The Hopf geometry of subsection 23.1.32 gives a nonclosed background for which a distinguished family of potentials has a complete geodesic metric. Its Perron envelope uses all subgeodesics, although the metric measures lengths within the distinguished family. The distinction matters: allowing additional smooth paths can decrease the length.
Fix $n\geq 3$ and put $d=n-1$. Write
\[ X=H_ n,\qquad Y=\mathbb P^ d,\qquad \pi =p_ n, \qquad \omega =\omega _ n=\alpha _ n+\beta _ n. \]
Let $\kappa $ be the mass-one Fubini–Study form on $Y$, so $\alpha _ n=\pi ^*\kappa $. On the cover, $r=\log |z|^2$ is defined modulo two and $\beta _ n=\mathrm dr\wedge \mathrm d^{\mathrm c}r$. The compact elliptic group
\[ E=\mathbb C/(\mathbb Z+2\pi \mathrm i\mathbb Z) \]
acts by scalar multiplication, preserves $\omega $, and acts transitively on each fiber. A potential is basic if it is invariant under this action; such a potential is exactly $\pi ^*v$.
These manifolds are not products of positive-dimensional compact complex manifolds. Indeed, $H_ n$ is diffeomorphic to $S^{2n-1}\times S^1$, whose intermediate positive even-degree real cohomology vanishes. A product with a compact complex factor of dimension $k$, $0<k<n$, would have a nonzero degree-$2k$ class: pullback of that factor's top class is injective because projection has a smooth section. Thus the fiber reduction below takes place on a nonproduct manifold.
Proposition 17.16.16.0290ABD0E348 Let $S$ be a strip and $V$ a genuine quasi-plurisubharmonic function on $Y\times S$. Then
\[ \operatorname {pr}_ X^*\omega +\mathrm{dd}^{\mathrm{c}}(\pi \times \mathrm{id})^*V\geq 0 \quad \Longleftrightarrow \quad \operatorname {pr}_ Y^*\kappa +\mathrm{dd}^{\mathrm{c}}V\geq 0. \]
For every $v\in \mathrm{PSH}(Y,\kappa )$, the extended non-pluripolar measure satisfies
4A3521DF9F72
17.16.17
\begin{equation} \operatorname{MA}^{\mathrm{ext}}_\omega (\pi ^*v) =n\beta _ n\wedge \pi ^*\operatorname{MA}_\kappa (v). \end{equation}
In particular, for every nonnegative Borel function $F$,
6D09A14331E6
17.16.18
\begin{equation} \int _ X\pi ^*F\, \operatorname{MA}^{\mathrm{ext}}_\omega (\pi ^*v) =2n\int _ YF\, \operatorname{MA}_\kappa (v). \end{equation}
If the base Perron family between $v_0,v_1$ is nonempty, the full Hermitian Perron envelope between $\pi ^*v_0,\pi ^*v_1$ is the pullback of the base envelope.
Proof.
Write $\alpha =\alpha _ n$ and $\beta =\beta _ n$. Locally $\beta =(\mathrm i/(2\pi ))\partial r\wedge \bar\partial r$, and $\partial r$ has a nonzero vertical component. Every vector on $Y\times S$ therefore has a lift annihilated by $\partial r$. Testing the current on these lifts removes $\beta $; integrating the resulting distributional inequality against a nonnegative compact fiber density gives positivity on the base. Conversely, pullback of a positive base current plus $\beta $ is positive. An invariant genuine potential descends canonically by restricting to local holomorphic sections, so the argument also applies to singular potentials.
For bounded $v$, the local product with smooth corrections from section 13.14 gives
\[ (\omega +\mathrm{dd}^{\mathrm{c}}\pi ^*v)^ n =(\beta +\pi ^*(\kappa +\mathrm{dd}^{\mathrm{c}}v))^ n =n\beta \wedge \pi ^*(\kappa +\mathrm{dd}^{\mathrm{c}}v)^ d. \]
All other terms vanish by $\beta ^2=0$ and the dimension of $Y$. Apply this identity to $\max (v,-j)$ and restrict to $\{ v>-j\} $. Increasing finite-level limits give (17.16.17), without assigning mass to the pole set. The identity $\int _{\pi ^{-1}(y)}\beta =2$ from subsection 23.1.32 gives (17.16.18).
The full Perron competitor family is preserved by $E$. Its supremum and upper semicontinuous regularization are invariant. The chord bound and a pulled-back base competitor make this envelope a genuine subgeodesic by proposition 17.16.6. Positivity descent makes its descended envelope a base competitor with the prescribed joint boundary upper bounds. Conversely, every base competitor pulls back to a full competitor. The two envelopes consequently agree; in submersion charts, upper semicontinuous regularization commutes with pullback.
$\square$
Let $\mathcal E^2(Y,\kappa )$ denote the canonical $\kappa $-plurisubharmonic potentials of non-pluripolar mass one and finite squared energy. Define
\[ \mathcal B_\infty =\{ \pi ^*v:v\in C^\infty (Y),\ \kappa +\mathrm{dd}^{\mathrm{c}}v>0\} , \qquad \mathcal B_2=\pi ^*\mathcal E^2(Y,\kappa ). \]
Thus $\mathcal B_2$ consists precisely of the basic potentials of extended mass $2n$ with $\int _ X|u|^2\operatorname{MA}^{\mathrm{ext}}_\omega (u)<\infty $. On $\mathcal B_\infty $, let $d_ B$ be the infimum of the lengths of smooth basic paths for the metric $G$ above.
A geodesic metric space $(Z,\delta )$ is CAT$(0)$ if every constant-speed geodesic $\gamma :[0,1]\to Z$ satisfies
\[ \begin{aligned} \delta (z,\gamma _ t)^2\leq {}& (1-t)\delta (z,\gamma _0)^2+t\delta (z,\gamma _1)^2\\ & -t(1-t)\delta (\gamma _0,\gamma _1)^2 \end{aligned} \]
for $z\in Z$ and $0\leq t\leq 1$. In particular, its geodesics are unique.
We use the classical Kähler completion theorem: the $L^2$ length metric for $\int _ Y\xi \eta (\kappa +\mathrm{dd}^{\mathrm{c}}v)^ d$ has completion $\mathcal E^2(Y,\kappa )$, is a CAT$(0)$ metric, and its geodesic between two points is their Perron segment. That segment has capacity endpoint limits and all its slices lie in $\mathcal E^2$. We also use the full-mass characterization $P_\kappa [v]=0$ for $v\in \mathcal E^2$. These are closed-background inputs on $Y$; their exact source and operator normalization are specified in 25.14.
Theorem 17.16.19.B0CA576B1836 The completion of $(\mathcal B_\infty ,d_ B)$ is $\mathcal B_2$. It is a CAT$(0)$ space, and its unique constant-speed geodesic between two points is their full Hermitian Perron segment. This segment attains both endpoints in $L^1(X,\mathrm dV)$ for every smooth positive volume form $\mathrm dV$. For every $u\in \mathcal B_2$ and $c\in \mathbb R$,
\[ d_ B(u,u+c)=\sqrt{2n}|c|. \]
Proof.
Fiber integration gives the exact restricted metric
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17.16.20
\begin{equation} G_{\pi ^*v}(\pi ^*\xi ,\pi ^*\eta ) =2n\int _ Y\xi \eta (\kappa +\mathrm{dd}^{\mathrm{c}}v)^ d. \end{equation}
The Kähler completion theorem therefore gives the stated completion and CAT$(0)$ property after multiplying base distances by $\sqrt{2n}$. If $\widetilde d_ D$ denotes the completed metric written with operator $\mathrm i\partial \bar\partial $ and background $\kappa $, the normalization is explicitly
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17.16.21
\begin{equation} d_ B(\pi ^*v,\pi ^*w) =2\pi \sqrt{2n}\, \widetilde d_ D(v/(2\pi ),w/(2\pi )), \end{equation}
because $\kappa +\mathrm i\partial \bar\partial (v/(2\pi ))=\kappa +\mathrm{dd}^{\mathrm{c}}v$. The preceding proposition identifies its unique geodesic with the full Hermitian Perron envelope. The base capacity endpoint limits give $L^1(Y)$ convergence. Since
\[ \int _ X|\pi ^*(v-w)|\frac{\omega ^ n}{2n} =\int _ Y|v-w|\kappa ^ d, \]
the same holds on $X$; every smooth positive $\mathrm dV$ has bounded density ratio with $\omega ^ n$ on compact $X$. For smooth $v$, the constant translation is a base geodesic with speed $|c|$. Smooth metric approximation and translation by the same constant extend the distance formula to every $v\in \mathcal E^2$.
$\square$
For an absolutely continuous curve $\gamma :[0,1]\to \mathcal B_2$, define its metric action by
\[ \mathcal A_ B(\gamma )=\frac12\int _0^1|\dot\gamma |_{d_ B}^{\, 2}\, \mathrm dt. \]
Here the metric derivative is the almost everywhere defined limit
\[ |\dot\gamma _ t|_{d_ B} =\lim _{h\to 0}\frac{d_ B(\gamma _{t+h},\gamma _ t)}{|h|}. \]
The action is defined even when the potentials are unbounded; it does not require a Riemannian tensor at a singular potential.
Proposition 17.16.22.2B828BD01DBB The unique action minimizer between two points of $\mathcal B_2$ is their constant-speed Perron segment, and its action is half their squared distance. On every smooth path in $\mathcal B_\infty $, $\mathcal A_ B$ equals the raw action $\mathscr S$. Every smooth basic Perron segment with strictly positive spatial forms is stationary for $\mathscr S$ under all smooth fixed-endpoint variations on $X$.
Proof.
The length inequality and Cauchy–Schwarz give
\[ d_ B(\gamma _0,\gamma _1)^2 \leq \left(\int _0^1|\dot\gamma |_{d_ B}\, \mathrm dt\right)^2 \leq 2\mathcal A_ B(\gamma ). \]
Equality holds for the constant-speed geodesic. Equality for a connecting curve forces constant speed and equality between length and endpoint distance, hence forces a geodesic; uniqueness follows from the CAT$(0)$ property.
For a smooth basic path $\pi ^*v_ s$, put $h=v_{s+\delta }-v_ s$. On its short base Perron segment, parametrized by $0\leq \tau \leq 1$, the chord bounds above, and
\[ v_ s+\tau h-C\delta ^2\tau (1-\tau ) \]
bounds below. Indeed, the spatial forms along the chord have a uniform positive lower bound, $\mathrm dh=O(\delta )$, and proposition 17.16.9 makes the displayed path a subgeodesic for a fixed sufficiently large $C$. The right initial velocity of the Perron segment is therefore $h+O(\delta ^2)$ uniformly. The classical smooth-endpoint Kähler distance formula gives
\[ \lim _{\delta \to 0}\frac{d_ B(\pi ^*v_ s,\pi ^*v_{s+\delta })^2}{\delta ^2} =2n\int _ Y\dot v_ s^{\, 2}(\kappa +\mathrm{dd}^{\mathrm{c}}v_ s)^ d. \]
Integration proves $\mathcal A_ B=\mathscr S$ along this path.
For stationarity, let $L(w)$ be the first variation at a smooth basic Perron path. Uniform spatial positivity allows every smooth endpoint-zero variation. Group invariance of the action and of the path gives $L(g^*w)=L(w)$ for $g\in E$. Average over Haar probability measure to obtain
\[ L(w)=L\left(\int _ Eg^*w\, \mathrm dg\right)=L(\pi ^*q) \]
for a smooth endpoint-zero base variation $q$. Fiber integration reduces this to $2n$ times the closed-base first variation. With $\eta _ t=\kappa +\mathrm{dd}^{\mathrm{c}}v_ t$, integration by parts gives that variation as
\[ \int _0^1\int _ Yq_ t\left[-\ddot v_ t\eta _ t^ d +d\, \mathrm d\dot v_ t\wedge \mathrm d^{\mathrm c}\dot v_ t\wedge \eta _ t^{d-1}\right]\mathrm dt. \]
The bracket vanishes by the smooth base homogeneous equation. Thus $L(w)=0$ for every variation on $X$.
$\square$
The metric action minimizes over curves in $\mathcal B_2$. The stationarity assertion allows all smooth variations. It does not imply minimization over all smooth Hermitian paths. Likewise, the initial-velocity formula used above has smooth endpoints; it is not a definition of the metric on all of $\mathcal B_2$.
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