope is their greatest member, so $q\circ g=q$. Descend $q=\pi ^*w$ as above. By lemma 5.37.7, its ideal equivalence to $\pi ^*v$ implies $w\sim _{\mathcal I}v$: a germ pulled back from the base is weighted-integrable in a product chart if and only if the base germ is, by Fubini. Hence $w$ is a base competitor. Every base competitor also lifts to an ambient competitor by the same lemma and the current identity. Taking the greatest members gives equality. $\square$
Fix real numbers $a<c$. For an envelope system $\boldsymbol \Gamma \in \mathscr S_{a,c}(h)$, scalar invariance alone supplies the whole transverse system, including its components over nonclosed base perturbations.
Lemma 16.29.21.2478F9CE0209 Suppose $\Gamma _\tau ^{\epsilon h}$ is $E$-invariant for every $\epsilon >0$ and $\tau <c$. There is a unique envelope system $\boldsymbol \Psi \in \mathscr S_{a,c}(\kappa )$ such that
\[ \Gamma _\tau ^{\epsilon h}=\pi ^*\Psi _\tau ^{\epsilon \kappa } \quad (\epsilon >0,\ \tau <c). \]
Writing $\Gamma =\mathsf R\boldsymbol \Gamma $ and $\psi =\mathsf R\boldsymbol \Psi $, we have $\Gamma _\tau =\pi ^*\psi _\tau $ for every $\tau <c$.
Proof. The descent in proposition 16.29.20 gives $\Gamma _\tau ^{\epsilon h}=\pi ^*\phi _\tau ^\epsilon $ with $\phi _\tau ^\epsilon \in \mathrm{PSH}(B,(1+\epsilon )\kappa )$. Normalization, decrease, concavity and constancy for $\tau \leq a$ descend pointwise. The fiber-envelope identity descends both ideal-model fixedness and scalar compatibility:
\[ \phi _\tau ^{\epsilon _2} =P_{(1+\epsilon _2)\kappa } [\phi _\tau ^{\epsilon _1}]_{\mathcal I} \quad (\epsilon _2\geq \epsilon _1>0). \]
In particular, the positive scalar slices have the same ideals at every weight.
For a Hermitian form $\xi >0$ on $B$, choose $\delta >0$ with $\delta \kappa \leq \xi $, fixed for the whole parameter interval, and define
\[ \Psi _\tau ^\xi =P_{\kappa +\xi } [\phi _\tau ^\delta ]_{\mathcal I}. \]
The definition is independent of $\delta $ because the scalar inputs have the same ideals. It reproduces the scalar components, and proposition 16.29.5 proves that each component is a curve in $\mathscr T_{a,c}(\kappa +\xi )$. Projection preserves the ideals, so comparing the complete competitor families proves compatibility for $\xi \geq \xi '$. Thus $\boldsymbol \Psi $ is a full envelope system. Every possible extension is forced by this formula, which proves uniqueness.
The retraction theorem applies to both systems. Taking their decreasing limits along $h/j$ and $\kappa /j$ in the scalar identity gives $\Gamma _\tau =\pi ^*\psi _\tau $. In particular the limits, including their values at poles, agree. For any other Hermitian metric $k$ on $B$, constants $0<l\leq r$ with $l\kappa \leq k\leq r\kappa $ give
\[ \phi _\tau ^{l/j}\leq \Psi _\tau ^{k/j}\leq \phi _\tau ^{r/j}. \]
Both outer sequences decrease to $\psi _\tau $. This also identifies the transverse limit directly, without assuming that it keeps the ideals of the positive scalar slices. $\square$
Theorem 16.29.22.2DB4FF5FB5C1 Let $\boldsymbol \Gamma \in \mathscr S_{a,c}(h)$ have $E$-invariant scalar components as in lemma 16.29.21. Let $\psi =\mathsf R\boldsymbol \Psi $ be the transverse limiting curve. If
\[ \underline{\operatorname {vol}}_\kappa (\psi _ a)>0, \]
then for every $\tau <c$ and every Hermitian form $\eta >0$ on $X$,
\[ \Gamma _\tau ^\eta =P_{h+\eta }[\pi ^*\psi _\tau ]_{\mathcal I}. \]
Thus $\boldsymbol \Gamma =\mathsf E(\mathsf R\boldsymbol \Gamma )$.
Proof. The closed Kähler background $\kappa $ is big and $B$ has the bounded mass property. Apply theorem 13.21.7 to the full transverse system. Its initial-volume hypothesis is exactly the displayed one, and it gives
\[ \Psi _\tau ^{\epsilon \kappa } =P_{(1+\epsilon )\kappa }[\psi _\tau ]_{\mathcal I} \quad (\epsilon >0,\ \tau <c). \]
The fiber-envelope identity yields $\Gamma _\tau ^{\epsilon h}=P_{(1+\epsilon )h}[\pi ^*\psi _\tau ]_{\mathcal I}$. For a general $\eta >0$, choose $\epsilon h\leq \eta $. Compatibility and the nested-background identity of proposition 16.29.5 then give
\[ \Gamma _\tau ^\eta =P_{h+\eta }[\Gamma _\tau ^{\epsilon h}]_{\mathcal I} =P_{h+\eta }[\pi ^*\psi _\tau ]_{\mathcal I}. \]
$\square$
The positivity condition is a lower ordinary-type volume on the closed base. It does not use an ambient upper-volume deficit or identify the ideals of the limit in advance. The conclusion holds at every finite $\tau <c$; it gives no uniform positive-volume bound as $\tau $ approaches $c$.
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